**2. Modeling and preliminaries**

## **2.1. Discrete-time modeling**

2 Will-be-set-by-IN-TECH

Also, an epidemic can be modeled by means of discrete-time. This is the case of the classical Reed-Frost model, which is a Markovian discrete-time SIR epidemic model. However, this modeling has two differences with the corresponding one based on counting processes. First, its latent period is assumed constant and equal to the time unit. Secondly, there are no deterministic counterpart based on differential equations as it is the case of an epidemic modeled by means of a Markovian counting process. Another type of population modeling, which is applied to metapopulations, has been introduced by some authors such [9, 10, 27] and other references therein. These researchers derive an approximation that preserves the discrete time structure and reduces the complexity of the models. Probably, these results could be applied to epidemic models and asymptotic inference on the parameters of these models,

This chapter is a compendium of two works by the author, whose references are [12, 13]. A wide class of discrete-time stochastic epidemic models is introduced and analyzed from a statistical point of view. Just as some models based on ordinary differential equations involve a natural alternative through Markovian counting processes, this class includes a counterpart based on differential equations. Unlike those epidemic models where transitions occur at random times, our proposal involves the advantage of being suitable for epidemics that cannot be observed for a long period of time, as in some epidemics where observations are done at previously determined times. This is the main reason for preferentially considering these kind of stochastic models instead of those based on continuous time. It is expected the smaller the periods of time between transitions and the bigger the population, the more similar the stochastic and deterministic models would become. Indeed, one of the main aims of this paper is to prove such a similarity. As a second aim, we are highly interested in carry out statistical analysis on the parameters of the modeling. For this purpose, martingale estimators for the parameters involve in the modeling are derived and their asymptotic

Since the results stated here do not assume a distribution for the process modeling the epidemic, it is not possible to derive a likelihood ratio and hence maximum likelihood estimators cannot be obtained. Even, in many cases when the process representing the model is Markovian, the maximum likelihood estimators cannot be obtained in a closed form, which makes difficult to carry out statistical inference on the parameter of the model. As pointed out in [7], likelihood functions corresponding to epidemic data are often very complicated. In these cases, parameter estimation based on martingale estimators use to be an appropriate alternative to work out this difficulty. This method arises as a natural way of estimation when no distribution in the model is assumed or, when the maximum likelihood estimators cannot

This chapter is organized as follows. The general form of the model and two preliminary lemmas are introduced in Section 2. Section 3 contains brief definitions of some typical models included in the biomathematical literature. The deterministic counterpart of the general model along with its relationships with it is presented in Section 4. Indeed, the convergence of the stochastic model to the deterministic one and the asymptotic behavior of the corresponding fluctuations are proved. Moreover, in Section 4 a version of the SIS epidemic model is presented and numerical simulations are carried out. The parameter estimators are defined in Section 5, and their asymptotic normality is proved. The General Epidemic Model along with the statistical analysis on the parameters is stated in Section 6.

could be carried out.

normality is proved.

be obtained in a closed form.

The model which we introduce here is defined as follows: a community of individuals divided into three different compartments is considered, namely, susceptible, infective and removed individuals.

Suppose the size population is *<sup>n</sup>*, and for each *<sup>t</sup>* <sup>≥</sup> 0, *<sup>S</sup>n*(*t*), *<sup>I</sup>n*(*t*) and *<sup>R</sup>n*(*t*) represent, respectively, the number of susceptible, infective and removed individuals at time *t*. Since it is assumed the population size is constant, then for each *<sup>t</sup>* <sup>≥</sup> 0, should be *<sup>S</sup>n*(*t*) + *<sup>I</sup>n*(*t*) + *<sup>R</sup>n*(*t*) = *n*. These processes are observed at discrete-time instants which are defined by the sequence {*tn <sup>k</sup>* }*k*∈**N**, where for each *<sup>k</sup>* ∈ **<sup>N</sup>**, *<sup>t</sup> n <sup>k</sup>* = *k*Δ/*n*, (Δ > 0), i.e. each time subinterval has length Δ/*n*. Let (Ω, F, IP) be a probability space. In the sequel, all stochastic processes and random variables are defined on this probability space and for a stochastic process *Z*, we denote Δ*Z*(*t<sup>n</sup> <sup>k</sup>* ) = *<sup>Z</sup>*(*t<sup>n</sup> <sup>k</sup>* ) <sup>−</sup> *<sup>Z</sup>*(*t<sup>n</sup> <sup>k</sup>*−1).

Let *M*� denote the transpose of a matrix *M* and *X<sup>n</sup>* = (*Sn*, *In*, *Rn*)�. Transitions of individuals among the three compartments are determined by *m* increasing stochastic processes *Z<sup>n</sup>* <sup>1</sup> ,..., *<sup>Z</sup><sup>n</sup> <sup>m</sup>*, and the number of individuals in each compartment is obtained, for each *t* ≥ 0, by means of

$$X^{\mathbb{N}}(t) = X^{\mathbb{N}}(0) + AZ^{\mathbb{N}}(t),\tag{1}$$

where *Zn*(*t*)=(*Z<sup>n</sup>* <sup>1</sup> (*t*),..., *<sup>Z</sup><sup>n</sup> <sup>m</sup>*(*t*))� and *A* is a 3 × *m*-incidence matrix.

It is assumed *Z<sup>n</sup>* <sup>1</sup> ,..., *<sup>Z</sup><sup>n</sup> <sup>m</sup>* take values in the set of non-negative integer numbers, have right-continuous trajectories and start at zero, i.e. *Z<sup>n</sup>* <sup>1</sup> (0) = ··· <sup>=</sup> *<sup>Z</sup><sup>n</sup> <sup>m</sup>*(0) = 0. Let <sup>F</sup>*<sup>n</sup> <sup>k</sup>* be the *σ*-field *σ*(*Zn*(*t n* <sup>1</sup> ),..., *<sup>Z</sup>n*(*<sup>t</sup> n <sup>k</sup>* )) generated by *<sup>Z</sup>n*(*<sup>t</sup> n* <sup>1</sup> ),..., *<sup>Z</sup>n*(*<sup>t</sup> n <sup>k</sup>* ). The stochastic processes *Zn* <sup>1</sup> ,..., *<sup>Z</sup><sup>n</sup> <sup>m</sup>* increase according to *m* density dependent transition rates, which are defined by means of *m* non-negative functions *a*1,..., *am*, respectively. The domain of these functions is and open set of **<sup>R</sup>**<sup>3</sup> containing the 3-simplex *<sup>E</sup>* <sup>=</sup> {(*u*, *<sup>v</sup>*, *<sup>w</sup>*)� <sup>∈</sup> [0, 1] <sup>3</sup> : *<sup>u</sup>* <sup>+</sup> *<sup>v</sup>* <sup>+</sup> *<sup>w</sup>* <sup>=</sup> <sup>1</sup>} and it is assumed the following condition holds:

$$\text{(C)}\quad \text{For each } k \in \mathbb{N}, \Delta Z\_1^n(t\_k^n), \dots, \Delta Z\_m^n(t\_k^n) \text{ are } \mathcal{F}\_{k-1}^n-\text{conditional}\text{) independent and satisfy}$$

$$\mathbb{E}(\Delta Z\_i^n(t\_k^n)|\mathcal{F}\_{k-1}^n) = a\_i(\chi^n(t\_{k-1}^n)), \quad (i \in \{1, \dots, m\}),$$

where *χn*(*t*)=(*σn*(*t*), *ι <sup>n</sup>*(*t*), *ρn*(*t*))�, *σn*(*t*) = *Sn*(*t*)/*n*, *ι <sup>n</sup>*(*t*) = *In*(*t*)/*n* and *ρn*(*t*) = *Rn*(*t*)/*n*.

A wide variety of stochastic models for epidemics satisfy condition (C). It is important to point out this condition does not determine the law or distribution of *χn*, i.e. there could be two or more processes satisfying this condition, though they have different transition probabilities. This fact enables this condition to be applied to a wide class of models, since in order to verify condition (C), the distribution of the process need not be known. Actually, a stochastic process satisfying condition (C) need not be Markovian. Nevertheless, some Markov chains, having density dependent transition rates, satisfy condition (C) and hence they may be included in our setting.

#### **2.2. Two preliminary lemmas**

In what follows, [*a*] stands for the integer part of a real number *a* and for each *i* = 1, . . . , *m*, we denote *L<sup>n</sup> <sup>i</sup>* (*t*) = <sup>∑</sup>[*nt*] *<sup>k</sup>*=<sup>1</sup> *<sup>ξ</sup><sup>n</sup> <sup>k</sup>* (*i*)Δ*t<sup>n</sup>* and *<sup>L</sup>n*(*t*)=(*L<sup>n</sup>* <sup>1</sup> (*t*),..., *<sup>L</sup><sup>n</sup> <sup>m</sup>*(*t*))�, (*<sup>t</sup>* <sup>≥</sup> 0), where *<sup>ξ</sup><sup>n</sup> <sup>k</sup>* (*i*) = Δ*Z<sup>n</sup> <sup>i</sup>* (*t<sup>n</sup> <sup>k</sup>* ) <sup>−</sup> *ai*(*χn*(*t<sup>n</sup> <sup>k</sup>*−1)) and <sup>Δ</sup>*t<sup>n</sup>* <sup>=</sup> 1/*n*. From Condition (C), it is obtained that, by defining G*n <sup>t</sup>* <sup>=</sup> <sup>F</sup>*<sup>n</sup>* [*nt*] , *<sup>L</sup><sup>n</sup>* <sup>=</sup> {*Ln*(*t*); *<sup>t</sup>* <sup>≥</sup> <sup>0</sup>} is an *<sup>m</sup>*-dimensional martingale with respect to {G*<sup>n</sup> <sup>t</sup>* ; *t* ≥ 0}.

Through this chapter, for each *i* = 1, . . . , *m*, *v<sup>n</sup> <sup>i</sup>* (*t*) and �*L<sup>n</sup> <sup>i</sup>* � stand for the random variable

$$v\_i^n(t) = \frac{1}{n} \sum\_{k=1}^{[nt]} \mathbb{E}(\xi\_k^n(i)^2 | \mathcal{F}\_{k-1}^n)^2$$

and the predictable quadratic variation of *L<sup>n</sup> <sup>i</sup>* , respectively.

**Lemma 2.1.** *For each t* <sup>≥</sup> <sup>0</sup>*, L<sup>n</sup>* <sup>1</sup> (*t*),..., *<sup>L</sup><sup>n</sup> <sup>m</sup>*(*t*) *are* <sup>G</sup>*<sup>n</sup> <sup>t</sup>*−*-conditionally independent random variables and, the predictable quadratic variation matrix of L<sup>n</sup> is given by*

$$
\langle L^n \rangle(t) = \frac{1}{n} \begin{pmatrix} v\_1^n(t) \cdot \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & v\_m^n(t) \end{pmatrix}, \quad (t \ge 0).
$$

*Proof.* For *<sup>t</sup>* <sup>≥</sup> 0, the <sup>G</sup>*<sup>n</sup> <sup>t</sup>*−-conditional independence of *<sup>L</sup><sup>n</sup>* <sup>1</sup> (*t*),..., *<sup>L</sup><sup>n</sup> <sup>m</sup>*(*t*) follows from Assumption (C) and it is clear that for each *i* = 1, . . . , *m*, the predictable quadratic variation of *L<sup>n</sup> <sup>i</sup>* is given by �*L<sup>n</sup> <sup>i</sup>* �(*t*) = <sup>1</sup> *<sup>n</sup>*<sup>2</sup> <sup>∑</sup>[*nt*] *<sup>k</sup>*=<sup>1</sup> IE(*ξ<sup>n</sup> <sup>k</sup>* (*i*)2|F*<sup>n</sup> <sup>k</sup>*−1). Hence �*L<sup>n</sup> <sup>i</sup>* �(*t*) = *<sup>v</sup><sup>n</sup> <sup>i</sup>* (*t*)/*n*, (*t* ≥ 0), which concludes the proof.

In the sequel, for each *<sup>d</sup>* <sup>∈</sup> **<sup>N</sup>**, �·� stands for the Euclidean vector norm in **<sup>R</sup>***d*.

**Lemma 2.2.** *Let T* <sup>&</sup>gt; <sup>0</sup> *and suppose for each i* <sup>=</sup> 1, . . . , *m,* { <sup>1</sup> *<sup>n</sup> <sup>v</sup><sup>n</sup> <sup>i</sup>* (*T*)}*n*∈**<sup>N</sup>** *converges in probability to zero, as n goes to* <sup>∞</sup>*. Then,* {sup0≤*t*≤*<sup>T</sup>* �*Ln*(*t*)�}*n*∈**<sup>N</sup>** *converges in probability to zero.*

For each *<sup>T</sup>* <sup>&</sup>gt; 0 and each *<sup>i</sup>* <sup>=</sup> 1, . . . , *<sup>m</sup>*, { <sup>1</sup> *<sup>n</sup> <sup>v</sup><sup>n</sup> <sup>i</sup>* (*T*)}*n*∈**<sup>N</sup>** converges in probability to zero, as *<sup>n</sup>* goes to ∞.

*Proof.* From Theorem 1 in [26], for any *�*, *η* > 0 we have

$$\mathbb{P}(\sup\_{0 \le t \le T} \|L^n(t)\|^2 > \epsilon) \le \frac{1}{\epsilon} \sum\_{i=1}^m \mathbb{E}(\langle L\_i^n \rangle(T) \wedge \eta) + \mathbb{P}(\sum\_{i=1}^m \langle L\_i^n \rangle(T) > \eta).$$

and hence, Lemma 2.1 implies

$$\mathbb{P}(\sup\_{0 \le t \le T} \|L^n(t)\|^2 > \epsilon) \le \frac{m\eta}{\epsilon} + \mathbb{P}(\sum\_{i=1}^m \frac{1}{n} v\_i^n(T) > \eta).$$

By assumption this lemma follows.

**3. Some epidemic models**

**3.1. The general epidemic model**

by *Z<sup>n</sup>*

*Zn*

<sup>1</sup> and *<sup>Z</sup><sup>n</sup>*

<sup>1</sup> and *<sup>Z</sup><sup>n</sup>*

**3.2. The SIRS Model**

**3.3. The SIS Model**

allowed, and determined by *Z<sup>n</sup>*

the modeling of this epidemic must satisfy (1) with

determined by an increasing stochastic process *Z<sup>n</sup>*

*A* =

⎛

⎞

⎟⎠

⎜⎝

infection and removal rates. This model is also known as SIR model.

infected again. The incidence matrix defining this model is given by

*A* =

<sup>1</sup> and *<sup>Z</sup><sup>n</sup>*

*A* =

⎛

Notice the compartment corresponding to removed cases is considered having no individuals.

⎞

⎟⎠ .

⎜⎝

and *a*2(*u*, *v*, *w*) = *γv*, respectively. In this case, the incidence matrix is

⎛

⎜⎝

Most of the typical models involved in the biomathematical literature have a version which belongs to the model class defined in this approach. Some of them are included below.

A deterministic version of this model based on differential equations, was introduced by [19], while the article by [18] was a pioneer in the stochastic version based on counting processes. This model has received the most attention in the literature, and its analysis can be found in [2], [1] and [23]. Two type of transitions are possible for any individual, from susceptible to infected and from infected to removed individuals. Thus under the perspective of this work,

Transitions from susceptible to infected and from infected to removed cases are described

*a*1(*u*, *v*, *w*) = *βuv* and *a*2(*u*, *v*, *w*) = *γv*, where *β* and *γ* are two parameters denoting the

This is a slight modification of the preceding model. Besides the transitions determined by

*a*3(*u*, *v*, *w*) = *δw*. In this case, some of the removed cases may become susceptible to be

−10 1 1 −1 0 0 1 −1

One of the simplest epidemic models is the SIS model, which uses to be suitable for infections resulting from bacteria such as gonorrhea, malaria, etc. In this case, only transitions from susceptible to infected individuals are allowed, as well as transitions from infected to susceptible individuals. According to the approach of this study, two transitions are

<sup>2</sup> in the SIR model, a transition from removed to susceptible case is allowed and

⎞

⎟⎠ .

and *Z<sup>n</sup>* =

<sup>2</sup> , respectively, and the functions defining their transition rates are given by

� *Zn* 1 *Zn* 2

� .

Discrete-Time Stochastic Epidemic Models and Their Statistical Inference 235

<sup>3</sup> , where its transition rate is defined by

<sup>2</sup> with transition rates defined by *a*1(*u*, *v*, *w*) = *βuv*

## **3. Some epidemic models**

4 Will-be-set-by-IN-TECH

In what follows, [*a*] stands for the integer part of a real number *a* and for each *i* = 1, . . . , *m*,

, *<sup>L</sup><sup>n</sup>* <sup>=</sup> {*Ln*(*t*); *<sup>t</sup>* <sup>≥</sup> <sup>0</sup>} is an *<sup>m</sup>*-dimensional martingale with respect to {G*<sup>n</sup>*

[*nt*] ∑ *k*=1

*<sup>m</sup>*(*t*) *are* <sup>G</sup>*<sup>n</sup>*

<sup>1</sup> (*t*) ··· 0

*. ... .*

<sup>0</sup> ··· *<sup>v</sup><sup>n</sup>*

*<sup>t</sup>*−-conditional independence of *<sup>L</sup><sup>n</sup>*

Assumption (C) and it is clear that for each *i* = 1, . . . , *m*, the predictable quadratic variation

*<sup>k</sup>* (*i*)2|F*<sup>n</sup>*

*<sup>n</sup> <sup>v</sup><sup>n</sup>*

*<sup>i</sup>* (*t*) and �*L<sup>n</sup>*

*<sup>i</sup>* , respectively.

*. .* ⎞

⎟⎟⎟⎟⎠ , (*<sup>t</sup>* <sup>≥</sup> <sup>0</sup>).

*<sup>m</sup>*(*t*)

*<sup>k</sup>*−1). Hence �*L<sup>n</sup>*

*<sup>n</sup> <sup>v</sup><sup>n</sup>*

*<sup>i</sup>* �(*T*) ∧ *η*) + IP(

1 *n vn*

IE(*ξ<sup>n</sup> <sup>k</sup>* (*i*) <sup>2</sup>|F*<sup>n</sup> <sup>k</sup>*−1)

<sup>1</sup> (*t*),..., *<sup>L</sup><sup>n</sup>*

*<sup>k</sup>*−1)) and <sup>Δ</sup>*t<sup>n</sup>* <sup>=</sup> 1/*n*. From Condition (C), it is obtained that, by defining

*<sup>m</sup>*(*t*))�, (*<sup>t</sup>* <sup>≥</sup> 0), where *<sup>ξ</sup><sup>n</sup>*

*<sup>i</sup>* � stand for the random variable

*<sup>t</sup>*−*-conditionally independent random variables*

<sup>1</sup> (*t*),..., *<sup>L</sup><sup>n</sup>*

*<sup>i</sup>* �(*t*) = *<sup>v</sup><sup>n</sup>*

*<sup>i</sup>* (*T*)}*n*∈**<sup>N</sup>** converges in probability to zero, as *<sup>n</sup>*

*m* ∑ *i*=1 �*Ln*

*<sup>i</sup>* (*T*) > *η*).

*<sup>k</sup>* (*i*) =

*<sup>t</sup>* ; *t* ≥ 0}.

*<sup>m</sup>*(*t*) follows from

*<sup>i</sup>* (*t*)/*n*, (*t* ≥ 0), which

*<sup>i</sup>* (*T*)}*n*∈**<sup>N</sup>** *converges in probability to*

*<sup>i</sup>* �(*T*) > *η*)

*<sup>k</sup>* (*i*)Δ*t<sup>n</sup>* and *<sup>L</sup>n*(*t*)=(*L<sup>n</sup>*

*vn <sup>i</sup>* (*t*) = <sup>1</sup> *n*

<sup>1</sup> (*t*),..., *<sup>L</sup><sup>n</sup>*

*n*

*<sup>k</sup>*=<sup>1</sup> IE(*ξ<sup>n</sup>*

In the sequel, for each *<sup>d</sup>* <sup>∈</sup> **<sup>N</sup>**, �·� stands for the Euclidean vector norm in **<sup>R</sup>***d*.

*zero, as n goes to* <sup>∞</sup>*. Then,* {sup0≤*t*≤*<sup>T</sup>* �*Ln*(*t*)�}*n*∈**<sup>N</sup>** *converges in probability to zero.*

1 *�*

�*Ln*(*t*)�<sup>2</sup> <sup>&</sup>gt; *�*) <sup>≤</sup>

*m* ∑ *i*=1

IE(�*L<sup>n</sup>*

*mη �*

+ IP( *m* ∑ *i*=1

⎛

*vn*

*. .*

⎜⎜⎜⎜⎝

*and, the predictable quadratic variation matrix of L<sup>n</sup> is given by*

�*Ln*�(*t*) = <sup>1</sup>

*<sup>n</sup>*<sup>2</sup> <sup>∑</sup>[*nt*]

**Lemma 2.2.** *Let T* <sup>&</sup>gt; <sup>0</sup> *and suppose for each i* <sup>=</sup> 1, . . . , *m,* { <sup>1</sup>

*Proof.* From Theorem 1 in [26], for any *�*, *η* > 0 we have

IP( sup 0≤*t*≤*T*

�*Ln*(*t*)�<sup>2</sup> <sup>&</sup>gt; *�*) <sup>≤</sup>

**2.2. Two preliminary lemmas**

*<sup>k</sup>* ) <sup>−</sup> *ai*(*χn*(*t<sup>n</sup>*

*<sup>i</sup>* (*t*) = <sup>∑</sup>[*nt*]

*<sup>k</sup>*=<sup>1</sup> *<sup>ξ</sup><sup>n</sup>*

Through this chapter, for each *i* = 1, . . . , *m*, *v<sup>n</sup>*

and the predictable quadratic variation of *L<sup>n</sup>*

*<sup>i</sup>* �(*t*) = <sup>1</sup>

For each *<sup>T</sup>* <sup>&</sup>gt; 0 and each *<sup>i</sup>* <sup>=</sup> 1, . . . , *<sup>m</sup>*, { <sup>1</sup>

IP( sup 0≤*t*≤*T*

and hence, Lemma 2.1 implies

By assumption this lemma follows.

**Lemma 2.1.** *For each t* <sup>≥</sup> <sup>0</sup>*, L<sup>n</sup>*

*Proof.* For *<sup>t</sup>* <sup>≥</sup> 0, the <sup>G</sup>*<sup>n</sup>*

*<sup>i</sup>* is given by �*L<sup>n</sup>*

concludes the proof.

we denote *L<sup>n</sup>*

Δ*Z<sup>n</sup> <sup>i</sup>* (*t<sup>n</sup>*

of *L<sup>n</sup>*

goes to ∞.

G*n <sup>t</sup>* <sup>=</sup> <sup>F</sup>*<sup>n</sup>* [*nt*] Most of the typical models involved in the biomathematical literature have a version which belongs to the model class defined in this approach. Some of them are included below.

### **3.1. The general epidemic model**

A deterministic version of this model based on differential equations, was introduced by [19], while the article by [18] was a pioneer in the stochastic version based on counting processes. This model has received the most attention in the literature, and its analysis can be found in [2], [1] and [23]. Two type of transitions are possible for any individual, from susceptible to infected and from infected to removed individuals. Thus under the perspective of this work, the modeling of this epidemic must satisfy (1) with

$$A = \begin{pmatrix} -1 & 0 \\ 1 & -1 \\ 0 & 1 \end{pmatrix} \quad \text{and} \quad Z^n = \begin{pmatrix} Z\_1^n \\ Z\_2^n \end{pmatrix}.$$

Transitions from susceptible to infected and from infected to removed cases are described by *Z<sup>n</sup>* <sup>1</sup> and *<sup>Z</sup><sup>n</sup>* <sup>2</sup> , respectively, and the functions defining their transition rates are given by *a*1(*u*, *v*, *w*) = *βuv* and *a*2(*u*, *v*, *w*) = *γv*, where *β* and *γ* are two parameters denoting the infection and removal rates. This model is also known as SIR model.

### **3.2. The SIRS Model**

This is a slight modification of the preceding model. Besides the transitions determined by *Zn* <sup>1</sup> and *<sup>Z</sup><sup>n</sup>* <sup>2</sup> in the SIR model, a transition from removed to susceptible case is allowed and determined by an increasing stochastic process *Z<sup>n</sup>* <sup>3</sup> , where its transition rate is defined by *a*3(*u*, *v*, *w*) = *δw*. In this case, some of the removed cases may become susceptible to be infected again. The incidence matrix defining this model is given by

$$A = \begin{pmatrix} -1 & 0 & 1 \\ 1 & -1 & 0 \\ 0 & 1 & -1 \end{pmatrix}.$$

#### **3.3. The SIS Model**

One of the simplest epidemic models is the SIS model, which uses to be suitable for infections resulting from bacteria such as gonorrhea, malaria, etc. In this case, only transitions from susceptible to infected individuals are allowed, as well as transitions from infected to susceptible individuals. According to the approach of this study, two transitions are allowed, and determined by *Z<sup>n</sup>* <sup>1</sup> and *<sup>Z</sup><sup>n</sup>* <sup>2</sup> with transition rates defined by *a*1(*u*, *v*, *w*) = *βuv* and *a*2(*u*, *v*, *w*) = *γv*, respectively. In this case, the incidence matrix is

$$A = \begin{pmatrix} -1 & 1\\ 1 & -1\\ 0 & 0 \end{pmatrix}.$$

Notice the compartment corresponding to removed cases is considered having no individuals.

#### **3.4. The modified SIR model**

This is a modification of the general epidemic model and aims to AIDS modeling. As in the general epidemic model, two transitions *Z<sup>n</sup>* <sup>1</sup> and *<sup>Z</sup><sup>n</sup>* <sup>2</sup> define the model with transition rates given by *a*1(*u*, *v*, *w*) = *βuv*/(*u* + *v*) and *a*2(*u*, *v*, *w*) = *γv*, respectively. As before, *β* and *γ* correspond to the model parameters. Some references concerning the deterministic version of this model are, for instance, [16, 17], while the stochastic version based on Markovian counting processes was introduced by [4]. The incidence matrix is defined as in the SIR model, i.e.

$$A = \begin{pmatrix} -1 & 0\\ 1 & -1\\ 0 & 1 \end{pmatrix}.$$

In general, as much in the SIR as the modified model, the transition rate due to infection is proportional to the susceptible density and the fraction of infected individuals with respect to individuals in circulation. Consequently, in the SIR model this fraction is *v*/(*u* + *v* + *w*) = *v*, while in an epidemic where to be removed is equivalent to be dead or out of circulation, this fraction is *v*/(*u* + *v*). In modeling AIDS, removed cases are presumed to be so ill with AIDS that they no longer take part in transmission.

### **4. The deterministic counterpart**

In this section, we examine the relationship between the model we are introducing here and an associated ordinary differential equation, which we call its deterministic counterpart.

Let *F*(*x*) = *Aa*(*x*), where *a*(*x*)=(*a*1(*x*),..., *am*(*x*))�, (*x* ∈ *E*), and consider the following ordinary differential equation:

$$\frac{d\chi}{dt}(t) = F(\chi(t)), \quad \chi(0) = \chi\_{0\prime} \tag{2}$$

**(4.1.2)** *For each T* <sup>&</sup>gt; <sup>0</sup> *and each i* <sup>=</sup> 1, . . . , *m,* { <sup>1</sup>

[*nt*] ∑ *k*=1

= *n* � *t* 0

*<sup>χ</sup>n*(*t*) = *<sup>χ</sup>n*(0) + � *<sup>t</sup>*

*<sup>χ</sup>n*(*t*) <sup>−</sup> *<sup>χ</sup>*(*t*) = *<sup>χ</sup>n*(0) <sup>−</sup> *<sup>χ</sup>*(0) + � *<sup>t</sup>*

*<sup>n</sup> <sup>F</sup>*(*χn*(*t*)).

*condition holds: For each T* > 0 *and each i* = 1, . . . , *m,*

D(*F*)(*x*) =

In the sequel, {*Yn*}*n*∈**<sup>N</sup>** is the sequence defined as *<sup>Y</sup><sup>n</sup>* <sup>=</sup> <sup>√</sup>*n*(*χ<sup>n</sup>* <sup>−</sup> *<sup>χ</sup>*).

⎛

*∂F*<sup>1</sup> *∂x*<sup>1</sup>

*∂F*<sup>2</sup> *∂x*<sup>1</sup>

*∂F*<sup>3</sup> *∂x*<sup>1</sup>

⎜⎝

*F* at *x* = (*x*1, *x*2, *x*3)�, that is,

From (3) and (L), for each *t* ≥ 0, we have

*ai*(*χn*(*t n <sup>k</sup>*−1))�*<sup>t</sup>*

0

*gn*(*t*) = sup

*Proof.* Since, for each *<sup>i</sup>* <sup>=</sup> 1, . . . , *<sup>m</sup>*, �*Z<sup>n</sup>*

*Zn <sup>i</sup>* (*t*) = *n*

*goes to* ∞*.*

and

Hence

the proof.

where *�n*(*t*) = (*nt*−[*nt*])

*<sup>n</sup> <sup>v</sup><sup>n</sup>*

*Then,* {*χn*}*n*∈**<sup>N</sup>** *converges in probability uniformly over compact subsets of* **<sup>R</sup>**<sup>+</sup> *to <sup>χ</sup>, i.e. for each*

*<sup>k</sup>* ) = *ai*(*χn*(*<sup>t</sup>*

*<sup>n</sup>* + *n*

*ai*(*χn*(*u*)) <sup>d</sup>*<sup>u</sup>* <sup>−</sup> (*nt* <sup>−</sup> [*nt*])*ai*(*χn*(*t*)) + *nL<sup>n</sup>*

*<sup>F</sup>*(*χn*(*u*)) <sup>d</sup>*<sup>u</sup>* <sup>+</sup> *ALn*(*t*) + (*nt* <sup>−</sup> [*nt*])

�*χn*(*s*) <sup>−</sup> *<sup>χ</sup>*(*s*)�, (*<sup>t</sup>* <sup>∈</sup> [0, *<sup>T</sup>*]).

*gn*(*u*) d*u*,

� *t* 0

where *<sup>α</sup><sup>n</sup>* <sup>=</sup> *<sup>g</sup>n*(0) + sup0≤*t*≤*<sup>T</sup>* �*ALn*(*t*)� <sup>+</sup> <sup>O</sup>(1/*n*). Since {*gn*(0)}*n*∈**<sup>N</sup>** converges in probability to zero and sup0≤*t*≤*<sup>T</sup>* �*ALn*(*t*)� ≤ |||*A*||| sup0≤*t*≤*<sup>T</sup>* �*Ln*(*t*)�, Lemma 2.1 and Gronwall's inequality imply {*gn*(*T*)}*n*∈**<sup>N</sup>** converges in probability to zero. This completes

**Remark 4.1.** *By Chebishev's inequality, Condition (4.1.2) is satisfied whenever the following stronger*

The following result aims to the problem of finding approximate confident bands for the solution to (2). Before stating it, for each *x* ∈ *E*, let D(*F*)(*x*) denote the Jacobian matrix of

> (*x*) *<sup>∂</sup>F*<sup>1</sup> *∂x*<sup>2</sup>

> (*x*) *<sup>∂</sup>F*<sup>2</sup> *∂x*<sup>2</sup>

> (*x*) *<sup>∂</sup>F*<sup>3</sup> *∂x*<sup>2</sup>

(*x*) *<sup>∂</sup>F*<sup>1</sup> *∂x*<sup>3</sup> (*x*)

(*x*) *<sup>∂</sup>F*<sup>2</sup> *∂x*<sup>3</sup> (*x*)

(*x*) *<sup>∂</sup>F*<sup>3</sup> *∂x*<sup>3</sup> (*x*)

*n <sup>k</sup>*−1)) + *<sup>ξ</sup><sup>n</sup>*

[*nt*] ∑ *k*=1 *ξn <sup>k</sup>* (*i*)�*t n*

*<sup>T</sup>* <sup>&</sup>gt; <sup>0</sup>*,* {sup0≤*t*≤*<sup>T</sup>* �*χn*(*t*) <sup>−</sup> *<sup>χ</sup>*(*t*)�}*n*∈**<sup>N</sup>** *converges in probability to zero, as n goes to* <sup>∞</sup>*.*

0

*<sup>g</sup>n*(*t*) <sup>≤</sup> *<sup>α</sup><sup>n</sup>* <sup>+</sup> *<sup>K</sup>*

For any matrix *<sup>B</sup>*, let us denote |||*B*||| <sup>=</sup> sup�*x*�=<sup>1</sup> �*Bx*�. Fix *<sup>T</sup>* <sup>&</sup>gt; 0 and let

0≤*s*≤*t*

*<sup>i</sup>* (*t n* *<sup>i</sup>* (*T*)}*n*∈**<sup>N</sup>** *converges in probability to zero, as n*

Discrete-Time Stochastic Epidemic Models and Their Statistical Inference 237

*<sup>k</sup>* (*i*), we have

*n*

lim*n*→<sup>∞</sup> IE(�*Ln*(*i*)�(*T*)) = 0. (4)

⎞

⎟⎠ .

[*F*(*χn*(*u*)) <sup>−</sup> *<sup>F</sup>*(*χ*(*u*))] <sup>d</sup>*<sup>u</sup>* <sup>+</sup> *ALn*(*t*) <sup>−</sup> *�n*(*t*), (3)

*<sup>i</sup>* (*t*),

*F*(*χn*(*t*)).

where *χ*<sup>0</sup> = (*σ*0, *ι*0, *ρ*0)� ∈ *E* is the initial condition.

In order to obtain existence and uniqueness of the solution to (2), it will be assumed the following usual Lipchitz condition holds:

$$\text{(L)}\quad||F(\mathfrak{x}) - F(\mathfrak{y})|| \le K||\mathfrak{x} - \mathfrak{y}||\_{\mathsf{V}}$$

for all *x*, *y* ∈ *E*, where *K* is a positive constant.

#### **4.1. Comparison between the stochastic and deterministic models**

The theorem below stated the consistency of the stochastic model with respect to the deterministic one and we will use it to study the asymptotic behavior of the estimators for the parameters of the model.

**Theorem 4.1.** *Let χ be the unique solution to (3) and assume conditions (C) and (L) are satisfied. Moreover, suppose the following two conditions hold:*

**(4.1.1)** *The sequence of initial conditions* {*χn*(0)}*n*∈**<sup>N</sup>** *converges in probability to <sup>χ</sup>*<sup>0</sup> = (*σ*0, *<sup>ι</sup>*0, *<sup>ρ</sup>*0)�*, as n goes to* ∞*.*

**(4.1.2)** *For each T* <sup>&</sup>gt; <sup>0</sup> *and each i* <sup>=</sup> 1, . . . , *m,* { <sup>1</sup> *<sup>n</sup> <sup>v</sup><sup>n</sup> <sup>i</sup>* (*T*)}*n*∈**<sup>N</sup>** *converges in probability to zero, as n goes to* ∞*.*

*Then,* {*χn*}*n*∈**<sup>N</sup>** *converges in probability uniformly over compact subsets of* **<sup>R</sup>**<sup>+</sup> *to <sup>χ</sup>, i.e. for each <sup>T</sup>* <sup>&</sup>gt; <sup>0</sup>*,* {sup0≤*t*≤*<sup>T</sup>* �*χn*(*t*) <sup>−</sup> *<sup>χ</sup>*(*t*)�}*n*∈**<sup>N</sup>** *converges in probability to zero, as n goes to* <sup>∞</sup>*.*

*Proof.* Since, for each *<sup>i</sup>* <sup>=</sup> 1, . . . , *<sup>m</sup>*, �*Z<sup>n</sup> <sup>i</sup>* (*t n <sup>k</sup>* ) = *ai*(*χn*(*<sup>t</sup> n <sup>k</sup>*−1)) + *<sup>ξ</sup><sup>n</sup> <sup>k</sup>* (*i*), we have

$$\begin{aligned} Z\_i^n(t) &= n \sum\_{k=1}^{\lfloor nt \rfloor} a\_i(\chi^n(t\_{k-1}^n)) \triangle t^n + n \sum\_{k=1}^{\lfloor nt \rfloor} \mathfrak{Z}\_k^n(i) \triangle t^n \\ &= n \int\_0^t a\_i(\chi^n(u)) \operatorname{d}u - (nt - \lfloor nt \rfloor) a\_i(\chi^n(t)) + n L\_i^n(t) \omega\_i \end{aligned}$$

and

6 Will-be-set-by-IN-TECH

This is a modification of the general epidemic model and aims to AIDS modeling. As in the

given by *a*1(*u*, *v*, *w*) = *βuv*/(*u* + *v*) and *a*2(*u*, *v*, *w*) = *γv*, respectively. As before, *β* and *γ* correspond to the model parameters. Some references concerning the deterministic version of this model are, for instance, [16, 17], while the stochastic version based on Markovian counting processes was introduced by [4]. The incidence matrix is defined as in the SIR model, i.e.

*A* =

⎛

⎜⎝

<sup>1</sup> and *<sup>Z</sup><sup>n</sup>*

In general, as much in the SIR as the modified model, the transition rate due to infection is proportional to the susceptible density and the fraction of infected individuals with respect to individuals in circulation. Consequently, in the SIR model this fraction is *v*/(*u* + *v* + *w*) = *v*, while in an epidemic where to be removed is equivalent to be dead or out of circulation, this fraction is *v*/(*u* + *v*). In modeling AIDS, removed cases are presumed to be so ill with AIDS

In this section, we examine the relationship between the model we are introducing here and an associated ordinary differential equation, which we call its deterministic counterpart.

Let *F*(*x*) = *Aa*(*x*), where *a*(*x*)=(*a*1(*x*),..., *am*(*x*))�, (*x* ∈ *E*), and consider the following

In order to obtain existence and uniqueness of the solution to (2), it will be assumed the

The theorem below stated the consistency of the stochastic model with respect to the deterministic one and we will use it to study the asymptotic behavior of the estimators for

**Theorem 4.1.** *Let χ be the unique solution to (3) and assume conditions (C) and (L) are satisfied.*

**(4.1.1)** *The sequence of initial conditions* {*χn*(0)}*n*∈**<sup>N</sup>** *converges in probability to <sup>χ</sup>*<sup>0</sup> = (*σ*0, *<sup>ι</sup>*0, *<sup>ρ</sup>*0)�*,*

(*t*) = *F*(*χ*(*t*)), *χ*(0) = *χ*0, (2)

d*χ* d*t*

**4.1. Comparison between the stochastic and deterministic models**

where *χ*<sup>0</sup> = (*σ*0, *ι*0, *ρ*0)� ∈ *E* is the initial condition.

following usual Lipchitz condition holds:

for all *x*, *y* ∈ *E*, where *K* is a positive constant.

*Moreover, suppose the following two conditions hold:*

(L) �*F*(*x*) − *F*(*y*)� ≤ *K*�*x* − *y*�,

the parameters of the model.

*as n goes to* ∞*.*

⎞

⎟⎠ .

<sup>2</sup> define the model with transition rates

**3.4. The modified SIR model**

general epidemic model, two transitions *Z<sup>n</sup>*

that they no longer take part in transmission.

**4. The deterministic counterpart**

ordinary differential equation:

$$
\chi^n(t) = \chi^n(0) + \int\_0^t F(\chi^n(u)) \, \mathrm{d}u + AL^n(t) + \frac{(nt - [nt])}{n} F(\chi^n(t)).
$$

Hence

$$\chi^{\mathfrak{n}}(t) - \chi(t) = \chi^{\mathfrak{n}}(0) - \chi(0) + \int\_{0}^{t} [F(\chi^{\mathfrak{n}}(u)) - F(\chi(u))] \, \mathrm{d}u + AL^{\mathfrak{n}}(t) - \varepsilon^{\mathfrak{n}}(t), \tag{3}$$

where *�n*(*t*) = (*nt*−[*nt*]) *<sup>n</sup> <sup>F</sup>*(*χn*(*t*)).

For any matrix *<sup>B</sup>*, let us denote |||*B*||| <sup>=</sup> sup�*x*�=<sup>1</sup> �*Bx*�. Fix *<sup>T</sup>* <sup>&</sup>gt; 0 and let

$$\|g^n(t) = \sup\_{0 \le s \le t} \|\chi^n(s) - \chi(s)\|\_{\prime} \quad (t \in [0, T]).$$

From (3) and (L), for each *t* ≥ 0, we have

$$g^n(t) \le \mathfrak{a}\_n + \mathcal{K} \int\_0^t g^n(u) \, \mathrm{d}u\_\prime$$

where *<sup>α</sup><sup>n</sup>* <sup>=</sup> *<sup>g</sup>n*(0) + sup0≤*t*≤*<sup>T</sup>* �*ALn*(*t*)� <sup>+</sup> <sup>O</sup>(1/*n*). Since {*gn*(0)}*n*∈**<sup>N</sup>** converges in probability to zero and sup0≤*t*≤*<sup>T</sup>* �*ALn*(*t*)� ≤ |||*A*||| sup0≤*t*≤*<sup>T</sup>* �*Ln*(*t*)�, Lemma 2.1 and Gronwall's inequality imply {*gn*(*T*)}*n*∈**<sup>N</sup>** converges in probability to zero. This completes the proof.

**Remark 4.1.** *By Chebishev's inequality, Condition (4.1.2) is satisfied whenever the following stronger condition holds: For each T* > 0 *and each i* = 1, . . . , *m,*

$$\lim\_{n \to \infty} \mathbb{E}(\langle L^{\mathbb{n}}(i) \rangle (T)) = 0. \tag{4}$$

The following result aims to the problem of finding approximate confident bands for the solution to (2). Before stating it, for each *x* ∈ *E*, let D(*F*)(*x*) denote the Jacobian matrix of *F* at *x* = (*x*1, *x*2, *x*3)�, that is,

$$\mathbf{D}(F)(\boldsymbol{\chi}) = \begin{pmatrix} \frac{\partial F\_1}{\partial \boldsymbol{x}\_1}(\boldsymbol{\chi}) \begin{array}{c} \frac{\partial F\_1}{\partial \boldsymbol{x}\_2}(\boldsymbol{\chi}) \ \frac{\partial F\_1}{\partial \boldsymbol{x}\_3}(\boldsymbol{\chi})\\ \frac{\partial F\_2}{\partial \boldsymbol{x}\_1}(\boldsymbol{\chi}) \ \frac{\partial F\_2}{\partial \boldsymbol{x}\_2}(\boldsymbol{\chi}) \ \frac{\partial F\_2}{\partial \boldsymbol{x}\_3}(\boldsymbol{\chi})\\ \frac{\partial F\_3}{\partial \boldsymbol{x}\_1}(\boldsymbol{\chi}) \ \frac{\partial F\_3}{\partial \boldsymbol{x}\_2}(\boldsymbol{\chi}) \ \frac{\partial F\_3}{\partial \boldsymbol{x}\_3}(\boldsymbol{\chi}) \end{array} . \end{pmatrix}.$$

In the sequel, {*Yn*}*n*∈**<sup>N</sup>** is the sequence defined as *<sup>Y</sup><sup>n</sup>* <sup>=</sup> <sup>√</sup>*n*(*χ<sup>n</sup>* <sup>−</sup> *<sup>χ</sup>*).

**Theorem 4.2.** *Let* {*Yn*}*n*∈**<sup>N</sup>** *be the sequence defined as Y<sup>n</sup>* <sup>=</sup> <sup>√</sup>*n*(*χ<sup>n</sup>* <sup>−</sup> *<sup>χ</sup>*) *and suppose the following three conditions hold:*

**(4.2.1)** *For each i*, *<sup>j</sup>* <sup>=</sup> 1, 2, 3*, the partial derivative <sup>∂</sup>Fi ∂xj* (*x*) *exists and it is continuous at x in an open set containing E.*

$$\begin{aligned} \text{(4.2.2)} \quad &\text{For each } \varepsilon > 0 \text{ and each } i = 1, \dots, m, \left\{ \frac{1}{n} \sum\_{k=1}^{n} \mathbb{E}(\mathfrak{f}\_{k}^{n}(i)^{2} \mathbf{I}\_{\{|\mathfrak{f}\_{k}^{n}(i)| > \varepsilon \sqrt{n}\}} | \mathcal{F}\_{k-1}^{n}) \right\}\_{n \in \mathbb{N}} \text{ converges.}\\ &\text{where } \varepsilon = \text{mod}(\mathfrak{f}\_{k}^{n}) \text{ is the } \varepsilon \text{-norm} \end{aligned}$$

*in probability to zero.*


*Then,* {*Yn*}*n*∈**<sup>N</sup>** *converges in law to the solution Y satisfying the following stochastic differential equation:*

$$\mathbf{d}\mathbf{Y}(t) = \mathbf{D}(F)(\chi(t))\mathbf{Y}(t)\mathbf{d}t + \mathbf{d}M(t), \quad \mathbf{Y}(0) = \eta,\tag{5}$$

and, from (18), for each *�* > 0 and any *t* ≥ 0,

are right continuous and left-hand limited.

*ω<sup>D</sup>*

of continuity *ω<sup>D</sup>*

From (6) we have

lim*δ*→<sup>0</sup> sup*n*∈**<sup>N</sup>** IP(*ω<sup>D</sup>*

Theorem 4.1 that

lim *δ*→0 sup *n*∈**N**

*<sup>T</sup>* : D([0, *<sup>T</sup>*], **<sup>R</sup>**3)×]0, <sup>∞</sup>[<sup>→</sup> **<sup>R</sup>** as

*<sup>T</sup>* (*Yn*, *<sup>δ</sup>*) <sup>≤</sup> *<sup>δ</sup>C*<sup>1</sup> sup

lim *δ*→0 sup *n*∈**N**

*<sup>Y</sup>*(*t*) = *<sup>Y</sup>*(0) + *<sup>t</sup>*

converges in distribution to *Y*, which concludes the proof.

**Remark 4.2.** *By Itô's rule, the unique solution to (5) is given by*

*where* Ψ *is the unique solution to the matrix differential equation*

epidemic model, where some simulations are carried out.

 *η* + *t* 0

*Y*(*t*) = Ψ(*t*)

Ψ�

*<sup>T</sup>* (*x*, *δ*) = inf

*ω<sup>D</sup>*

IP( sup 0≤*u*≤*t*

{*ti*}

0≤*t*≤*T*

where the infimum extends over the finite sets {*ti*} of points satisfying 

max <sup>0</sup><*i*≤*<sup>r</sup>* sup *ti*−<sup>1</sup>≤*s*,*t*<*ti*

In order to prove the convergence in law of {*Yn*}*n*∈**N**, fix *<sup>T</sup>* <sup>&</sup>gt; 0 and let us define the modulus

0 = *t*<sup>0</sup> < *t*<sup>1</sup> < ··· < *tr* = *T*, *ti* − *ti*−<sup>1</sup> > *<sup>δ</sup>*, *<sup>i</sup>* = 1, . . . ,*r*. Here, D([0, *T*], **R**3) stands for the Skorohod space of all functions from [0, *T*] into **R**3, which

�*Yn*(*t*)� <sup>+</sup> *<sup>ω</sup><sup>D</sup>*

*<sup>T</sup>* (*Mn*, *<sup>δ</sup>*) <sup>&</sup>gt; *�*) = 0. Hence, from (9) and (10), for each *�* <sup>&</sup>gt; 0, we have

D(*F*)(*χ*(*s*))*Y*(*s*) d*s* + *M*(*t*).

, 0 ≤ *t* ≤ 1,

Since {*Mn*}*n*∈**<sup>N</sup>** converges in law to *<sup>M</sup>*, it follows from Theorem 15.2 by [8] that for each *�* <sup>&</sup>gt; 0,

Conditions (8) and (11) imply the sequence {*Yn*}*n*∈**<sup>N</sup>** satisfies the hypotheses of Theorem 15.2 in [8] and hence, the sequence of probabilities measures {IP(*Y<sup>n</sup>* ∈ ·)}*n*∈**<sup>N</sup>** is tight. This fact, along Theorem 6.1 in [8], imply that for the convergence in law, {*Yn*}*n*∈**<sup>N</sup>** is relatively compact. Let *<sup>Y</sup>* be a process and {*Ynk*}*k*∈**<sup>N</sup>** a subsequence of {*Yn*}*n*∈**<sup>N</sup>** such that {*Ynk*}*k*∈**<sup>N</sup>** converges in law to *<sup>Y</sup>*. Since {sup*t*≥<sup>0</sup> �*Un*(*t*)�}*n*∈**<sup>N</sup>** converges to zero, it follows from (6), (4.2.1) and

Moreover, since {*Yn*(0)}*n*∈**<sup>N</sup>** converges in distribution to *<sup>η</sup>* and *<sup>Y</sup>*(0) equals *<sup>η</sup>* in distribution, we have *<sup>Y</sup>* is a solution to (5). Finally, uniqueness of solutions to (5) implies {*Yn*}*n*∈**<sup>N</sup>**

Ψ(*s*)−<sup>1</sup> d*M*(*s*)

(*t*) = D(*F*)(*χ*(*t*))Ψ(*t*), Ψ(0) = *identity matrix*.

The stochastic process {*Y*(*t*);*t* ≥ 0} allows us to give confidence bands for the deterministic model defined by the solution *χ* to (2). In Section 6 such band is constructed for the SIS

IP(*ω<sup>D</sup>*

0

�*Yn*(*u*)� <sup>&</sup>gt; *�*/*δ*) = 0, (9)

Discrete-Time Stochastic Epidemic Models and Their Statistical Inference 239

�*x*(*s*) − *x*(*t*)�,

*<sup>T</sup>* (*Mn*, *<sup>δ</sup>*) + <sup>2</sup>*C*2/

*<sup>T</sup>* (*Yn*, *<sup>δ</sup>*) <sup>&</sup>gt; *�*) = 0. (11)

<sup>√</sup>*n*. (10)

*where M is a continuous martingale with predictable quadratic variation given by the matrix*

$$\langle M \rangle(t) = A \cdot \text{Diag}(v\_1(t), \dots, v\_m(t)) \cdot A^{\perp}$$

*and* Diag(*v*1(*t*),..., *vm*(*t*)) *stands for the diagonal matrix with entries v*1(*t*),..., *vm*(*t*) *at its diagonal.*

*Proof.* Let *<sup>M</sup><sup>n</sup>* <sup>=</sup> <sup>√</sup>*nALn*. By making use of Corollary 12, Chapter II in [29], (4.2.2) and (4.2.4) imply that { <sup>√</sup>*nLn*}*n*∈**<sup>N</sup>** converges in law to a continuous martingale *<sup>Q</sup>* with predictable quadratic variation �*Q*� given by �*Q*�(*t*) = Diag(*v*1(*t*),..., *vm*(*t*)). Consequently, {*Mn*}*n*∈**<sup>N</sup>** converges in law to a continuous martingale *M* = *AQ* with predictable quadratic variation �*M*� given, for each *<sup>t</sup>* <sup>≥</sup> 0, by �*M*�(*t*) = *<sup>A</sup>* · Diag(*v*1(*t*),..., *vm*(*t*)) · *<sup>A</sup>*T.

From (3), we have

$$Y^{\mathfrak{n}}(t) = Y^{\mathfrak{n}}(0) + \int\_0^t \mathbf{D}(F)(\theta^{\mathfrak{n}}(s))Y^{\mathfrak{n}}(s) \, \mathrm{d}s + M^{\mathfrak{n}}(t) - \mathcal{U}^{\mathfrak{n}}(t), \tag{6}$$

where *<sup>θ</sup>n*(*s*) is between *<sup>χ</sup>n*(*s*) and *<sup>χ</sup>*(*s*), and *<sup>U</sup>n*(*t*) = <sup>√</sup>*n�n*(*t*) = (*nt*−[*nt*]) <sup>√</sup>*<sup>n</sup> <sup>F</sup>*(*χn*(*t*)).

Put *<sup>C</sup>*<sup>1</sup> <sup>=</sup> sup*x*∈*<sup>E</sup>* �D(*F*)(*x*))� and let *<sup>C</sup>*<sup>2</sup> <sup>&</sup>gt; 0 such that sup*t*≥<sup>0</sup> �*Un*(*t*)� ≤ *<sup>C</sup>*2. We have

$$\sup\_{0 \le u \le t} \|Y^{\mathbb{I}}(u)\| \le \|Y^{\mathbb{I}}(0)\| + \mathbb{C}\_1 \int\_0^t \sup\_{0 \le u \le s} \|Y^{\mathbb{I}}(u)\| \, \mathrm{d}s + \sup\_{0 \le u \le t} \|M^{\mathbb{I}}(u)\| + \mathbb{C}\_2.$$

Consequently, from a standard application of the Gronwall inequality, we obtain

$$\sup\_{0 \le u \le t} \|Y^{\mathbb{N}}(u)\| \le \left(\|Y^{\mathbb{N}}(0)\| + \sup\_{0 \le u \le t} \|M^{\mathbb{N}}(u)\| + \mathcal{C}\_2\right) \mathbf{e}^{\mathbb{C}\_1 t}.\tag{7}$$

Since {*Yn*(0)}*n*∈**<sup>N</sup>** and {sup0≤*t*≤<sup>1</sup> �*Mn*(*t*)�}*n*∈**<sup>N</sup>** are sequences converging in distribution, Theorem 6.2 in [8] implies

$$\lim\_{a \to \infty} \sup\_{n \in \mathbb{N}} \mathbb{P}(\|Y^{\prime\prime}(0)\| > a) = 0,\tag{8}$$

and, from (18), for each *�* > 0 and any *t* ≥ 0,

$$\lim\_{\delta \to 0} \sup\_{n \in \mathbb{N}} \mathbb{P}(\sup\_{0 \le u \le t} ||Y^{\mathbb{N}}(u)|| > \varepsilon/\delta) = 0,\tag{9}$$

In order to prove the convergence in law of {*Yn*}*n*∈**N**, fix *<sup>T</sup>* <sup>&</sup>gt; 0 and let us define the modulus of continuity *ω<sup>D</sup> <sup>T</sup>* : D([0, *<sup>T</sup>*], **<sup>R</sup>**3)×]0, <sup>∞</sup>[<sup>→</sup> **<sup>R</sup>** as

$$\omega\_T^D(\mathfrak{x}, \delta) = \inf\_{\{t\_l\}} \max\_{0 < i \le r} \sup\_{t\_{l-1} \le s, t < t\_l} ||\mathfrak{x}(\mathfrak{s}) - \mathfrak{x}(t)||\_{\mathcal{H}}$$

where the infimum extends over the finite sets {*ti*} of points satisfying

$$\begin{cases} 0 = t\_0 < t\_1 < \dots < t\_r = T\_r\\ t\_i - t\_{i-1} > \delta\_r \quad i = 1, \dots, r. \end{cases}$$

Here, D([0, *T*], **R**3) stands for the Skorohod space of all functions from [0, *T*] into **R**3, which are right continuous and left-hand limited.

From (6) we have

8 Will-be-set-by-IN-TECH

**Theorem 4.2.** *Let* {*Yn*}*n*∈**<sup>N</sup>** *be the sequence defined as Y<sup>n</sup>* <sup>=</sup> <sup>√</sup>*n*(*χ<sup>n</sup>* <sup>−</sup> *<sup>χ</sup>*) *and suppose the following*

*n*

*each i* = 1, . . . , *m, vi* : [0, ∞[→ **R** *is an increasing continuous function such that vi*(0) = 0*.*

*Then,* {*Yn*}*n*∈**<sup>N</sup>** *converges in law to the solution Y satisfying the following stochastic differential*

�*M*�(*t*) = *A* · Diag(*v*1(*t*),..., *vm*(*t*)) · *A*� *and* Diag(*v*1(*t*),..., *vm*(*t*)) *stands for the diagonal matrix with entries v*1(*t*),..., *vm*(*t*) *at its*

*Proof.* Let *<sup>M</sup><sup>n</sup>* <sup>=</sup> <sup>√</sup>*nALn*. By making use of Corollary 12, Chapter II in [29], (4.2.2) and

quadratic variation �*Q*� given by �*Q*�(*t*) = Diag(*v*1(*t*),..., *vm*(*t*)). Consequently, {*Mn*}*n*∈**<sup>N</sup>** converges in law to a continuous martingale *M* = *AQ* with predictable quadratic variation

�*M*� given, for each *<sup>t</sup>* <sup>≥</sup> 0, by �*M*�(*t*) = *<sup>A</sup>* · Diag(*v*1(*t*),..., *vm*(*t*)) · *<sup>A</sup>*T.

 *t* 0

where *<sup>θ</sup>n*(*s*) is between *<sup>χ</sup>n*(*s*) and *<sup>χ</sup>*(*s*), and *<sup>U</sup>n*(*t*) = <sup>√</sup>*n�n*(*t*) = (*nt*−[*nt*])

Put *<sup>C</sup>*<sup>1</sup> <sup>=</sup> sup*x*∈*<sup>E</sup>* �D(*F*)(*x*))� and let *<sup>C</sup>*<sup>2</sup> <sup>&</sup>gt; 0 such that sup*t*≥<sup>0</sup> �*Un*(*t*)� ≤ *<sup>C</sup>*2. We have

sup 0≤*u*≤*s*

Since {*Yn*(0)}*n*∈**<sup>N</sup>** and {sup0≤*t*≤<sup>1</sup> �*Mn*(*t*)�}*n*∈**<sup>N</sup>** are sequences converging in distribution,

0≤*u*≤*t*

 *t* 0

Consequently, from a standard application of the Gronwall inequality, we obtain

�*Yn*(*u*)� ≤ (�*Yn*(0)� <sup>+</sup> sup

lim*a*→<sup>∞</sup> sup *n*∈**N**

*Yn*(*t*) = *Yn*(0) +

�*Yn*(*u*)�≤�*Yn*(0)� <sup>+</sup> *<sup>C</sup>*<sup>1</sup>

sup 0≤*u*≤*t*

*where M is a continuous martingale with predictable quadratic variation given by the matrix*

**(4.2.3)** {*Yn*(0)}*n*∈**<sup>N</sup>** *converges in distribution to a three-variate random vector <sup>η</sup>.*

*∂xj*

*n* ∑ *k*=1

IE(*ξ<sup>n</sup>*

*<sup>k</sup>* (*i*)2I{|*ξ<sup>n</sup>*

d*Y*(*t*) = D(*F*)(*χ*(*t*))*Y*(*t*)d*t* + d*M*(*t*), *Y*(0) = *η*, (5)

<sup>√</sup>*nLn*}*n*∈**<sup>N</sup>** converges in law to a continuous martingale *<sup>Q</sup>* with predictable

<sup>D</sup>(*F*)(*θn*(*s*))*Yn*(*s*) <sup>d</sup>*<sup>s</sup>* <sup>+</sup> *<sup>M</sup>n*(*t*) <sup>−</sup> *<sup>U</sup>n*(*t*), (6)

0≤*u*≤*t*

IP(�*Yn*(0)� <sup>&</sup>gt; *<sup>a</sup>*) = 0, (8)

�*Yn*(*u*)� <sup>d</sup>*<sup>s</sup>* <sup>+</sup> sup

<sup>√</sup>*<sup>n</sup> <sup>F</sup>*(*χn*(*t*)).

�*Mn*(*u*)� <sup>+</sup> *<sup>C</sup>*2) <sup>e</sup>*C*1*<sup>t</sup>* . (7)

�*Mn*(*u*)� <sup>+</sup> *<sup>C</sup>*2.

*<sup>k</sup>* (*i*)|>*�*

(*x*) *exists and it is continuous at x in an open*

<sup>√</sup>*n*}|F*<sup>n</sup>*

*<sup>i</sup>* (*t*)}*n*∈**<sup>N</sup>** *converges in probability to vi*(*t*)*, where for*

*<sup>k</sup>*−1)}*n*∈**<sup>N</sup>** *converges*

*three conditions hold:*

*set containing E.*

*in probability to zero.*

*equation:*

*diagonal.*

(4.2.4) imply that {

From (3), we have

sup 0≤*u*≤*t*

Theorem 6.2 in [8] implies

**(4.2.1)** *For each i*, *<sup>j</sup>* <sup>=</sup> 1, 2, 3*, the partial derivative <sup>∂</sup>Fi*

**(4.2.2)** *For each �* <sup>&</sup>gt; <sup>0</sup> *and each i* <sup>=</sup> 1, . . . , *m,* { <sup>1</sup>

**(4.2.4)** *For each t* <sup>≥</sup> <sup>0</sup> *and each i* <sup>=</sup> 1, . . . , *m,* {*v<sup>n</sup>*

$$
\omega\_T^D(Y^\eta, \delta) \le \delta \mathbb{C}\_1 \sup\_{0 \le t \le T} \|Y^\eta(t)\| + \omega\_T^D(M^\eta, \delta) + 2\mathbb{C}\_2/\sqrt{n}.\tag{10}
$$

Since {*Mn*}*n*∈**<sup>N</sup>** converges in law to *<sup>M</sup>*, it follows from Theorem 15.2 by [8] that for each *�* <sup>&</sup>gt; 0, lim*δ*→<sup>0</sup> sup*n*∈**<sup>N</sup>** IP(*ω<sup>D</sup> <sup>T</sup>* (*Mn*, *<sup>δ</sup>*) <sup>&</sup>gt; *�*) = 0. Hence, from (9) and (10), for each *�* <sup>&</sup>gt; 0, we have

$$\lim\_{\delta \to 0} \sup\_{n \in \mathbb{N}} \mathbb{P}(\omega\_T^D(Y^n, \delta) > \varepsilon) = 0.\tag{11}$$

Conditions (8) and (11) imply the sequence {*Yn*}*n*∈**<sup>N</sup>** satisfies the hypotheses of Theorem 15.2 in [8] and hence, the sequence of probabilities measures {IP(*Y<sup>n</sup>* ∈ ·)}*n*∈**<sup>N</sup>** is tight. This fact, along Theorem 6.1 in [8], imply that for the convergence in law, {*Yn*}*n*∈**<sup>N</sup>** is relatively compact. Let *<sup>Y</sup>* be a process and {*Ynk*}*k*∈**<sup>N</sup>** a subsequence of {*Yn*}*n*∈**<sup>N</sup>** such that {*Ynk*}*k*∈**<sup>N</sup>** converges in law to *<sup>Y</sup>*. Since {sup*t*≥<sup>0</sup> �*Un*(*t*)�}*n*∈**<sup>N</sup>** converges to zero, it follows from (6), (4.2.1) and Theorem 4.1 that

$$Y(t) = Y(0) + \int\_0^t \mathbf{D}(F)(\chi(s))Y(s) \, \mathrm{d}s + M(t).$$

Moreover, since {*Yn*(0)}*n*∈**<sup>N</sup>** converges in distribution to *<sup>η</sup>* and *<sup>Y</sup>*(0) equals *<sup>η</sup>* in distribution, we have *<sup>Y</sup>* is a solution to (5). Finally, uniqueness of solutions to (5) implies {*Yn*}*n*∈**<sup>N</sup>** converges in distribution to *Y*, which concludes the proof.

**Remark 4.2.** *By Itô's rule, the unique solution to (5) is given by*

$$Y(t) = \Psi(t) \left[ \eta + \int\_0^t \Psi(s)^{-1} \, \mathrm{d}M(s) \right], \quad 0 \le t \le 1,$$

*where* Ψ *is the unique solution to the matrix differential equation*

$$\Psi'(t) = \mathcal{D}(F)(\chi(t))\Psi(t), \quad \Psi(0) = \text{identity matrix}.$$

The stochastic process {*Y*(*t*);*t* ≥ 0} allows us to give confidence bands for the deterministic model defined by the solution *χ* to (2). In Section 6 such band is constructed for the SIS epidemic model, where some simulations are carried out.

#### **4.2. Numerical simulations for the SIS epidemic model**

In this section, our attention is focused on the SIS epidemic model. As explained in Subsection 3.3, it is assumed that only susceptible and infective individuals are in a closed homogeneously mixing population. Let *σ*(*t*) and *ι*(*t*) be the densities of susceptible and infective individuals, respectively. The deterministic model is defined by the following system of ordinary differential equations:

$$\frac{\mathrm{d}\sigma}{\mathrm{d}t}(t) = -\beta\sigma(t)\iota(t) + \gamma\iota(t)$$

$$\frac{\mathrm{d}\iota}{\mathrm{d}t}(t) = \beta\sigma(t)\iota(t) - \gamma\iota(t).$$

Since for each *t* ≥ 0, *σ*(*t*) + *ι*(*t*) = 1, this model is completely determined by the ordinary differential equation:

$$\frac{\mathbf{d}\iota}{\mathbf{d}t}(t) = \beta(1 - \iota(t))\iota(t) - \gamma\iota(t)\iota$$

which, given *ι*(0) = *ι*<sup>0</sup> ∈]0, 1[, has the unique solution

$$u(t) = \begin{cases} \frac{\frac{t\eta}{t\otimes\beta+1}}{t\otimes\beta+1} & \text{if } \beta = \gamma \\\\ \frac{\iota\_0(\beta-\gamma)\mathbf{e}^{(\beta-\gamma)t}}{\beta-\gamma+\iota\_0\theta(\mathbf{e}^{(\beta-\gamma)t}-1)} & \text{if } \beta \neq \gamma. \end{cases} \tag{12}$$

0 2 4 6 8 10

deterministic model stochastic model

time

where *<sup>a</sup>*(*s*) = *<sup>β</sup>*<sup>−</sup> *<sup>γ</sup>* <sup>−</sup>2*βy*(*s*) and *<sup>b</sup>*(*u*) = *<sup>β</sup>*(<sup>1</sup> <sup>−</sup> *<sup>y</sup>*(*u*))*y*(*u*) + *<sup>γ</sup>y*(*u*). Hence, *<sup>y</sup>*(*t*) has a normal

*b*(*u*)<sup>2</sup> e<sup>2</sup>

 *t* 0

In this case, a simple expression for the variance of *y*(*t*) can be obtained and a confidence interval for *ι*(*t*) derived. Indeed, for each *t* ≥ 0, *y*(*t*) has normal distribution with mean zero

*<sup>n</sup>*(*t*) <sup>±</sup> *<sup>w</sup>α*/2

*<sup>α</sup>* (*t*), *<sup>u</sup>*<sup>+</sup>

confidence bands for the solution *ι* given by (12). In turn, by defining *u*±

*<sup>n</sup>*(*t*) <sup>∈</sup> [*u*<sup>−</sup>

1 − Φ(*wα*/2) = *α*/2 and Φ is the cumulative distribution function of a standard normal random variable. The random bounds given by (14) allow to construct (nonuniform)

*<sup>β</sup>* (<sup>1</sup> <sup>−</sup> <sup>e</sup>−2(*β*−*γ*)*<sup>t</sup>*

*t*

*<sup>u</sup> <sup>a</sup>*(*s*)d*<sup>s</sup>* d*u*.

e−(*β*−*γ*)(*t*−*u*) d*W*(*u*).

).

*<sup>n</sup>*(*t*) differs from *ι*(*t*) at most *wα*/2

*<sup>n</sup>* is carried out for *β* = 3, *γ* = 1 and *n* = 10, and we note in this

Var(*y*(*t*))/*n*, (14)

*<sup>α</sup>* (*t*)] with an approximate probability 1 − *α*

*<sup>α</sup>*,*n*(*t*)] is a confidence interval for *ι*(*t*), with an approximate confidence

<sup>√</sup>*τ*/*<sup>n</sup>* with an

*<sup>α</sup>* (*t*) = *ι*(*t*) ±

*<sup>n</sup>* starts close to the endemic level, i.e. *<sup>ι</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>τ</sup>*.

Discrete-Time Stochastic Epidemic Models and Their Statistical Inference 241

0

0.3 0.4 0.5 0.6 0.7

**Figure 1.** Comparing the deterministic and stochastic models.

Consequently, *ι*(*t*) = 1 − *τ* for all *t* ≥ 0, and (13) becomes

*<sup>y</sup>*(*t*) =

case random fluctuations are important to be neglected.

*u*± *<sup>α</sup>*,*n*(*t*) = *ι*

Var(*y*(*t*)) = *<sup>t</sup>*

2*γ*(1 − *γ*/*β*)

Var(*y*(*t*)) = *<sup>γ</sup>*

distribution with mean zero and variance

Suppose *τ* < 1 and the process *nι*

Since for each *t* > 0 Var(*y*(*t*)) ≤ *τ* < 1, *ι*

approximate probability equals 1 − *α*.

*<sup>α</sup>*,*n*(*t*), *<sup>u</sup>*<sup>+</sup>

In Figure 2, a simulation of *ι*

*<sup>w</sup>α*/2Var(*y*(*t*))/*n*, we have *<sup>ι</sup>*

and variance

For *α* ∈]0, 1[, [*u*<sup>−</sup>

level 1 − *α*, where

for large values of *n*.

deterministic and stochastic model

The relative removal-rate, see for instance [2] and [3], is defined as *τ* = *γ*/*β*, where *γ* and *β* represent the removal and infection rates, respectively. We note that *ι*(*t*) → 0 as *t* → ∞ if *τ* ≥ 1, while *ι*(*t*) → 1 − *τ* > 0 as *t* → ∞ if *τ* < 1. For this reason, *n*(1 − *τ*) is called the endemic level of the process.

Let *σn*(*t*) and *ι <sup>n</sup>*(*t*) be the densities of susceptible and infective individuals in the stochastic version of the SIS epidemic model. According to our setting, *σ<sup>n</sup>* and *ι <sup>n</sup>* are defined as

$$\sigma^n(t) = \sigma^n(0) + \frac{1}{n}(Z\_2^n(t) - Z\_1^n(t))$$

$$\mu^n(t) = \mu^n(0) + \frac{1}{n}(Z\_1^n(t) - Z\_2^n(t))\_\prime$$

$$\sigma^{n+1} = \sigma^{n+1}(\mu^n(\mu^n)) = 1.1\pi(\Lambda \Im(\mu^n)) \qquad \text{for } \mu^n$$

where IE(Δ*Z<sup>n</sup>* <sup>1</sup> (*t<sup>n</sup> <sup>k</sup>* )|F*<sup>n</sup> <sup>k</sup>*−1) = *βσ*(*t<sup>n</sup> <sup>k</sup>*−1)*ι*(*<sup>t</sup> n <sup>k</sup>*−1) and IE(Δ*Z<sup>n</sup>* <sup>2</sup> (*t<sup>n</sup> <sup>k</sup>* )) = *γι*(*t<sup>n</sup> <sup>k</sup>*−1).

Let us assume *nι <sup>n</sup>*(0)=[*nι*0]. From Theorem 4.1, {*<sup>ι</sup> <sup>n</sup>*}*n*∈**<sup>N</sup>** converges uniformly in probability to *ι*, over compact subsets of **R**+, whenever *ι*(0) = *ι*0. It is worth noting 1 − *τ* is an asymptotically stable equilibrium state for the deterministic model and not for the stochastic model. However, it is expected *ι <sup>n</sup>*(*t*) is close to this value due to Theorem 4.1, for a large enough *n*. In Figure 1, the deterministic and stochastic models, for *β* = 3, *γ* = 1, *ι*<sup>0</sup> = (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)/2 and *<sup>n</sup>* <sup>=</sup> 1, 000, are compared. Let *<sup>y</sup><sup>n</sup>* <sup>=</sup> <sup>√</sup>*n*(*<sup>ι</sup> <sup>n</sup>* <sup>−</sup> *<sup>ι</sup>*). It follows from Theorem 4.2 that {*yn*}*n*∈**<sup>N</sup>** converges in law to *<sup>y</sup>*, the solution to the following Langevin equation:

$$\mathbf{d}\mathbf{d}y(t) = (\beta - \gamma - 2\beta \iota(t))y(t)\,\mathrm{d}t + \sqrt{\beta(1 - \iota(t))\iota(t) + \gamma\iota(t)}\,\mathrm{d}W(t), \quad \iota(0) = 0,$$

where *W* is a one dimensional standard Brownian motion. From Remark 4.2, the solution to this equation is

$$\mathbf{y}(t) = \int\_{0}^{t} b(u) \, \mathbf{e}^{\int\_{u}^{t} a(s) \, \mathrm{d}s} \, \mathrm{d}\mathcal{W}(u), \tag{13}$$

**Figure 1.** Comparing the deterministic and stochastic models.

10 Will-be-set-by-IN-TECH

In this section, our attention is focused on the SIS epidemic model. As explained in Subsection 3.3, it is assumed that only susceptible and infective individuals are in a closed homogeneously mixing population. Let *σ*(*t*) and *ι*(*t*) be the densities of susceptible and infective individuals, respectively. The deterministic model is defined by the following system

<sup>d</sup>*<sup>t</sup>* (*t*) = −*βσ*(*t*)*ι*(*t*) + *γι*(*t*)

<sup>d</sup>*t*(*t*) = *βσ*(*t*)*ι*(*t*) − *γι*(*t*). Since for each *t* ≥ 0, *σ*(*t*) + *ι*(*t*) = 1, this model is completely determined by the ordinary

(*t*) = *β*(1 − *ι*(*t*))*ι*(*t*) − *γι*(*t*),

*<sup>ι</sup>*0*βt*+<sup>1</sup> if *β* = *γ*

*<sup>n</sup>*(*t*) be the densities of susceptible and infective individuals in the stochastic

<sup>2</sup> (*t*) <sup>−</sup> *<sup>Z</sup><sup>n</sup>*

<sup>1</sup> (*t*) <sup>−</sup> *<sup>Z</sup><sup>n</sup>*

<sup>2</sup> (*t<sup>n</sup>*

<sup>1</sup> (*t*))

<sup>2</sup> (*t*)),

*<sup>k</sup>* )) = *γι*(*t<sup>n</sup>*

*<sup>n</sup>*(*t*) is close to this value due to Theorem 4.1, for a large

*β*(1 − *ι*(*t*))*ι*(*t*) + *γι*(*t*) d*W*(*t*), *ι*(0) = 0,

*<sup>u</sup> <sup>a</sup>*(*s*)d*<sup>s</sup>* d*W*(*u*), (13)

*<sup>k</sup>*−1).

*<sup>n</sup>*}*n*∈**<sup>N</sup>** converges uniformly in probability

*<sup>n</sup>* <sup>−</sup> *<sup>ι</sup>*). It follows from Theorem 4.2

*<sup>n</sup>* (*Z<sup>n</sup>*

*<sup>n</sup>* (*Z<sup>n</sup>*

*<sup>k</sup>*−1) and IE(Δ*Z<sup>n</sup>*

to *ι*, over compact subsets of **R**+, whenever *ι*(0) = *ι*0. It is worth noting 1 − *τ* is an asymptotically stable equilibrium state for the deterministic model and not for the stochastic

enough *n*. In Figure 1, the deterministic and stochastic models, for *β* = 3, *γ* = 1, *ι*<sup>0</sup> =

where *W* is a one dimensional standard Brownian motion. From Remark 4.2, the solution to

that {*yn*}*n*∈**<sup>N</sup>** converges in law to *<sup>y</sup>*, the solution to the following Langevin equation:

�

*b*(*u*) e � *t* (12)

*<sup>n</sup>* are defined as

*<sup>β</sup>*−*γ*+*ι*0*β*(e(*β*−*γ*)*<sup>t</sup>*−1) if *<sup>β</sup>* �<sup>=</sup> *<sup>γ</sup>*.

*ι*0

*<sup>ι</sup>*0(*β*−*γ*) <sup>e</sup>(*β*−*γ*)*<sup>t</sup>*

The relative removal-rate, see for instance [2] and [3], is defined as *τ* = *γ*/*β*, where *γ* and *β* represent the removal and infection rates, respectively. We note that *ι*(*t*) → 0 as *t* → ∞ if *τ* ≥ 1, while *ι*(*t*) → 1 − *τ* > 0 as *t* → ∞ if *τ* < 1. For this reason, *n*(1 − *τ*) is called the endemic

**4.2. Numerical simulations for the SIS epidemic model**

d*σ*

d*ι*

d*ι* d*t*

*ι*(*t*) =

⎧ ⎪⎨

⎪⎩

version of the SIS epidemic model. According to our setting, *σ<sup>n</sup>* and *ι*

*<sup>k</sup>*−1)*ι*(*<sup>t</sup> n*

*<sup>n</sup>*(0)=[*nι*0]. From Theorem 4.1, {*<sup>ι</sup>*

*y*(*t*) =

� *t* 0

*ι <sup>n</sup>*(*t*) = *ι*

*<sup>k</sup>*−1) = *βσ*(*t<sup>n</sup>*

(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)/2 and *<sup>n</sup>* <sup>=</sup> 1, 000, are compared. Let *<sup>y</sup><sup>n</sup>* <sup>=</sup> <sup>√</sup>*n*(*<sup>ι</sup>*

d*y*(*t*)=(*β* − *γ* − 2*βι*(*t*))*y*(*t*) d*t* +

*σn*(*t*) = *σn*(0) + <sup>1</sup>

*<sup>n</sup>*(0) + <sup>1</sup>

which, given *ι*(0) = *ι*<sup>0</sup> ∈]0, 1[, has the unique solution

of ordinary differential equations:

differential equation:

level of the process. Let *σn*(*t*) and *ι*

where IE(Δ*Z<sup>n</sup>*

Let us assume *nι*

this equation is

1 (*t n <sup>k</sup>* )|F*<sup>n</sup>*

model. However, it is expected *ι*

where *<sup>a</sup>*(*s*) = *<sup>β</sup>*<sup>−</sup> *<sup>γ</sup>* <sup>−</sup>2*βy*(*s*) and *<sup>b</sup>*(*u*) = *<sup>β</sup>*(<sup>1</sup> <sup>−</sup> *<sup>y</sup>*(*u*))*y*(*u*) + *<sup>γ</sup>y*(*u*). Hence, *<sup>y</sup>*(*t*) has a normal distribution with mean zero and variance

$$\text{Var}(y(t)) = \int\_0^t b(u)^2 \,\mathbf{e}^{2\int\_u^t a(s)\,\text{ds}}\,\mathbf{d}u.$$

Suppose *τ* < 1 and the process *nι <sup>n</sup>* starts close to the endemic level, i.e. *<sup>ι</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>τ</sup>*. Consequently, *ι*(*t*) = 1 − *τ* for all *t* ≥ 0, and (13) becomes

$$y(t) = \sqrt{2\gamma(1-\gamma/\beta)} \int\_0^t \mathbf{e}^{-(\beta-\gamma)(t-u)} \, \text{d}\mathcal{W}(u).$$

In this case, a simple expression for the variance of *y*(*t*) can be obtained and a confidence interval for *ι*(*t*) derived. Indeed, for each *t* ≥ 0, *y*(*t*) has normal distribution with mean zero and variance

$$\text{Var}(y(t)) = \frac{\gamma}{\beta} (1 - \mathbf{e}^{-2(\beta - \gamma)t}).$$

Since for each *t* > 0 Var(*y*(*t*)) ≤ *τ* < 1, *ι <sup>n</sup>*(*t*) differs from *ι*(*t*) at most *wα*/2 <sup>√</sup>*τ*/*<sup>n</sup>* with an approximate probability equals 1 − *α*.

In Figure 2, a simulation of *ι <sup>n</sup>* is carried out for *β* = 3, *γ* = 1 and *n* = 10, and we note in this case random fluctuations are important to be neglected.

For *α* ∈]0, 1[, [*u*<sup>−</sup> *<sup>α</sup>*,*n*(*t*), *<sup>u</sup>*<sup>+</sup> *<sup>α</sup>*,*n*(*t*)] is a confidence interval for *ι*(*t*), with an approximate confidence level 1 − *α*, where

$$u\_{\mathfrak{a},n}^{\pm}(t) = \iota^n(t) \pm w\_{\mathfrak{a}/2} \sqrt{\text{Var}(y(t))/n},\tag{14}$$

1 − Φ(*wα*/2) = *α*/2 and Φ is the cumulative distribution function of a standard normal random variable. The random bounds given by (14) allow to construct (nonuniform) confidence bands for the solution *ι* given by (12). In turn, by defining *u*± *<sup>α</sup>* (*t*) = *ι*(*t*) ± *<sup>w</sup>α*/2Var(*y*(*t*))/*n*, we have *<sup>ι</sup> <sup>n</sup>*(*t*) <sup>∈</sup> [*u*<sup>−</sup> *<sup>α</sup>* (*t*), *<sup>u</sup>*<sup>+</sup> *<sup>α</sup>* (*t*)] with an approximate probability 1 − *α* for large values of *n*.

**5.1. Preliminaries**

Since for each *i* = 1, . . . , *m*, *L<sup>n</sup>*

is a martingale estimator of *βi*.

*Proof.* Note that for each *i* = 1, . . . , *m*,

For each *T* > 0, let *β*

From Theorem 4.1, {∑[*nT*]

Therefore, the proof is complete.

{*β*

*β*.

*sequence* {*β*

*Zn*

*<sup>i</sup>* (*t*)/*n* = *β<sup>i</sup>*

*βn*

*<sup>n</sup>*(*T*)=(*<sup>β</sup>*

*<sup>n</sup>*(*T*)}*n*∈**<sup>N</sup>** *converges in probability to <sup>β</sup>.*

*β n*

*<sup>k</sup>*=<sup>1</sup> *bi*(*χn*(*t<sup>n</sup>*

and from (4.1.1) along with Lemma 2.2 imply {*L<sup>n</sup>*

**5.2. Asymptotic normality of estimators**

*Moreover, suppose the following three conditions hold:*

**(5.1.2)** *For each �* <sup>&</sup>gt; <sup>0</sup> *and each i* <sup>=</sup> 1, . . . , *m,* { <sup>1</sup>

*converges in probability to zero, as n goes to* ∞*.*

*vi*(0) = <sup>0</sup> *and for each t* <sup>&</sup>gt; <sup>0</sup>*,* {*v<sup>n</sup>*

Let us suppose for each *i* = 1, . . . , *m*, *ai* splits as *ai* = *βibi*, where *β<sup>i</sup>* is a parameter of the model

*<sup>n</sup>* + *L<sup>n</sup>*

*<sup>i</sup>* (*T*)

*<sup>k</sup>*=<sup>1</sup> *bi*(*χn*(*t<sup>n</sup>*

*n*

**Proposition 5.1.** *Let assume the hypotheses of Theorem 4.1 are satisfied. Then, for each T* > 0 *the*

<sup>∑</sup>[*nT*]

In this subsection we state two asymptotic normality results for the martingale estimators for

**Theorem 5.1.** *Let χ be the unique solution to (3) and assume conditions (C) and (L) are satisfied.*

**(5.1.3)** *For each i* = 1, . . . , *m, there exists a continuous increasing function vi* : [0, ∞[→ **R** *such that*

*<sup>n</sup>* <sup>∑</sup>*<sup>n</sup>*

*<sup>k</sup>*=<sup>1</sup> IE(*ξ<sup>n</sup>*

*<sup>i</sup>* (*t*)}*n*∈**<sup>N</sup>** *converges in probability to vi*(*t*)*, as n goes to* <sup>∞</sup>*.*

*<sup>k</sup>* (*i*)2I{|*ξ<sup>n</sup>*

*<sup>k</sup>* (*i*)|>*�*

<sup>√</sup>*n*}|F*<sup>n</sup>*

*<sup>k</sup>*−1)}*n*∈**<sup>N</sup>**

**(5.1.1)** {*χn*(0)}*n*∈**<sup>N</sup>** *converges in probability to <sup>χ</sup>*(0)=(*σ*0, *<sup>ι</sup>*0, *<sup>ρ</sup>*0)�*, as n goes to* <sup>∞</sup>*.*

*Ln <sup>i</sup>* (*T*)

*<sup>k</sup>*−1))Δ*t<sup>n</sup>*

*<sup>k</sup>*−1))Δ*tn*}*n*∈**<sup>N</sup>** converges in probability to *<sup>T</sup>*

.

*<sup>k</sup>*=<sup>1</sup> *bi*(*χn*(*t<sup>n</sup>*

*<sup>i</sup>* (*t*), (*t* > 0),

Discrete-Time Stochastic Epidemic Models and Their Statistical Inference 243

*<sup>k</sup>*−1))Δ*t<sup>n</sup>* (15)

<sup>0</sup> *bi*(*χ*(*t*)) d*t*,

*<sup>m</sup>*(*T*))�. Proposition 5.1 below states that

*<sup>i</sup>* (*T*)}*n*∈**<sup>N</sup>** converges in probability to zero.

and *bi* is a non-negative continuous function defined on an open set containing *E*.

*bi*(*χn*(*t n <sup>k</sup>*−1))Δ*<sup>t</sup>*

*<sup>i</sup>* (*T*) = *<sup>Z</sup><sup>n</sup>*

*<sup>n</sup>* <sup>∑</sup>[*nT*]

<sup>1</sup> (*T*),..., *β*

*<sup>i</sup>* is a martingale and

[*nt*] ∑ *k*=1

by observing the epidemic through the time interval [0, *T*], (*T* > 0), we have

*n*

*<sup>n</sup>*(*T*)}*n*∈**<sup>N</sup>** is a consistent sequence of estimators for *<sup>β</sup>* = (*β*1,..., *<sup>β</sup>m*)�.

*<sup>i</sup>* (*T*) = *β<sup>i</sup>* +

**Figure 2.** Simulation for *β* = 3, *γ* = 1 and *n* = 10.

In Figure 3, another simulation of *ι <sup>n</sup>* is carried out for *β* = 3 and *γ* = 1. However, in order to appreciate the convergence of *ι <sup>n</sup>* to the equilibrium 1 <sup>−</sup> *<sup>τ</sup>* for the deterministic model, a population of size *n* = 5, 000 is now considered. The bounds *u*− *<sup>α</sup>* and *<sup>u</sup>*<sup>+</sup> *<sup>α</sup>* are pictured with dash lines for *α* = .05, and hence, in this case *wα*/2 = 1.96.

**Figure 3.** Simulation for *β* = 3, *γ* = 1 and *n* = 5, 000.

Figure 3 confirms |*ι <sup>n</sup>*(*t*) <sup>−</sup> *<sup>ι</sup>*(*t*)| ≤ *<sup>C</sup>* with an approximate probability bigger than 1 <sup>−</sup> *<sup>α</sup>* <sup>=</sup> .95, where *C* = *wα*/2 <sup>√</sup>*τ*/*<sup>n</sup>* <sup>=</sup> .0226.

#### **5. Estimators and their asymptotic behavior**

The main results of this article are stated in this section; however their proofs are deferred to the last section.

#### **5.1. Preliminaries**

12 Will-be-set-by-IN-TECH

0 2 4 6 8 10

time

0 2 4 6 8 10

y(t) bounds

time

The main results of this article are stated in this section; however their proofs are deferred to

*<sup>n</sup>*(*t*) <sup>−</sup> *<sup>ι</sup>*(*t*)| ≤ *<sup>C</sup>* with an approximate probability bigger than 1 <sup>−</sup> *<sup>α</sup>* <sup>=</sup> .95,

*<sup>n</sup>* is carried out for *β* = 3 and *γ* = 1. However, in order

*<sup>n</sup>* to the equilibrium 1 <sup>−</sup> *<sup>τ</sup>* for the deterministic model, a

*<sup>α</sup>* and *<sup>u</sup>*<sup>+</sup>

*<sup>α</sup>* are pictured with

0.0 0.2 0.4 0.6 0.8 1.0

population of size *n* = 5, 000 is now considered. The bounds *u*−

dash lines for *α* = .05, and hence, in this case *wα*/2 = 1.96.

0.46

 0.50

dynamical behavior

**Figure 3.** Simulation for *β* = 3, *γ* = 1 and *n* = 5, 000.

<sup>√</sup>*τ*/*<sup>n</sup>* <sup>=</sup> .0226.

**5. Estimators and their asymptotic behavior**

Figure 3 confirms |*ι*

where *C* = *wα*/2

the last section.

 0.54

dynamical behavior

**Figure 2.** Simulation for *β* = 3, *γ* = 1 and *n* = 10.

In Figure 3, another simulation of *ι*

to appreciate the convergence of *ι*

Let us suppose for each *i* = 1, . . . , *m*, *ai* splits as *ai* = *βibi*, where *β<sup>i</sup>* is a parameter of the model and *bi* is a non-negative continuous function defined on an open set containing *E*.

Since for each *i* = 1, . . . , *m*, *L<sup>n</sup> <sup>i</sup>* is a martingale and

$$Z\_i^n(t)/n = \beta\_i \sum\_{k=1}^{\left[nt\right]} b\_i(\chi^n(t\_{k-1}^n)) \Delta t^n + L\_i^n(t), \quad (t > 0),$$

by observing the epidemic through the time interval [0, *T*], (*T* > 0), we have

$$\widehat{\beta}\_i^{\mathbb{R}}(T) = \frac{Z\_i^{\mathbb{R}}(T)}{n \sum\_{k=1}^{\left[nT\right]} b\_i(\chi^n(t\_{k-1}^n)) \Delta t^n} \tag{15}$$

is a martingale estimator of *βi*.

For each *T* > 0, let *β <sup>n</sup>*(*T*)=(*<sup>β</sup> n* <sup>1</sup> (*T*),..., *β n <sup>m</sup>*(*T*))�. Proposition 5.1 below states that {*β <sup>n</sup>*(*T*)}*n*∈**<sup>N</sup>** is a consistent sequence of estimators for *<sup>β</sup>* = (*β*1,..., *<sup>β</sup>m*)�.

**Proposition 5.1.** *Let assume the hypotheses of Theorem 4.1 are satisfied. Then, for each T* > 0 *the sequence* {*β <sup>n</sup>*(*T*)}*n*∈**<sup>N</sup>** *converges in probability to <sup>β</sup>.*

*Proof.* Note that for each *i* = 1, . . . , *m*,

$$
\widehat{\beta\_i^n}(T) = \beta\_i + \frac{L\_i^n(T)}{\sum\_{k=1}^{\lceil nT \rceil} b\_i(\chi^n(t\_{k-1}^n)) \Delta t^n}.
$$

From Theorem 4.1, {∑[*nT*] *<sup>k</sup>*=<sup>1</sup> *bi*(*χn*(*t<sup>n</sup> <sup>k</sup>*−1))Δ*tn*}*n*∈**<sup>N</sup>** converges in probability to *<sup>T</sup>* <sup>0</sup> *bi*(*χ*(*t*)) d*t*, and from (4.1.1) along with Lemma 2.2 imply {*L<sup>n</sup> <sup>i</sup>* (*T*)}*n*∈**<sup>N</sup>** converges in probability to zero. Therefore, the proof is complete.

#### **5.2. Asymptotic normality of estimators**

In this subsection we state two asymptotic normality results for the martingale estimators for *β*.

**Theorem 5.1.** *Let χ be the unique solution to (3) and assume conditions (C) and (L) are satisfied. Moreover, suppose the following three conditions hold:*

**(5.1.1)** {*χn*(0)}*n*∈**<sup>N</sup>** *converges in probability to <sup>χ</sup>*(0)=(*σ*0, *<sup>ι</sup>*0, *<sup>ρ</sup>*0)�*, as n goes to* <sup>∞</sup>*.*


#### 14 Will-be-set-by-IN-TECH 244 Stochastic Modeling and Control Discrete-Time Stochastic Epidemic Models and their Statistical Inference <sup>15</sup>

*Then, for each T* > 0*,* { <sup>√</sup>*n*(*<sup>β</sup> <sup>n</sup>*(*T*) <sup>−</sup> *<sup>β</sup>*)}*n*∈**<sup>N</sup>** *converges in distribution, as n goes to* <sup>∞</sup>*, to a normal random vector N*(0, <sup>Σ</sup>) *having mean zero and variance-covariance matrix* <sup>Σ</sup> = {*σij*(*T*)}1≤*i*,*j*≤*<sup>m</sup> satisfying*

$$\sigma\_{\hat{l}\hat{\jmath}}(T) = \begin{cases} \,\,\, v\_{\hat{\imath}}(T) / [\int\_0^T b\_{\hat{\imath}}(\chi(t)) \,\,\mathbf{d}t]^2 & \text{if } \,\,\mathbf{i} = \mathbf{j} \\ & \mathbf{0} & \text{if } \,\,\mathbf{i} \neq \mathbf{j}. \end{cases}$$

*Proof.* Let *Q<sup>n</sup>* = (*Q<sup>n</sup>* <sup>1</sup> ,..., *<sup>Q</sup><sup>n</sup> <sup>m</sup>*)�, where *Q<sup>n</sup> <sup>i</sup>* <sup>=</sup> <sup>√</sup>*nL<sup>n</sup> <sup>i</sup>* , (*i* = 1, . . . , *m*). From Lemma 2.1, for each *<sup>t</sup>* <sup>≥</sup> 0, *<sup>Q</sup><sup>n</sup>* <sup>1</sup> (*t*),..., *<sup>Q</sup><sup>n</sup> <sup>m</sup>*(*t*) are <sup>G</sup>*<sup>n</sup> <sup>t</sup>*−-conditionally independent random variables and for each *i* = 1, . . . , *m*, the predictable quadratic variation of the martingale *Q<sup>n</sup> <sup>i</sup>* is given by �*Q<sup>n</sup> <sup>i</sup>* � <sup>=</sup> *<sup>v</sup><sup>n</sup> i* . Condition (5.1.3) indicates for each *<sup>i</sup>* <sup>=</sup> 1, . . . , *<sup>m</sup>*, {�*Q<sup>n</sup> <sup>i</sup>* �(*t*)}*n*∈**<sup>N</sup>** converges in probability to *vi*(*t*). This fact and Condition (5.1.2) enable to conclude that the hypotheses of Corollary 12 in Chapter II in (Rebolledo, 1979) hold. Consequently, {*Qn*(*T*)}*n*∈**<sup>N</sup>** converges in distribution to a normal random vector *Q*(*T*)=(*Q*1(*T*),..., *Qm*(*T*))� with mean zero and satisfying

$$\mathbb{E}(Q\_i(T)Q\_j(T)) = \begin{cases} v\_i(T) \text{ if } i = j \\ 0 & \text{if } i \neq j. \end{cases}$$

We have√*n*(*<sup>β</sup> <sup>n</sup>*(*T*) <sup>−</sup> *<sup>β</sup>*) = *<sup>D</sup>nQn*(*T*), where *<sup>D</sup><sup>n</sup>* <sup>=</sup> *Diag*(*d<sup>n</sup>* <sup>1</sup> ,..., *<sup>d</sup><sup>n</sup> <sup>m</sup>*) is the diagonal random matrix with entries *d<sup>n</sup>* <sup>1</sup> ,..., *<sup>d</sup><sup>n</sup> <sup>m</sup>* in its diagonal given by

$$d\_i^n = \frac{1}{\sum\_{k=1}^{\left[nT\right]} b\_i(\boldsymbol{\chi}^n(t\_{k-1}^n)) \Delta t^n}.$$

Condition (5.1.3) implies Condition (4.1.2) and consequently, the hypotheses of Theorem 4.1 are satisfied. Thus, for each *<sup>i</sup>* <sup>=</sup> 1, . . . , *<sup>m</sup>*, {∑[*nT*] *<sup>k</sup>*=<sup>1</sup> *bi*(*χn*(*<sup>t</sup> n <sup>k</sup>*−1))Δ*tn*}*n*∈**<sup>N</sup>** converges in probability to *<sup>T</sup>* <sup>0</sup> *bi*(*χ*(*t*)) d*t* and Slutzky's theorem (see Theorem 5.1.6 in [25], for instance) enables us to conclude {*DnQn*(*T*)}*n*∈**<sup>N</sup>** converges in distribution to a normal random vector with mean zero and variance-covariance matrix <sup>Σ</sup> = {*σij*}1≤*i*,*j*≤*m*, which satisfies

$$\sigma\_{ij} = \begin{cases} v\_i(T) / \left[ \int\_0^T b\_i(\chi(t)) \, \mathrm{d}t \right]^2 & \text{if } i = j \\ 0 & \text{if } i \neq j. \end{cases}$$

This concludes the proof of Theorem 5.1.

**Remark 5.1.** *Note that Condition (5.1.2) holds whenever for each �* > 0 *and each i* = 1, . . . , *m,*

$$\{\max\_{k \le n} \mathbb{E}(\xi\_k^n(i)^2 \mathbf{I}\_{\{|\xi\_k^n(i)| > \varepsilon \sqrt{n}\}})\}\_{n \in \mathbb{N}}$$

*converges in probability to zero, as n goes to* ∞*. In particular, this condition is satisfied when the double sequence* {*ξ<sup>n</sup> <sup>k</sup>* (*i*)2; 0 <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> *<sup>n</sup>*, *<sup>n</sup>* <sup>∈</sup> **<sup>N</sup>**} *is uniformly integrable.*

## **6. The general epidemic model**

In order to carry out asymptotical inference for a great number of epidemic models, results stated in Section 5 can be applied. In this subsection, statistical inference for the General Epidemic Model is developed.

**6.1. The model**

processes, which we denote by *A<sup>n</sup>* and *Bn*, so that

of *β* and *γ* in [0, *T*], are respectively given by

*<sup>n</sup>*(*T*) = *<sup>A</sup>n*(*T*) <sup>∑</sup>[*nT*] *<sup>k</sup>*=<sup>1</sup> *<sup>σ</sup>n*(*t<sup>n</sup>*

*<sup>n</sup>*(*T*) and *<sup>γ</sup>*

*β*

may produce an infection in *t<sup>n</sup>*

*n*

*n <sup>k</sup>* )|F*<sup>n</sup>*

*<sup>k</sup>*−1),(*β*/*n*)*<sup>ι</sup>*

*<sup>k</sup>* ) and <sup>Δ</sup>*Bn*(*t<sup>n</sup>*

IE(Δ*An*(*t*

In order to verify *β*

(*β*/*n*)*ι*

Δ*An*(*t<sup>n</sup>*

and *γ*

is obtained.

*<sup>n</sup>*(*t n*

parameters (*Sn*(*t*

Note that it satisfies

As mentioned in Subsection 3.1, this model contains two parameters *β* and *γ* denoting the infection and removal rate, respectively, and it is defined by two increasing integer valued

*<sup>I</sup>n*(*t*) = *<sup>I</sup>n*(0) + *<sup>A</sup>n*(*t*) <sup>−</sup> *<sup>B</sup>n*(*t*),

From (15) and the definition of this model given in Subsection 3.1, the martingale estimators

By taking into account some heuristic considerations, which are related to the infection spreading, the distribution of the process can be determined. This fact is sufficient, although not necessary as mentioned previously, to obtain Condition (C). Let *β* denote the average of effective contacts per time unit between an infected person and any other individual in the population. This constant is known as the contact rate, cf. [15]. Hence, *β*/*n* is the average number of effective contacts per time unit per capita of an infected, and it is natural to assume

*<sup>k</sup>*−1). On the other hand, since the total number of adequate contacts in *<sup>t</sup>*

*n*

*<sup>k</sup>*−1)) and IE(Δ*Bn*(*<sup>t</sup>*

*<sup>n</sup>*(*T*) are martingales estimators. Also, in the next subsection, we see that this approach

Regarding the initial state of the epidemic, two natural assumptions can be made and both of them satisfy Condition (5.1.1) in Theorem 5.1. The first consists in assuming {*χn*(0)}*n*∈**<sup>N</sup>** is a deterministic sequence converging to *χ*(0) in *E*. This assumption is quite reasonable whenever a good knowledge about the initial numbers of susceptible, infected and removed individuals is involved. For instance, this assumption can be done when the population is small and the proportion of individuals belonging to each compartment can be observed and calculated at time zero. The second possible assumption consists in assuming (*Sn*(0), *In*(0), *Rn*(0))� has multinomial distribution with parameters *n*, *p*1, *p*<sup>2</sup> and 1 − *p*<sup>1</sup> − *p*2, (0 < *p*<sup>1</sup> + *p*<sup>2</sup> < 1), i.e. for

*s*!*i*!(*n* − *s* − *i*)!

*ps* 1 *pi*

where *a*1(*u*, *v*, *w*) = *βuv* and *a*2(*u*, *v*, *w*) = *γv*. Hence, conditions (C) and (L) hold and, *β*

*<sup>k</sup>* equals the susceptible number *<sup>S</sup>n*(*t<sup>n</sup>*

the probability of a susceptible individual to become infective in a time interval ]*t<sup>n</sup>*

*<sup>k</sup>*−1)) and (*In*(*<sup>t</sup>*

*n*

satisfies the hypotheses of Theorem 5.1 and hence, asymptotic normality of *β*

*<sup>n</sup>*(0) = *<sup>i</sup>*) = *<sup>n</sup>*!

and *γ*

*<sup>n</sup>*(*T*) = *<sup>B</sup>n*(*T*) <sup>∑</sup>[*nT*] *<sup>k</sup>*=<sup>1</sup> *<sup>ι</sup>n*(*t<sup>n</sup>*

Discrete-Time Stochastic Epidemic Models and Their Statistical Inference 245

*<sup>k</sup>*−<sup>1</sup> have independent Binomial distribution with

*<sup>k</sup>*−1) = *<sup>a</sup>*2(*χn*(*<sup>t</sup>*

<sup>2</sup>(<sup>1</sup> <sup>−</sup> *<sup>p</sup>*<sup>1</sup> <sup>−</sup> *<sup>p</sup>*2)*n*−*s*−*<sup>i</sup>*

.

*<sup>n</sup>*(*T*) are martingale estimators, Condition (C) has to be hold.

*<sup>k</sup>*−1), *<sup>γ</sup>*Δ*tn*), respectively.

*n <sup>k</sup>* )|F*<sup>n</sup>* *<sup>k</sup>*−1)

. (16)

*<sup>k</sup>*−1, *<sup>t</sup><sup>n</sup> <sup>k</sup>* ] is

*<sup>n</sup>*(*T*)

*<sup>n</sup>*(*T*)

*n <sup>k</sup>*−<sup>1</sup> that

*<sup>k</sup>*−1), it is assumed that

*n <sup>k</sup>*−1)),

*<sup>n</sup>*(*T*) and *<sup>γ</sup>*

*<sup>S</sup>n*(*t*) = *<sup>S</sup>n*(0) <sup>−</sup> *<sup>A</sup>n*(*t*),

*Rn*(*t*) = *Rn*(0) + *Bn*(*t*).

*<sup>k</sup>*−1)*<sup>ι</sup> <sup>n</sup>*(*t<sup>n</sup> <sup>k</sup>*−1)

*<sup>k</sup>* ) conditionally on <sup>F</sup>*<sup>n</sup>*

*<sup>n</sup>*(*t n*

*<sup>k</sup>*−1) = *<sup>a</sup>*1(*χn*(*<sup>t</sup>*

each (*s*, *i*) ∈ {0, . . . , *n*}×{0, . . . , *n*} such that *s* + *i* ≤ *n*,

IP(*Sn*(0) = *s*, *I*

#### **6.1. The model**

14 Will-be-set-by-IN-TECH

*random vector N*(0, <sup>Σ</sup>) *having mean zero and variance-covariance matrix* <sup>Σ</sup> = {*σij*(*T*)}1≤*i*,*j*≤*<sup>m</sup>*

<sup>0</sup> *bi*(*χ*(*t*)) d*t*]

*T*

*<sup>i</sup>* <sup>=</sup> <sup>√</sup>*nL<sup>n</sup>*

*vi*(*t*). This fact and Condition (5.1.2) enable to conclude that the hypotheses of Corollary 12 in Chapter II in (Rebolledo, 1979) hold. Consequently, {*Qn*(*T*)}*n*∈**<sup>N</sup>** converges in distribution to a normal random vector *Q*(*T*)=(*Q*1(*T*),..., *Qm*(*T*))� with mean zero and satisfying

*<sup>n</sup>*(*T*) <sup>−</sup> *<sup>β</sup>*)}*n*∈**<sup>N</sup>** *converges in distribution, as n goes to* <sup>∞</sup>*, to a normal*

0 *if i* �= *j*.

*vi*(*T*) if *i* = *j* 0 if *i* �= *j*.

*<sup>k</sup>*−1))Δ*t<sup>n</sup>*

*n*

0 if *i* �= *j*.

<sup>√</sup>*n*})}*n*∈**<sup>N</sup>**

<sup>2</sup> *if i* = *j*

*<sup>t</sup>*−-conditionally independent random variables and for each

<sup>1</sup> ,..., *<sup>d</sup><sup>n</sup>*

.

<sup>2</sup> if *i* = *j*

*<sup>i</sup>* , (*i* = 1, . . . , *m*). From Lemma 2.1, for each

*<sup>i</sup>* �(*t*)}*n*∈**<sup>N</sup>** converges in probability to

*<sup>k</sup>*−1))Δ*tn*}*n*∈**<sup>N</sup>** converges in probability

*<sup>i</sup>* is given by �*Q<sup>n</sup>*

*<sup>m</sup>*) is the diagonal random

*<sup>i</sup>* � <sup>=</sup> *<sup>v</sup><sup>n</sup> i* .

*Then, for each T* > 0*,* {

*Proof.* Let *Q<sup>n</sup>* = (*Q<sup>n</sup>*

<sup>1</sup> (*t*),..., *<sup>Q</sup><sup>n</sup>*

*satisfying*

*<sup>t</sup>* <sup>≥</sup> 0, *<sup>Q</sup><sup>n</sup>*

We have√*n*(*<sup>β</sup>*

to *<sup>T</sup>*

*sequence* {*ξ<sup>n</sup>*

matrix with entries *d<sup>n</sup>*

<sup>√</sup>*n*(*<sup>β</sup>*

<sup>1</sup> ,..., *<sup>Q</sup><sup>n</sup>*

*<sup>m</sup>*(*t*) are <sup>G</sup>*<sup>n</sup>*

<sup>1</sup> ,..., *<sup>d</sup><sup>n</sup>*

are satisfied. Thus, for each *<sup>i</sup>* <sup>=</sup> 1, . . . , *<sup>m</sup>*, {∑[*nT*]

This concludes the proof of Theorem 5.1.

**6. The general epidemic model**

Epidemic Model is developed.

Condition (5.1.3) indicates for each *<sup>i</sup>* <sup>=</sup> 1, . . . , *<sup>m</sup>*, {�*Q<sup>n</sup>*

*σij*(*T*) =

*<sup>m</sup>*)�, where *Q<sup>n</sup>*

*i* = 1, . . . , *m*, the predictable quadratic variation of the martingale *Q<sup>n</sup>*

IE(*Qi*(*T*)*Qj*(*T*)) =

*<sup>n</sup>*(*T*) <sup>−</sup> *<sup>β</sup>*) = *<sup>D</sup>nQn*(*T*), where *<sup>D</sup><sup>n</sup>* <sup>=</sup> *Diag*(*d<sup>n</sup>*

*dn*

zero and variance-covariance matrix <sup>Σ</sup> = {*σij*}1≤*i*,*j*≤*m*, which satisfies

*vi*(*T*)/[

*<sup>T</sup>*

**Remark 5.1.** *Note that Condition (5.1.2) holds whenever for each �* > 0 *and each i* = 1, . . . , *m,*

*<sup>k</sup>* (*i*)2I{|*ξ<sup>n</sup>*

*converges in probability to zero, as n goes to* ∞*. In particular, this condition is satisfied when the double*

In order to carry out asymptotical inference for a great number of epidemic models, results stated in Section 5 can be applied. In this subsection, statistical inference for the General

{max *<sup>k</sup>*≤*<sup>n</sup>* IE(*ξ<sup>n</sup>*

*<sup>k</sup>* (*i*)2; 0 <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> *<sup>n</sup>*, *<sup>n</sup>* <sup>∈</sup> **<sup>N</sup>**} *is uniformly integrable.*

*σij* =

*<sup>m</sup>* in its diagonal given by

*<sup>i</sup>* <sup>=</sup> <sup>1</sup> <sup>∑</sup>[*nT*]

*<sup>k</sup>*=<sup>1</sup> *bi*(*χn*(*t<sup>n</sup>*

Condition (5.1.3) implies Condition (4.1.2) and consequently, the hypotheses of Theorem 4.1

*<sup>k</sup>*=<sup>1</sup> *bi*(*χn*(*<sup>t</sup>*

<sup>0</sup> *bi*(*χ*(*t*)) d*t* and Slutzky's theorem (see Theorem 5.1.6 in [25], for instance) enables us to conclude {*DnQn*(*T*)}*n*∈**<sup>N</sup>** converges in distribution to a normal random vector with mean

<sup>0</sup> *bi*(*χ*(*t*)) d*t*]

*<sup>k</sup>* (*i*)|>*�*

*vi*(*T*)/[

As mentioned in Subsection 3.1, this model contains two parameters *β* and *γ* denoting the infection and removal rate, respectively, and it is defined by two increasing integer valued processes, which we denote by *A<sup>n</sup>* and *Bn*, so that

$$\begin{aligned} \mathcal{S}^n(t) &= \mathcal{S}^n(0) - A^n(t), \\ I^\eta(t) &= I^\eta(0) + A^\eta(t) - B^\eta(t), \\ R^\eta(t) &= R^\eta(0) + B^\eta(t). \end{aligned}$$

From (15) and the definition of this model given in Subsection 3.1, the martingale estimators of *β* and *γ* in [0, *T*], are respectively given by

$$\widehat{\beta^{\boldsymbol{n}}}(T) = \frac{A^{\boldsymbol{n}}(T)}{\sum\_{k=1}^{\lfloor \boldsymbol{n}T \rfloor} \sigma^{\boldsymbol{n}}(t\_{k-1}^{\boldsymbol{n}}) \iota^{\boldsymbol{n}}(t\_{k-1}^{\boldsymbol{n}})} \quad \text{and} \quad \widehat{\gamma^{\boldsymbol{n}}}(T) = \frac{B^{\boldsymbol{n}}(T)}{\sum\_{k=1}^{\lfloor \boldsymbol{n}T \rfloor} \iota^{\boldsymbol{n}}(t\_{k-1}^{\boldsymbol{n}})}. \tag{16}$$

In order to verify *β <sup>n</sup>*(*T*) and *<sup>γ</sup> <sup>n</sup>*(*T*) are martingale estimators, Condition (C) has to be hold. By taking into account some heuristic considerations, which are related to the infection spreading, the distribution of the process can be determined. This fact is sufficient, although not necessary as mentioned previously, to obtain Condition (C). Let *β* denote the average of effective contacts per time unit between an infected person and any other individual in the population. This constant is known as the contact rate, cf. [15]. Hence, *β*/*n* is the average number of effective contacts per time unit per capita of an infected, and it is natural to assume the probability of a susceptible individual to become infective in a time interval ]*t<sup>n</sup> <sup>k</sup>*−1, *<sup>t</sup><sup>n</sup> <sup>k</sup>* ] is (*β*/*n*)*ι <sup>n</sup>*(*t n <sup>k</sup>*−1). On the other hand, since the total number of adequate contacts in *<sup>t</sup> n <sup>k</sup>*−<sup>1</sup> that may produce an infection in *t<sup>n</sup> <sup>k</sup>* equals the susceptible number *<sup>S</sup>n*(*t<sup>n</sup> <sup>k</sup>*−1), it is assumed that Δ*An*(*t<sup>n</sup> <sup>k</sup>* ) and <sup>Δ</sup>*Bn*(*t<sup>n</sup> <sup>k</sup>* ) conditionally on <sup>F</sup>*<sup>n</sup> <sup>k</sup>*−<sup>1</sup> have independent Binomial distribution with parameters (*Sn*(*t n <sup>k</sup>*−1),(*β*/*n*)*<sup>ι</sup> <sup>n</sup>*(*t n <sup>k</sup>*−1)) and (*In*(*<sup>t</sup> n <sup>k</sup>*−1), *<sup>γ</sup>*Δ*tn*), respectively.

Note that it satisfies

$$\mathbb{E}(\Delta A^n(t\_k^n)|\mathcal{F}\_{k-1}^n) = a\_1(\chi^n(t\_{k-1}^n)) \quad \text{and} \quad \mathbb{E}(\Delta B^n(t\_k^n)|\mathcal{F}\_{k-1}^n) = a\_2(\chi^n(t\_{k-1}^n)),$$

where *a*1(*u*, *v*, *w*) = *βuv* and *a*2(*u*, *v*, *w*) = *γv*. Hence, conditions (C) and (L) hold and, *β <sup>n</sup>*(*T*) and *γ <sup>n</sup>*(*T*) are martingales estimators. Also, in the next subsection, we see that this approach satisfies the hypotheses of Theorem 5.1 and hence, asymptotic normality of *β <sup>n</sup>*(*T*) and *<sup>γ</sup> <sup>n</sup>*(*T*) is obtained.

Regarding the initial state of the epidemic, two natural assumptions can be made and both of them satisfy Condition (5.1.1) in Theorem 5.1. The first consists in assuming {*χn*(0)}*n*∈**<sup>N</sup>** is a deterministic sequence converging to *χ*(0) in *E*. This assumption is quite reasonable whenever a good knowledge about the initial numbers of susceptible, infected and removed individuals is involved. For instance, this assumption can be done when the population is small and the proportion of individuals belonging to each compartment can be observed and calculated at time zero. The second possible assumption consists in assuming (*Sn*(0), *In*(0), *Rn*(0))� has multinomial distribution with parameters *n*, *p*1, *p*<sup>2</sup> and 1 − *p*<sup>1</sup> − *p*2, (0 < *p*<sup>1</sup> + *p*<sup>2</sup> < 1), i.e. for each (*s*, *i*) ∈ {0, . . . , *n*}×{0, . . . , *n*} such that *s* + *i* ≤ *n*,

$$\mathbb{P}(S^{\mathbb{n}}(0) = \operatorname{s}, I^{\mathbb{n}}(0) = i) = \frac{n!}{\operatorname{s!i!}(n-s-i)!} p\_1^s p\_2^i (1-p\_1-p\_2)^{n-s-i}.$$

This second assumption will be held from now on, and although the parameters of this distribution can be estimated by taking a sample at time zero, in the sequel it will be assumed these parameters are known.

Note that the solution *χ* = (*σ*, *ι*, *ρ*)� to (3) satisfies:

$$\begin{aligned} \frac{d\sigma}{dt}(t) &= \quad -\beta\sigma(t)\iota(t) \\\\ \frac{d\mathbf{d}}{dt}(t) &= \beta\sigma(t)\iota(t) - \gamma\iota(t) \\\\ \frac{d\rho}{dt}(t) &= \quad \gamma\iota(t) \end{aligned} \tag{17}$$

and

and

We have,

Since 0 <sup>≤</sup> *<sup>v</sup><sup>n</sup>*

*vn*

respectively.

matrix

*vn <sup>B</sup>*(*t*) = <sup>1</sup> *n*

*vn <sup>A</sup>*(*t*) = <sup>1</sup> *n*

*<sup>A</sup>*(*t*) <sup>≤</sup> *<sup>β</sup><sup>t</sup>* and 0 <sup>≤</sup> *<sup>v</sup><sup>n</sup>*

*βσn*(*u*)*ι*

*vA*(*t*) = *<sup>t</sup>*

0

Thus, as a consequence of Theorem 5.1, we have that {

<sup>Σ</sup>(*T*) =

<sup>Σ</sup>(*T*) =

*Qn*(*T*) = <sup>√</sup>*<sup>n</sup>*

*β*/ *<sup>T</sup>*

*<sup>T</sup>*

subsets of **R**+. This fact along with

allow to conclude for each *<sup>t</sup>* <sup>≥</sup> 0, {*v<sup>n</sup>*

From (6) we have *σ*(0) − *σ*(*T*) = *β*

and this concludes the proof.

**Corollary 6.1.** *Let T* > 0 *and*

*covariance matrix*

*<sup>A</sup>*(*t*) = *<sup>t</sup>* 0

[*nt*] ∑ *k*=1

[*nt*] ∑ *k*=1

*vn <sup>B</sup>*(*t*) = <sup>1</sup> *n*

IE([Δ*Bn*(*t*

*βσn*(*t n <sup>k</sup>*−1)*<sup>ι</sup> <sup>n</sup>*(*t n*

> [*nt*] ∑ *k*=1

*<sup>n</sup>*(*u*) d*u* + O(1/*n*) and *v<sup>n</sup>*

*<sup>A</sup>*(*t*)}*n*∈**<sup>N</sup>** and {*v<sup>n</sup>*

*βσ*(*u*)*ι*(*u*) <sup>d</sup>*<sup>u</sup>* and *vB*(*t*) = *<sup>t</sup>*

distribution to a normal random bivariate vector having mean zero and variance-covariance

*<sup>β</sup>*2/(*σ*(0) <sup>−</sup> *<sup>σ</sup>*(*T*)) <sup>0</sup>

Since Proposition 6.1 involves a limit distribution depending on the solution of (17), this proposition may be of limited utility. However, Proposition 4.1 and Slutsky's theorem have, as a consequence, the following corollary, which is an alternative to this possible difficulty.

(*σ*(0) <sup>−</sup> *<sup>σ</sup>n*(*T*))1/2(*<sup>β</sup>*

(*ρn*(*T*) <sup>−</sup> *<sup>ρ</sup>*(0))1/2(*<sup>γ</sup>*

 .

*Then,* {*Qn*(*T*)}*n*∈**<sup>N</sup>** *converges in distribution to a normal random bivariate vector with mean zero and*

*I* = *β*<sup>2</sup> 0 0 *γ*<sup>2</sup>

<sup>0</sup> *σ*(*u*)*ι*(*u*) d*u* 0 0 *γ*/

*n <sup>k</sup>* ) − *γI*

*γιn*(*t n*

*<sup>B</sup>*(*t*) <sup>≤</sup> *<sup>γ</sup>t*, the sequences {*v<sup>n</sup>*

converge to zero and hence, Condition (4.1.2) holds. Thus, assumptions of Theorem 4.1 are satisfied and consequently, {*χn*}*n*∈**<sup>N</sup>** converges to *<sup>χ</sup>* uniformly in probability over compact

*n*(*t n <sup>k</sup>*−1)Δ*tn*]

*<sup>k</sup>*−1)(<sup>1</sup> <sup>−</sup> *βιn*(*<sup>t</sup>*

*<sup>k</sup>*−1)(<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*Δ*tn*).

*<sup>B</sup>*(*t*) = *<sup>t</sup>* 0

0

<sup>√</sup>*n*(*<sup>γ</sup>*

*<sup>n</sup>*(*T*) − *<sup>β</sup>*)

*<sup>n</sup>*(*T*) − *<sup>γ</sup>*)

 *T* <sup>0</sup> *ι*(*u*) d*u*

<sup>0</sup> *σ*(*u*)*ι*(*u*) d*u* and *ρ*(*T*) − *ρ*(0) = *γ*

<sup>0</sup> *<sup>γ</sup>*2/(*ρ*(*T*) <sup>−</sup> *<sup>ρ</sup>*(0))

<sup>2</sup>|F*<sup>n</sup> <sup>k</sup>*−1).

Discrete-Time Stochastic Epidemic Models and Their Statistical Inference 247

*<sup>A</sup>*(*t*)/*n*}*n*∈**<sup>N</sup>** and {*v<sup>n</sup>*

*<sup>B</sup>*(*t*)}*n*∈**<sup>N</sup>** converge in probability to

*γι*(*u*) d*u*,

 .

*<sup>T</sup>*

,

 .

*γιn*(*u*) d*u* + O(1/*n*),

*<sup>n</sup>*(*T*) <sup>−</sup> *<sup>γ</sup>*)}*n*∈**<sup>N</sup>** converges in

<sup>0</sup> *ι*(*u*) d*u*. Hence,

*<sup>B</sup>*(*t*)/*n*}*n*∈**<sup>N</sup>**

*n <sup>k</sup>*−1)Δ*tn*),

with initial condition (*σ*(0), *ι*(0), *ρ*(0)) = (*p*1, *p*2, 1 − *p*<sup>1</sup> − *p*2).

#### **6.2. Asymptotic normality**

As a consequence of Theorem 5.1, the following proposition is stated, which shows the parameter estimators for the SIR epidemic model are asymptotically normal. In this subsection all notations and facts given on the preceding subsection will be maintained.

**Proposition 6.1.** *Let λ* = (*β*, *γ*)�*, T* > 0 *and λ* �*<sup>n</sup>* = (*<sup>β</sup>* �*n*, *<sup>γ</sup>* �*n*)�*. Then,* { <sup>√</sup>*n*(*<sup>λ</sup>* �*n*(*T*) <sup>−</sup> *<sup>λ</sup>*)}*n*∈**<sup>N</sup>** *converges in distribution to a normal random bivariate vector with mean zero and variance-covariance matrix*

$$
\Sigma(T) = \begin{pmatrix}
\beta^2/(\sigma(0) - \sigma(T)) & 0 \\
& 0 & \gamma^2/(\rho(T) - \rho(0))
\end{pmatrix}.
$$

*Proof.* From the Kolmogorov Law of Large Numbers, {*χn*(0)}*n*∈**<sup>N</sup>** converges almost sure to *χ*<sup>0</sup> = (*p*1, *p*2, 1 − *p*<sup>1</sup> − *p*2)� and hence, Condition (5.1.1) holds.

Let *ξ<sup>n</sup> <sup>k</sup>* (1) = <sup>Δ</sup>*An*(*<sup>t</sup> n <sup>k</sup>* ) <sup>−</sup> *βσn*(*<sup>t</sup> n <sup>k</sup>*−1)*<sup>ι</sup> <sup>n</sup>*(*t<sup>n</sup> <sup>k</sup>*−1) and *<sup>ξ</sup><sup>n</sup> <sup>k</sup>* (2) = <sup>Δ</sup>*Bn*(*t<sup>n</sup> <sup>k</sup>* ) <sup>−</sup> *γιn*(*<sup>t</sup> n <sup>k</sup>*−1). In order to verify Condition (5.1.2), it suffices to prove for each *<sup>i</sup>* <sup>=</sup> 1, . . . , *<sup>m</sup>*, the double sequence {*ξ<sup>n</sup> <sup>k</sup>* (*i*)2; 0 <sup>≤</sup> *k* ≤ *n*, *n* ∈ **N**} is uniformly integrable. For this purpose, we note that, for a random variable *X* with Binomial distribution, the following inequality holds:

$$\mathbb{E}(|X - \mathbb{E}(X)|^3) \le 8\,\mathbb{E}(X)^3 + 6\,\mathbb{E}(X)^2 + \mathbb{E}(X).$$

Consequently, according to our approach, for each *i* = 1, 2, we have

$$\sup \{ \mathbb{E}(|\tilde{\zeta}\_k^{\eta}(i)|^3) : n, k \in \mathbb{N} \} \le 8\zeta^3 + 6\zeta^2 + \zeta\_{\nu}$$

where

$$\zeta = \begin{cases} \beta \text{ if } i = 1 \\ \gamma \text{ if } i = 2. \end{cases}$$

Hence, {*ξ<sup>n</sup> <sup>k</sup>* (*i*)2}*n*,*k*∈**<sup>N</sup>** is uniformly integrable and Condition (5.1.2) holds.

Let

$$v\_A^n(t) = \frac{1}{n} \sum\_{k=1}^{\lfloor nt \rfloor} \mathbb{E}( [\Delta A^n(t\_k^n) - \beta S^n(t\_{k-1}^n) \iota^n(t\_{k-1}^n) \Delta t\_n]^2 | \mathcal{F}\_{k-1}^n)$$

and

16 Will-be-set-by-IN-TECH

This second assumption will be held from now on, and although the parameters of this distribution can be estimated by taking a sample at time zero, in the sequel it will be assumed

<sup>d</sup>*<sup>t</sup>* (*t*) = −*βσ*(*t*)*ι*(*t*)

<sup>d</sup>*<sup>t</sup>* (*t*) = *γι*(*t*),

<sup>d</sup>*t*(*t*) = *βσ*(*t*)*ι*(*t*) − *γι*(*t*)

As a consequence of Theorem 5.1, the following proposition is stated, which shows the parameter estimators for the SIR epidemic model are asymptotically normal. In this subsection all notations and facts given on the preceding subsection will be maintained.

*converges in distribution to a normal random bivariate vector with mean zero and variance-covariance*

*<sup>β</sup>*2/(*σ*(0) <sup>−</sup> *<sup>σ</sup>*(*T*)) <sup>0</sup>

*Proof.* From the Kolmogorov Law of Large Numbers, {*χn*(0)}*n*∈**<sup>N</sup>** converges almost sure to

*k* ≤ *n*, *n* ∈ **N**} is uniformly integrable. For this purpose, we note that, for a random variable

*<sup>k</sup>*−1) and *<sup>ξ</sup><sup>n</sup>*

Condition (5.1.2), it suffices to prove for each *<sup>i</sup>* <sup>=</sup> 1, . . . , *<sup>m</sup>*, the double sequence {*ξ<sup>n</sup>*

<sup>3</sup>) <sup>≤</sup> 8 IE(*X*)

⎧ ⎨ ⎩ �*<sup>n</sup>* = (*<sup>β</sup>*

�*n*, *<sup>γ</sup>*

<sup>0</sup> *<sup>γ</sup>*2/(*ρ*(*T*) <sup>−</sup> *<sup>ρ</sup>*(0))

*<sup>k</sup>* (2) = <sup>Δ</sup>*Bn*(*t<sup>n</sup>*

<sup>3</sup>) : *<sup>n</sup>*, *<sup>k</sup>* <sup>∈</sup> **<sup>N</sup>**} ≤ <sup>8</sup>*ζ*<sup>3</sup> <sup>+</sup> <sup>6</sup>*ζ*<sup>2</sup> <sup>+</sup> *<sup>ζ</sup>*,

*β* if *i* = 1

*γ* if *i* = 2.

*n <sup>k</sup>*−1)*<sup>ι</sup> n*(*t n <sup>k</sup>*−1)Δ*tn*]

<sup>3</sup> + 6 IE(*X*)<sup>2</sup> + IE(*X*).

�*n*)�*. Then,* {

*<sup>k</sup>* ) <sup>−</sup> *γιn*(*<sup>t</sup>*

⎞ ⎠ .

*n*

<sup>2</sup>|F*<sup>n</sup> <sup>k</sup>*−1)

<sup>√</sup>*n*(*<sup>λ</sup>*

�*n*(*T*) <sup>−</sup> *<sup>λ</sup>*)}*n*∈**<sup>N</sup>**

*<sup>k</sup>*−1). In order to verify

*<sup>k</sup>* (*i*)2; 0 <sup>≤</sup>

these parameters are known.

**6.2. Asymptotic normality**

*matrix*

Let *ξ<sup>n</sup>*

where

Hence, {*ξ<sup>n</sup>*

Let

*<sup>k</sup>* (1) = <sup>Δ</sup>*An*(*<sup>t</sup>*

*n*

*vn <sup>A</sup>*(*t*) = <sup>1</sup> *n*

Note that the solution *χ* = (*σ*, *ι*, *ρ*)� to (3) satisfies:

**Proposition 6.1.** *Let λ* = (*β*, *γ*)�*, T* > 0 *and λ*

Σ(*T*) =

*<sup>k</sup>* ) <sup>−</sup> *βσn*(*t<sup>n</sup>*

d*σ*

d*ι*

d*ρ*

with initial condition (*σ*(0), *ι*(0), *ρ*(0)) = (*p*1, *p*2, 1 − *p*<sup>1</sup> − *p*2).

⎛ ⎝

*χ*<sup>0</sup> = (*p*1, *p*2, 1 − *p*<sup>1</sup> − *p*2)� and hence, Condition (5.1.1) holds.

*<sup>k</sup>*−1)*<sup>ι</sup> <sup>n</sup>*(*t<sup>n</sup>*

*X* with Binomial distribution, the following inequality holds:

IE(|*X* − IE(*X*)|

sup{IE(|*ξ<sup>n</sup>*

[*nt*] ∑ *k*=1

Consequently, according to our approach, for each *i* = 1, 2, we have

*<sup>k</sup>* (*i*)|

IE([Δ*An*(*t*

*ζ* =

*<sup>k</sup>* (*i*)2}*n*,*k*∈**<sup>N</sup>** is uniformly integrable and Condition (5.1.2) holds.

*n*

*<sup>k</sup>* ) <sup>−</sup> *<sup>β</sup>Sn*(*<sup>t</sup>*

$$v\_B^n(t) = \frac{1}{n} \sum\_{k=1}^{\lceil nt \rceil} \mathbb{E}( [\Delta B^n(t\_k^n) - \gamma I^n(t\_{k-1}^n) \Delta t\_n]^2 | \mathcal{F}\_{k-1}^n).$$

We have,

$$w\_A^n(t) = \frac{1}{n} \sum\_{k=1}^{\lfloor nt \rfloor} \beta \sigma^n(t\_{k-1}^n) \iota^n(t\_{k-1}^n) (1 - \beta \iota^n(t\_{k-1}^n) \Delta t\_n).$$

and

(17)

$$v\_B^n(t) = \frac{1}{n} \sum\_{k=1}^{\left[nt\right]} \gamma \iota^n(t\_{k-1}^n)(1 - \gamma \Delta t\_n).$$

Since 0 <sup>≤</sup> *<sup>v</sup><sup>n</sup> <sup>A</sup>*(*t*) <sup>≤</sup> *<sup>β</sup><sup>t</sup>* and 0 <sup>≤</sup> *<sup>v</sup><sup>n</sup> <sup>B</sup>*(*t*) <sup>≤</sup> *<sup>γ</sup>t*, the sequences {*v<sup>n</sup> <sup>A</sup>*(*t*)/*n*}*n*∈**<sup>N</sup>** and {*v<sup>n</sup> <sup>B</sup>*(*t*)/*n*}*n*∈**<sup>N</sup>** converge to zero and hence, Condition (4.1.2) holds. Thus, assumptions of Theorem 4.1 are satisfied and consequently, {*χn*}*n*∈**<sup>N</sup>** converges to *<sup>χ</sup>* uniformly in probability over compact subsets of **R**+. This fact along with

$$v\_A^n(t) = \int\_0^t \beta v^n(u) i^n(u) \,\mathrm{d}u + \mathcal{O}(1/n) \quad \text{and} \quad v\_B^n(t) = \int\_0^t \gamma u^n(u) \,\mathrm{d}u + \mathcal{O}(1/n),$$

allow to conclude for each *<sup>t</sup>* <sup>≥</sup> 0, {*v<sup>n</sup> <sup>A</sup>*(*t*)}*n*∈**<sup>N</sup>** and {*v<sup>n</sup> <sup>B</sup>*(*t*)}*n*∈**<sup>N</sup>** converge in probability to

$$v\_A(t) = \int\_0^t \beta \sigma(\boldsymbol{u}) \iota(\boldsymbol{u}) \, \mathrm{d}\boldsymbol{u} \quad \text{and} \quad v\_B(t) = \int\_0^t \gamma \iota(\boldsymbol{u}) \, \mathrm{d}\boldsymbol{u} \, \boldsymbol{u}$$

respectively.

Thus, as a consequence of Theorem 5.1, we have that { <sup>√</sup>*n*(*<sup>γ</sup> <sup>n</sup>*(*T*) <sup>−</sup> *<sup>γ</sup>*)}*n*∈**<sup>N</sup>** converges in distribution to a normal random bivariate vector having mean zero and variance-covariance matrix

$$
\Sigma(T) = \begin{pmatrix}
\beta/\int\_0^T \sigma(u)\iota(u)\,\mathrm{d}u & 0\\0 & \gamma/\int\_0^T \iota(u)\,\mathrm{d}u
\end{pmatrix}.
$$

From (6) we have *σ*(0) − *σ*(*T*) = *β <sup>T</sup>* <sup>0</sup> *σ*(*u*)*ι*(*u*) d*u* and *ρ*(*T*) − *ρ*(0) = *γ <sup>T</sup>* <sup>0</sup> *ι*(*u*) d*u*. Hence,

$$
\Sigma(T) = \begin{pmatrix}
\beta^2/(\sigma(0) - \sigma(T)) & 0 \\
0 & \gamma^2/(\rho(T) - \rho(0))
\end{pmatrix},
$$

and this concludes the proof.

Since Proposition 6.1 involves a limit distribution depending on the solution of (17), this proposition may be of limited utility. However, Proposition 4.1 and Slutsky's theorem have, as a consequence, the following corollary, which is an alternative to this possible difficulty.

**Corollary 6.1.** *Let T* > 0 *and*

$$Q\_n(T) = \sqrt{n} \begin{pmatrix} (\sigma(0) - \sigma^n(T))^{1/2} (\widehat{\beta^n}(T) - \beta) \\ (\rho^n(T) - \rho(0))^{1/2} (\widehat{\gamma^n}(T) - \gamma) \end{pmatrix}.$$

*Then,* {*Qn*(*T*)}*n*∈**<sup>N</sup>** *converges in distribution to a normal random bivariate vector with mean zero and covariance matrix*

$$I = \begin{pmatrix} \beta^2 & 0\\ 0 & \gamma^2 \end{pmatrix}.$$

#### **6.3. Hypothesis test for the infection rate**

In this section, we are interested in proving whether the parameter *β* of this SIR epidemic model belongs to a subset of the parametric set. This analysis, which is known as hypothesis test, will intend to define whether the infection rate is bigger than a fixed value *β*<sup>0</sup> > 0, i.e. the null hypothesis is stated as H0 : *β* = *β*<sup>0</sup> against the one-side alternative H1 : *β* > *β*0. Since under H0, {*β <sup>n</sup>*(*T*)}*n*∈**<sup>N</sup>** converges in probability to *<sup>β</sup>*0, H0 shall be rejected in favor of H1 when *β <sup>n</sup>*(*T*) is too large, i.e. when *<sup>β</sup> <sup>n</sup>*(*T*) > *un*, being *un* determined by IP(*<sup>β</sup> <sup>n</sup>*(*T*) > *un*|H0) ≤ *α*. Here, *α* is the preassigned level of significance which controls the probability of falsely rejecting H0 when H0 is true.

occurrence of an epidemic outbreak. Actually, due to the fact that in the deterministic version

(*t*) = *βσ*(*t*)*ι*(*t*) − *γι*(*t*),

by assuming *ι*(0) > 0, one has the derivative of *ι* at zero is bigger than zero, if and only if, *τ* < *σ*(0). Consequently, unless the initial density of susceptible individuals is bigger than the relative removal-rate, no epidemics can start. It is worth pointing out that this parameter is connected with the basic reproduction number, which is defined as the number of cases generated by one infective over the period of infectivity when that infective was introduced into a large population of susceptible individuals. See for instance, [16], or, [14]. Actually, it becomes *R*<sup>0</sup> = *σ*(0)/*τ*. Notice *σ*(0) ≈ 1 when a large population of susceptible individuals is considered. Hence, an epidemic can start with a positive probability, if *R*<sup>0</sup> > 1, and it will die out quickly if *R*<sup>0</sup> ≤ 1. The knowledge of the basic reproduction number allows the establishment of vaccination policies when necessary, in order to reduce the number of susceptible individuals in a population to such a level that *R*<sup>0</sup> is brought below the unity

H0 : *R*<sup>0</sup> = 1 against H1 : *R*<sup>0</sup> > 1.

preassigned level of significance controlling the probability of falsely rejecting H0 when H0 is

<sup>0</sup> (*T*) > *un*}, where *R*

<sup>√</sup>*n*,

1 *ρ*(*T*) − *ρ*(0)

*<sup>n</sup>*(*T*)) − *<sup>f</sup>*(*σ*(0)*β*, *<sup>γ</sup>*))

*γ*3 *n*

*<sup>n</sup>*(*T*)) and (*σ*(0)*β*, *<sup>γ</sup>*).

<sup>√</sup>*n*(*<sup>γ</sup>*

√*n*

(*γ*

*γ*2

*<sup>n</sup>*(*T*) is asymptotically normal with mean <sup>Δ</sup> and variance *<sup>v</sup>*<sup>2</sup>

*n*

*n*

Discrete-Time Stochastic Epidemic Models and Their Statistical Inference 249

1/2 .

<sup>0</sup> (*T*) − *R*0) + Δ. Let *f*(*x*, *y*) = *x*/*y*, (*x*, *y* ∈ **R**, *y* �= 0).

*<sup>n</sup>*(*T*) − *<sup>γ</sup>*) + *En*,

*<sup>n</sup>*(*T*) <sup>−</sup> *<sup>γ</sup>*)<sup>2</sup>

<sup>0</sup> (*T*) = *σ*(0)*β*

*n*

<sup>0</sup> (*T*) > *un*|H0) ≤ *α*, being *α* a

*<sup>n</sup>*(*T*)/*<sup>γ</sup>*

<sup>0</sup> (*T*) − 1)*. Then, under*

*<sup>n</sup>*(*T*)

<sup>0</sup>(*T*).

*n*

true. The following proposition allows us to carry out the mentioned hypothesis test.

*<sup>n</sup>*(*T*) *has as asymptotically normal distribution with mean* <sup>Δ</sup> *and variance v*0(*T*)2*, where*

*<sup>σ</sup>*(0) <sup>−</sup> *<sup>σ</sup>*(*T*) <sup>+</sup>

*n*

<sup>√</sup>*n*(*<sup>β</sup>*

*<sup>n</sup>*(*T*) − *<sup>β</sup>*)(*<sup>γ</sup>*

*<sup>n</sup>*(*T*), *<sup>γ</sup>*

*<sup>n</sup>*(*T*) <sup>−</sup> *<sup>β</sup>*) <sup>−</sup> *<sup>σ</sup>*(0)*<sup>β</sup>*

*<sup>n</sup>*(*T*), *<sup>γ</sup>*

From Propositions 5.1 and 6.1, {*En*}*n*∈**<sup>N</sup>** converges in probability to zero. Hence, it follows

*<sup>n</sup>*(*T*) <sup>−</sup> *<sup>γ</sup>*) + *<sup>σ</sup>*(0)2*δ<sup>n</sup>*

H<sup>Δ</sup> : *R*<sup>0</sup> = 1 + Δ/

d*ι* d*t*

threshold. Moreover, it is useful to carry out the following hypothesis test:

**Proposition 6.2.** *Let* <sup>Δ</sup> *be a non-negative real number and <sup>U</sup><sup>n</sup>*(*T*) = <sup>√</sup>*n*(*<sup>R</sup>*

*<sup>v</sup>*0(*T*) = <sup>1</sup>

*<sup>n</sup>*(*T*) = <sup>√</sup>*n*(*<sup>R</sup>*

<sup>0</sup> (*T*) <sup>−</sup> *<sup>R</sup>*0) = <sup>√</sup>*n*(*f*(*σ*(0)*<sup>β</sup>*

√*n γ*2 *n*

and (*δn*, *γn*) is a random point between (*σ*(0)*β*

(*β*

<sup>=</sup> *<sup>σ</sup>*(0) *γ*

of this model, we have (see (17))

A natural critical region for the test is {*R*

*local alternatives having the form*

*Proof.* Under H<sup>Δ</sup> we have *U*

from Proposition 6.1 that *U*

Therefore, this proof is complete.

By Taylor's theorem, we have <sup>√</sup>*n*(*<sup>R</sup> n*

*En* <sup>=</sup> <sup>−</sup> *<sup>σ</sup>*(0)

*U*

where

and *un* is a constant which should be determined by IP(*R*

By means of Corollary 6.1, it is possible to obtain an approximate value *un* when *n* is large. For this purpose, IP(*β <sup>n</sup>*(*T*) > *un*|H0) ≤ *<sup>α</sup>* is replaced by the following weaker requirement:

$$\mathbb{P}(\bar{\beta}^n(T) > \mathfrak{u}\_n | \mathcal{H}\_0) \to \mathfrak{a}\_\prime \quad \text{as} \quad n \to \infty.$$

From the Proposition 6.1, under H0, { <sup>√</sup>*n*(*σ*(0) <sup>−</sup> *<sup>σ</sup>*(*T*))1/2(*<sup>β</sup> <sup>n</sup>*(*T*) <sup>−</sup> *<sup>β</sup>*0)/*β*0}*n*∈**<sup>N</sup>** converges in distribution to a random variable having mean zero and variance one. Therefore, with an asymptotic level of significance *α*,

$$(\sqrt{n}(\sigma(0) - \sigma(T))^{1/2}(u\_n - \beta\_0)/\beta\_0 \to t\_\alpha \text{ as } n \to \infty)$$

where 1 − Φ(*tα*) = *α*. Here, Φ is the cumulative distribution function of a standard normal variable. Hence, *un* = *β*<sup>0</sup> + *β*0*tα*/ *<sup>n</sup>*(*σ*(0) <sup>−</sup> *<sup>σ</sup>*(*T*)) + <sup>o</sup>(1/*n*). In particular, we can choose *un* = *β*<sup>0</sup> + *β*0*tα*/ *<sup>n</sup>*(*σ*(0) <sup>−</sup> *<sup>σ</sup>*(*T*)) and therefore, with an asymptotic level of significance *<sup>α</sup>*, a critical region for the test is

$$R\_1(n) = \left\{ \sqrt{n}(\sigma(0) - \sigma(T))^{1/2} (\widehat{\beta^n}(T) - \beta\_0) / \beta\_0 \ge t\_a \right\}.$$

Under H0, *σ*(*T*) is known but it can not be explicitly obtained due to the fact that (6) does not admit a closed-form solution. However, we can take advantage of the fact that {*σn*(*T*)}*n*∈**<sup>N</sup>** converges in probability to *σ*(*T*), and by Slutzky's theorem, it can be obtained that

$$R\_2(n) = \left\{ \sqrt{n} (\sigma(0) - \sigma^n(T))^{1/2} (\widehat{\beta^n}(T) - \beta\_0) / \beta\_0 \ge t\_\alpha \right\}$$

is a critical region for the test with an asymptotic level of significance *α*.

Let Δ > 0 and let us consider the alternative hypothesis

$$\mathcal{H}\_1: \mathcal{J} = \mathcal{J}\_0 + \Delta / \sqrt{n}.$$

In this case, under H1 the power of the test *πn*(*β*) = IP(*R*(*n*)|H1), where *R*(*n*) = *R*1(*n*) or *<sup>R</sup>*(*n*) = *<sup>R</sup>*2(*n*), converges to <sup>Φ</sup>((*σ*(0)−*σ*(*T*))1/2Δ−*tαβ*<sup>0</sup> *<sup>β</sup>* ).

#### **6.4. Relative removal-rate and basic reproduction number**

As pointed out in Subsection 4.2, a fundamental concept resulting from the mathematical theory of the general deterministic epidemic model is the relative removal-rate, see for instance [2] and [3]. This number is defined as *τ* = *γ*/*β*, where *γ* and *β* represent the removal and infection rates, respectively, and it plays a crucial part in determining the probable occurrence of an epidemic outbreak. Actually, due to the fact that in the deterministic version of this model, we have (see (17))

$$\frac{\mathbf{d}\iota}{\mathbf{d}t}(t) = \beta \sigma(t)\iota(t) - \gamma \iota(t)\iota$$

by assuming *ι*(0) > 0, one has the derivative of *ι* at zero is bigger than zero, if and only if, *τ* < *σ*(0). Consequently, unless the initial density of susceptible individuals is bigger than the relative removal-rate, no epidemics can start. It is worth pointing out that this parameter is connected with the basic reproduction number, which is defined as the number of cases generated by one infective over the period of infectivity when that infective was introduced into a large population of susceptible individuals. See for instance, [16], or, [14]. Actually, it becomes *R*<sup>0</sup> = *σ*(0)/*τ*. Notice *σ*(0) ≈ 1 when a large population of susceptible individuals is considered. Hence, an epidemic can start with a positive probability, if *R*<sup>0</sup> > 1, and it will die out quickly if *R*<sup>0</sup> ≤ 1. The knowledge of the basic reproduction number allows the establishment of vaccination policies when necessary, in order to reduce the number of susceptible individuals in a population to such a level that *R*<sup>0</sup> is brought below the unity threshold. Moreover, it is useful to carry out the following hypothesis test:

$$H\_0: \mathcal{R}\_0 = 1 \quad \text{against} \quad H\_1: \mathcal{R}\_0 > 1.$$

A natural critical region for the test is {*R n* <sup>0</sup> (*T*) > *un*}, where *R n* <sup>0</sup> (*T*) = *σ*(0)*β <sup>n</sup>*(*T*)/*<sup>γ</sup> <sup>n</sup>*(*T*) and *un* is a constant which should be determined by IP(*R n* <sup>0</sup> (*T*) > *un*|H0) ≤ *α*, being *α* a preassigned level of significance controlling the probability of falsely rejecting H0 when H0 is true. The following proposition allows us to carry out the mentioned hypothesis test.

**Proposition 6.2.** *Let* <sup>Δ</sup> *be a non-negative real number and <sup>U</sup><sup>n</sup>*(*T*) = <sup>√</sup>*n*(*<sup>R</sup> n* <sup>0</sup> (*T*) − 1)*. Then, under local alternatives having the form*

$$\mathcal{H}\_{\Delta}: \mathcal{R}\_0 = 1 + \Delta / \sqrt{n}\_r$$

*U <sup>n</sup>*(*T*) *has as asymptotically normal distribution with mean* <sup>Δ</sup> *and variance v*0(*T*)2*, where*

$$v\_0(T) = \left(\frac{1}{\sigma(0) - \sigma(T)} + \frac{1}{\rho(T) - \rho(0)}\right)^{1/2}.\ .$$

*Proof.* Under H<sup>Δ</sup> we have *U <sup>n</sup>*(*T*) = <sup>√</sup>*n*(*<sup>R</sup> n* <sup>0</sup> (*T*) − *R*0) + Δ. Let *f*(*x*, *y*) = *x*/*y*, (*x*, *y* ∈ **R**, *y* �= 0). By Taylor's theorem, we have

$$\sqrt{n}(\widehat{R}\_0^{\overrightarrow{n}}(T) - R\_0) = \sqrt{n}(f(\sigma(0)\widehat{\beta}^{\overrightarrow{n}}(T), \widehat{\gamma}^{\overrightarrow{n}}(T)) - f(\sigma(0)\beta, \gamma))$$

$$= \frac{\sigma(0)}{\gamma}\sqrt{n}(\widehat{\beta}^{\overrightarrow{n}}(T) - \beta) - \frac{\sigma(0)\beta}{\gamma^2}\sqrt{n}(\widehat{\gamma}^{\overrightarrow{n}}(T) - \gamma) + E\_{n\beta}$$

where

18 Will-be-set-by-IN-TECH

In this section, we are interested in proving whether the parameter *β* of this SIR epidemic model belongs to a subset of the parametric set. This analysis, which is known as hypothesis test, will intend to define whether the infection rate is bigger than a fixed value *β*<sup>0</sup> > 0, i.e. the null hypothesis is stated as H0 : *β* = *β*<sup>0</sup> against the one-side alternative H1 : *β* > *β*0. Since

*α*. Here, *α* is the preassigned level of significance which controls the probability of falsely

By means of Corollary 6.1, it is possible to obtain an approximate value *un* when *n* is large.

*<sup>n</sup>*(*T*) > *un*|H0) → *<sup>α</sup>*, as *<sup>n</sup>* → <sup>∞</sup>.

in distribution to a random variable having mean zero and variance one. Therefore, with an

<sup>√</sup>*n*(*σ*(0) <sup>−</sup> *<sup>σ</sup>*(*T*))1/2(*un* <sup>−</sup> *<sup>β</sup>*0)/*β*<sup>0</sup> <sup>→</sup> *<sup>t</sup><sup>α</sup>* as *<sup>n</sup>* <sup>→</sup> <sup>∞</sup>,

where 1 − Φ(*tα*) = *α*. Here, Φ is the cumulative distribution function of a standard normal

Under H0, *σ*(*T*) is known but it can not be explicitly obtained due to the fact that (6) does not admit a closed-form solution. However, we can take advantage of the fact that {*σn*(*T*)}*n*∈**<sup>N</sup>**

<sup>√</sup>*n*(*σ*(0) <sup>−</sup> *<sup>σ</sup>*(*T*))1/2(*<sup>β</sup>*

converges in probability to *σ*(*T*), and by Slutzky's theorem, it can be obtained that

<sup>√</sup>*n*(*σ*(0) <sup>−</sup> *<sup>σ</sup>n*(*T*))1/2(*<sup>β</sup>*

H1 : *β* = *β*<sup>0</sup> + Δ/

In this case, under H1 the power of the test *πn*(*β*) = IP(*R*(*n*)|H1), where *R*(*n*) = *R*1(*n*) or

*<sup>β</sup>* ).

As pointed out in Subsection 4.2, a fundamental concept resulting from the mathematical theory of the general deterministic epidemic model is the relative removal-rate, see for instance [2] and [3]. This number is defined as *τ* = *γ*/*β*, where *γ* and *β* represent the removal and infection rates, respectively, and it plays a crucial part in determining the probable

is a critical region for the test with an asymptotic level of significance *α*.

**6.4. Relative removal-rate and basic reproduction number**

<sup>√</sup>*n*(*σ*(0) <sup>−</sup> *<sup>σ</sup>*(*T*))1/2(*<sup>β</sup>*

*<sup>n</sup>*(*T*)}*n*∈**<sup>N</sup>** converges in probability to *<sup>β</sup>*0, H0 shall be rejected in favor of H1

*<sup>n</sup>*(*T*) > *un*, being *un* determined by IP(*<sup>β</sup>*

*<sup>n</sup>*(*T*) > *un*|H0) ≤ *<sup>α</sup>* is replaced by the following weaker requirement:

*<sup>n</sup>*(*σ*(0) <sup>−</sup> *<sup>σ</sup>*(*T*)) and therefore, with an asymptotic level of significance *<sup>α</sup>*, a

<sup>√</sup>*n*.

*<sup>n</sup>*(*σ*(0) <sup>−</sup> *<sup>σ</sup>*(*T*)) + <sup>o</sup>(1/*n*). In particular, we can choose

*<sup>n</sup>*(*T*) − *<sup>β</sup>*0)/*β*<sup>0</sup> ≥ *<sup>t</sup>α*}.

*<sup>n</sup>*(*T*) − *<sup>β</sup>*0)/*β*<sup>0</sup> ≥ *<sup>t</sup>α*}

*<sup>n</sup>*(*T*) > *un*|H0) ≤

*<sup>n</sup>*(*T*) <sup>−</sup> *<sup>β</sup>*0)/*β*0}*n*∈**<sup>N</sup>** converges

**6.3. Hypothesis test for the infection rate**

*<sup>n</sup>*(*T*) is too large, i.e. when *<sup>β</sup>*

From the Proposition 6.1, under H0, {

asymptotic level of significance *α*,

variable. Hence, *un* = *β*<sup>0</sup> + *β*0*tα*/

*R*1(*n*) = {

*R*2(*n*) = {

*<sup>R</sup>*(*n*) = *<sup>R</sup>*2(*n*), converges to <sup>Φ</sup>((*σ*(0)−*σ*(*T*))1/2Δ−*tαβ*<sup>0</sup>

Let Δ > 0 and let us consider the alternative hypothesis

critical region for the test is

IP(*β*

rejecting H0 when H0 is true.

For this purpose, IP(*β*

*un* = *β*<sup>0</sup> + *β*0*tα*/

under H0, {*β*

when *β*

$$E\_n = -\frac{\sigma(0)\sqrt{n}}{\gamma\_n^2} (\widehat{\beta^n}(T) - \beta)(\widehat{\gamma^n}(T) - \gamma) + \frac{\sigma(0)^2 \delta\_n \sqrt{n}}{\gamma\_n^3} (\widehat{\gamma^n}(T) - \gamma)^2$$

and (*δn*, *γn*) is a random point between (*σ*(0)*β <sup>n</sup>*(*T*), *<sup>γ</sup> <sup>n</sup>*(*T*)) and (*σ*(0)*β*, *<sup>γ</sup>*).

From Propositions 5.1 and 6.1, {*En*}*n*∈**<sup>N</sup>** converges in probability to zero. Hence, it follows from Proposition 6.1 that *U <sup>n</sup>*(*T*) is asymptotically normal with mean <sup>Δ</sup> and variance *<sup>v</sup>*<sup>2</sup> <sup>0</sup>(*T*). Therefore, this proof is complete.

The constant *un* can be chosen as *un* = 1 + *tαv*0(*T*)/ <sup>√</sup>*<sup>n</sup>* <sup>+</sup> <sup>o</sup>(1/√*n*), where *<sup>t</sup><sup>α</sup>* satisfies 1 − Φ(*tα*) = *α*. In particular, we can choose *un* = 1 + *tαv*0(*T*)/ √*n*, and therefore, with an asymptotic level of significance *α*, a critical region for the test is {*U*�*n*(*T*)/*v*0(*T*) > *tα*}. Note also that, under HΔ, the power of the test *πn*(Δ) = IP(*U*�*n*(*T*)/*v*0(*T*) > *tα*|*H*Δ) converges to *π*(Δ) = Φ(Δ/*v*0(*T*) − *tα*).

*n* [*t*0, *t*1[[*t*1, *t*2[[*t*2, *t*3[[*t*3, *t*4[[*t*4, *t*5[[*t*5, *t*6[[*t*6, *t*7[[*t*7, *t*8[[*t*8, *t*9[[*t*9, *t*10[ SD 30 10.7 8.9 8.3 9.0 7.4 7.1 9.1 8.4 9.1 22.0 43.3 50 10.8 7.6 10.0 9.3 7.8 10.0 10.6 9.1 8.3 16.5 25.4 100 11.0 9.6 10.6 9.3 10.3 10.2 8.3 8.1 10.8 11.8 11.8 300 10.0 10.0 9.3 8.6 8.9 10.3 10.0 9.8 10.8 12.3 10.4 500 9.3 10.8 9.2 9.2 9.9 11.0 9.5 10.2 11.3 9.6 7.9 1000 9.4 9.3 10.0 10.3 10.8 10.1 9.8 9.0 9.9 11.4 7.2 **Table 1.** Percentages of observations of *Xn*(*T*)<sup>2</sup> for the indicated population size, *<sup>β</sup>* <sup>=</sup> 2, *<sup>γ</sup>* <sup>=</sup> 1 and

By carrying out five simulations of a random variable *χ*2(2), 1, 000 times each, the percentages of values that resulted in the corresponding subintervals for each of the five simulations, with the corresponding SD at the last column, are showed in Table 2 below. From Tables 1 and *N*[*t*0, *t*1[[*t*1, *t*2[[*t*2, *t*3[[*t*3, *t*4[[*t*4, *t*5[[*t*5, *t*6[[*t*6, *t*7[[*t*7, *t*8[[*t*8, *t*9[ [*t*9, *t*10[ SD 1 10.3 10.3 11.4 9.3 9.7 7.8 8.8 11.2 11.5 9.6 12.0 2 10.4 9.6 9.8 10.4 10.3 11.4 10.5 8.7 10.2 8.4 9.1 3 10.5 11.3 10.3 8.8 9.7 11.0 11.0 9.4 9.4 8.6 9.6 4 9.7 12.3 8.8 9.5 9.7 9.8 8.9 11.7 9.8 9.8 11.2 5 10.6 10.4 10.3 10.1 9.5 8.1 9.9 10.3 9.7 11.1 8.1

**Table 2.** Percentages of *χ*2(2) observations into each subintervals for 5 series of 1, 000 trials each.

*n βn*(*T*) *γ*

*<sup>n</sup>*(*T*) and *<sup>R</sup>*

simulated values is also quite close to the standard normal density.

*n*

2, we can conclude that, for a population size over 300, the distribution of *Xn*(*T*)<sup>2</sup> is quite approximate to the *χ*2(2) distribution. In Table 3, estimates of *β*, *γ* and *R*<sup>0</sup> have been simulated for different population size. Table 3 corroborates the convergence in probability demands

> 100 2.87 0.82 1.76 200 1.73 1.17 0.74 500 1.82 0.99 0.93 1,000 1.91 0.98 0.99 2,000 1.95 1.01 0.97 5,000 2.01 0.99 1.01 10,000 2.00 1.00 1.01

bigger population size in order to obtain a suitable approximation of the parameters. We appreciate, appropriate estimates for *β*, *γ* and *R*<sup>0</sup> are obtained for population sizes over *n* =

For *β* = 2, *γ* = 1 and *T* = 10, *Xn*(*T*) is simulated 1, 000 times again. Let *Xn*,1(*T*) and *Xn*,2(*T*) the first and second row of *Xn*(*T*), i.e. *Xn*(*T*)=(*Xn*,1(*T*), *Xn*,2(*T*)). In Figure 4 below, histograms of *Xn*,1(*T*) and *Xn*,2(*T*) are showed for *n* = 300. It is observed that their corresponding frequencies of simulated values are quite close to the standard normal density. Let *U<sup>n</sup>*(*T*) and *v*0(*T*) be as in Proposition 6.2 and denote *Xn*,3(*T*) = *U<sup>n</sup>*(*T*)/(*R*0*v*0(*T*)). By observing in Figure 5 the histogram of *Xn*,3(10), we notice the graphical frequency of its

*<sup>n</sup>*(*T*) *<sup>R</sup> n* <sup>0</sup> (*T*) Average SD:10.0

Discrete-Time Stochastic Epidemic Models and Their Statistical Inference 251

<sup>0</sup> (*T*) for the indicated population size, *β* = 2, *γ* = 1 and

*T* = 10.

**Table 3.** Estimates for *βn*(*T*), *γ*

*T* = 10.

5, 000.

**Remark 6.1.** *An application of the preceding proposition is to calculate the approximate power of the test, with a level of significance α, relative to*

$$\mathbf{H}\_0: \mathcal{R}\_0 = 1 \quad \text{against} \quad \mathbf{H}\_1: \mathcal{R}\_0 = r\_{\prime\prime}$$

*where r* > 1*.*

*We interpret* <sup>Δ</sup> *in* <sup>H</sup><sup>Δ</sup> *as* <sup>√</sup>*n*(*<sup>r</sup>* <sup>−</sup> <sup>1</sup>) *and approximate the power of the test by means of <sup>π</sup>* <sup>=</sup> IP(<sup>N</sup> <sup>&</sup>gt; *tα*|*H*Δ)*, where* N *is a normal random variable with mean* <sup>√</sup>*n*(*r*−1) *rv*0(*T*) *and variance* <sup>1</sup>*. Consequently, the power of the test can be approximated by*

$$
\pi = \Phi(\frac{\sqrt{n}(r-1)}{r v\_0(T)} - t\_a).
$$

## **7. Numerical simulations for the general epidemic model**

In order to carry out asymptotical inference for a great number of epidemic models, results stated in Section 5 can be applied. In this subsection, statistical inference for the General Epidemic Model is developed. In this section we maintain notations used in Section 6.

#### **7.1. Validating the population size**

In this subsection, some numerical simulations are carried out in order to validate the appropriate population size under which the results contained in this work are applicable.

Let

$$X\_{\mathfrak{n}}(T) = \begin{pmatrix} \frac{\sqrt{\sigma(0) - \sigma(T)}}{\beta} & 0\\ 0 & \frac{\sqrt{\rho(T) - \rho(0)}}{\gamma} \end{pmatrix} \begin{pmatrix} \widehat{\beta^{\vec{n}}}(T) \\ \widehat{\gamma^{\vec{n}}}(T) \end{pmatrix}.$$

According to Proposition 6.1, the square Euclidean norm of *Xn*(*T*), namely �*Xn*(*T*)�2, has asymptotically *χ*2(2) distribution. The positive part of the real straight line is partitioned in *<sup>m</sup>* subintervals, which are determined by 0 = *<sup>t</sup>*<sup>0</sup> < *<sup>t</sup>*<sup>1</sup> < ··· < *tm*−<sup>1</sup> < <sup>∞</sup>. Let *<sup>F</sup>* denote the *<sup>χ</sup>*2(2) distribution function. For *<sup>β</sup>* <sup>=</sup> 2, *<sup>γ</sup>* <sup>=</sup> 1 and *<sup>T</sup>* <sup>=</sup> 10, �*Xn*(*T*)�<sup>2</sup> is simulated 1, 000 times with *m* = 10, where *t*1,..., *t*<sup>9</sup> have been chosen in such a way that *F*(*ti*) = *i*/10, i.e., *t*<sup>1</sup> = 0.21, *t*<sup>2</sup> = 0.45, *t*<sup>3</sup> = 0.71, *t*<sup>4</sup> = 1.02, *t*<sup>5</sup> = 1.39, *t*<sup>6</sup> = 1.83, *t*<sup>7</sup> = 2.41, *t*<sup>8</sup> = 3.22 and *t*<sup>9</sup> = 4.61. In order to *F*(*t*10) = 1, *t*<sup>10</sup> is defined as ∞. For different population sizes (*n* = 20, 50, 100, 300, 500, 1000), percentages obtained into the corresponding subintervals are presented in Table 1. From Proposition 6.1, better approximations correspond to percentages close to 10%. These closeness are measured by means of the standard deviation (SD) of the observations corresponding to the different population sizes, which are indicated at the last column of Table 1 below.


20 Will-be-set-by-IN-TECH

asymptotic level of significance *α*, a critical region for the test is {*U*�*n*(*T*)/*v*0(*T*) > *tα*}. Note also that, under HΔ, the power of the test *πn*(Δ) = IP(*U*�*n*(*T*)/*v*0(*T*) > *tα*|*H*Δ) converges to

**Remark 6.1.** *An application of the preceding proposition is to calculate the approximate power of the*

H0 : *R*<sup>0</sup> = 1 *against* H1 : *R*<sup>0</sup> = *r*,

*We interpret* <sup>Δ</sup> *in* <sup>H</sup><sup>Δ</sup> *as* <sup>√</sup>*n*(*<sup>r</sup>* <sup>−</sup> <sup>1</sup>) *and approximate the power of the test by means of <sup>π</sup>* <sup>=</sup> IP(<sup>N</sup> <sup>&</sup>gt;

<sup>√</sup>*n*(*<sup>r</sup>* <sup>−</sup> <sup>1</sup>)

In order to carry out asymptotical inference for a great number of epidemic models, results stated in Section 5 can be applied. In this subsection, statistical inference for the General Epidemic Model is developed. In this section we maintain notations used in Section 6.

In this subsection, some numerical simulations are carried out in order to validate the appropriate population size under which the results contained in this work are applicable.

*<sup>β</sup>* 0

According to Proposition 6.1, the square Euclidean norm of *Xn*(*T*), namely �*Xn*(*T*)�2, has asymptotically *χ*2(2) distribution. The positive part of the real straight line is partitioned in *<sup>m</sup>* subintervals, which are determined by 0 = *<sup>t</sup>*<sup>0</sup> < *<sup>t</sup>*<sup>1</sup> < ··· < *tm*−<sup>1</sup> < <sup>∞</sup>. Let *<sup>F</sup>* denote the *<sup>χ</sup>*2(2) distribution function. For *<sup>β</sup>* <sup>=</sup> 2, *<sup>γ</sup>* <sup>=</sup> 1 and *<sup>T</sup>* <sup>=</sup> 10, �*Xn*(*T*)�<sup>2</sup> is simulated 1, 000 times with *m* = 10, where *t*1,..., *t*<sup>9</sup> have been chosen in such a way that *F*(*ti*) = *i*/10, i.e., *t*<sup>1</sup> = 0.21, *t*<sup>2</sup> = 0.45, *t*<sup>3</sup> = 0.71, *t*<sup>4</sup> = 1.02, *t*<sup>5</sup> = 1.39, *t*<sup>6</sup> = 1.83, *t*<sup>7</sup> = 2.41, *t*<sup>8</sup> = 3.22 and *t*<sup>9</sup> = 4.61. In order to *F*(*t*10) = 1, *t*<sup>10</sup> is defined as ∞. For different population sizes (*n* = 20, 50, 100, 300, 500, 1000), percentages obtained into the corresponding subintervals are presented in Table 1. From Proposition 6.1, better approximations correspond to percentages close to 10%. These closeness are measured by means of the standard deviation (SD) of the observations corresponding to the different population sizes, which are indicated at the last

<sup>√</sup>*ρ*(*T*)−*ρ*(0) *γ*

⎞

�*βn*(*T*)

⎞ ⎠ .

*γ* �*n*(*T*)

⎟⎠ ⎛ ⎝

*rv*0(*T*) <sup>−</sup> *<sup>t</sup>α*).

*π* = Φ(

**7. Numerical simulations for the general epidemic model**

<sup>√</sup>*n*(*r*−1)

<sup>√</sup>*<sup>n</sup>* <sup>+</sup> <sup>o</sup>(1/√*n*), where *<sup>t</sup><sup>α</sup>* satisfies

*rv*0(*T*) *and variance* <sup>1</sup>*. Consequently, the*

√*n*, and therefore, with an

The constant *un* can be chosen as *un* = 1 + *tαv*0(*T*)/

*tα*|*H*Δ)*, where* N *is a normal random variable with mean*

*π*(Δ) = Φ(Δ/*v*0(*T*) − *tα*).

*where r* > 1*.*

Let

*test, with a level of significance α, relative to*

*power of the test can be approximated by*

**7.1. Validating the population size**

column of Table 1 below.

*Xn*(*T*) =

⎛

<sup>√</sup>*σ*(0)−*σ*(*T*)

0

⎜⎝

1 − Φ(*tα*) = *α*. In particular, we can choose *un* = 1 + *tαv*0(*T*)/

**Table 1.** Percentages of observations of *Xn*(*T*)<sup>2</sup> for the indicated population size, *<sup>β</sup>* <sup>=</sup> 2, *<sup>γ</sup>* <sup>=</sup> 1 and *T* = 10.

By carrying out five simulations of a random variable *χ*2(2), 1, 000 times each, the percentages of values that resulted in the corresponding subintervals for each of the five simulations, with the corresponding SD at the last column, are showed in Table 2 below. From Tables 1 and


**Table 2.** Percentages of *χ*2(2) observations into each subintervals for 5 series of 1, 000 trials each.

2, we can conclude that, for a population size over 300, the distribution of *Xn*(*T*)<sup>2</sup> is quite approximate to the *χ*2(2) distribution. In Table 3, estimates of *β*, *γ* and *R*<sup>0</sup> have been simulated for different population size. Table 3 corroborates the convergence in probability demands


**Table 3.** Estimates for *βn*(*T*), *γ <sup>n</sup>*(*T*) and *<sup>R</sup> n* <sup>0</sup> (*T*) for the indicated population size, *β* = 2, *γ* = 1 and *T* = 10.

bigger population size in order to obtain a suitable approximation of the parameters. We appreciate, appropriate estimates for *β*, *γ* and *R*<sup>0</sup> are obtained for population sizes over *n* = 5, 000.

For *β* = 2, *γ* = 1 and *T* = 10, *Xn*(*T*) is simulated 1, 000 times again. Let *Xn*,1(*T*) and *Xn*,2(*T*) the first and second row of *Xn*(*T*), i.e. *Xn*(*T*)=(*Xn*,1(*T*), *Xn*,2(*T*)). In Figure 4 below, histograms of *Xn*,1(*T*) and *Xn*,2(*T*) are showed for *n* = 300. It is observed that their corresponding frequencies of simulated values are quite close to the standard normal density. Let *U<sup>n</sup>*(*T*) and *v*0(*T*) be as in Proposition 6.2 and denote *Xn*,3(*T*) = *U<sup>n</sup>*(*T*)/(*R*0*v*0(*T*)). By observing in Figure 5 the histogram of *Xn*,3(10), we notice the graphical frequency of its simulated values is also quite close to the standard normal density.

In Table 4 below, *<sup>M</sup>* <sup>=</sup> 1, 000 simulations of *<sup>π</sup><sup>n</sup>*(Δ) are carried out for a population size equals 300 and for 150 values of Δ between 0 and 14.9, with step sizes of equal length Δ = 0.1. Since Δ .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 0.000 0.001 0.001 0.004 0.002 0.005 0.002 0.007 0.008 0.005 0.004 0.014 0.004 0.021 0.010 0.015 0.015 0.016 0.018 0.020 0.023 0.040 0.032 0.036 0.042 0.047 0.052 0.059 0.063 0.060 0.078 0.075 0.104 0.085 0.120 0.105 0.136 0.140 0.143 0.150 0.151 0.183 0.190 0.202 0.201 0.210 0.247 0.267 0.289 0.282 0.296 0.329 0.343 0.357 0.342 0.376 0.415 0.431 0.427 0.446 0.467 0.463 0.509 0.536 0.524 0.558 0.579 0.570 0.604 0.595 0.632 0.628 0.655 0.663 0.703 0.706 0.723 0.719 0.730 0.775 0.762 0.787 0.765 0.807 0.814 0.798 0.805 0.842 0.845 0.846 0.881 0.854 0.874 0.880 0.890 0.900 0.913 0.901 0.913 0.904 0.911 0.927 0.934 0.937 0.941 0.941 0.944 0.942 0.958 0.957 0.972 0.961 0.969 0.967 0.988 0.973 0.978 0.986 0.988 0.981 0.982 0.987 0.982 0.987 0.988 0.989 0.981 0.995 0.990 0.996 0.991 0.992 0.993 0.998 0.996 0.998 0.998 0.997 0.994 0.997 0.997 1.000 1.000 0.999 0.999 0.999 0.999 1.000 0.998 0.998

Discrete-Time Stochastic Epidemic Models and Their Statistical Inference 253

for <sup>Δ</sup> <sup>=</sup> 0 the power of the test should be close to 0.05, the random function *<sup>π</sup><sup>n</sup>* cannot be considered a good estimator for *πn*, at least for *n* = 300. However, the asymptotic power function *π* seems to give a better approximation for *πn*. The values of *π*(Δ) are given in Table 5 below for 150 values of Δ between 0 and 14.9, with step sizes of equal length Δ = 0.1. In Δ .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 0.050 0.053 0.056 0.059 0.063 0.067 0.071 0.075 0.079 0.083 0.088 0.093 0.098 0.103 0.108 0.114 0.119 0.125 0.132 0.138 0.144 0.151 0.158 0.165 0.173 0.180 0.188 0.196 0.205 0.213 0.222 0.230 0.239 0.249 0.258 0.268 0.277 0.287 0.297 0.307 0.318 0.328 0.339 0.350 0.361 0.372 0.383 0.394 0.405 0.417 0.428 0.440 0.451 0.463 0.475 0.486 0.498 0.510 0.521 0.533 0.545 0.556 0.568 0.579 0.591 0.602 0.613 0.625 0.636 0.647 0.657 0.668 0.679 0.689 0.699 0.709 0.719 0.729 0.739 0.748 0.757 0.766 0.775 0.784 0.792 0.801 0.809 0.817 0.824 0.832 0.839 0.846 0.853 0.860 0.866 0.872 0.878 0.884 0.890 0.895 0.900 0.905 0.910 0.915 0.919 0.923 0.928 0.931 0.935 0.939 0.942 0.946 0.949 0.952 0.955 0.957 0.960 0.962 0.965 0.967 0.969 0.971 0.973 0.974 0.976 0.978 0.979 0.981 0.982 0.983 0.984 0.985 0.986 0.987 0.988 0.989 0.990 0.991 0.991 0.992 0.992 0.993 0.994 0.994 0.994 0.995 0.995 0.996 0.996 0.996

Figure 6 below, a graphical comparison between *<sup>π</sup>* and *<sup>π</sup><sup>n</sup>* is given for a population size equals

**Table 4.** Frequency of times *U<sup>n</sup>*(*T*)/*v*0(*T*) > 1.644853 for different values of Δ.

**Table 5.** Values of *π*(Δ) for different values of Δ.

*n* = 300.

**Figure 4.** Histograms of *Xn*,1(10) and *Xn*,2(10) for a population size equals 300.

**Histogram associated to the**

**Figure 5.** Histograms of *Xn*,3(10) for a population size equals 300.

#### **7.2. The power function of the test for the basic reproduction number**

According to Proposition 6.2, the power function *πn*(Δ) converges to the asymptotic power function (apf) *π*(Δ) = Φ(Δ/*v*0(*T*) − *tα*). The level of significance of the test is assumed to be *<sup>α</sup>* <sup>=</sup> 0.05 and consequently, *<sup>t</sup><sup>α</sup>* <sup>=</sup> 1.644853. Notice *<sup>π</sup>n*(Δ) can be estimated by *<sup>π</sup><sup>n</sup>*(Δ), the frequency H0 is rejected (rf). Hence, *<sup>π</sup><sup>n</sup>*(Δ) is the frequency of times that *<sup>U</sup><sup>n</sup>*(*T*)/*v*0(*T*) <sup>&</sup>gt; 1.644853, under H<sup>Δ</sup> : *R*<sup>0</sup> = 1 + Δ/ √*n*. By the Law of Large Number, a strong consistent estimator of *πn*(Δ) is given by

$$\widehat{\pi}\_{\mathcal{U}}(\Delta) = \frac{1}{M} \sum\_{k=1}^{M} \mathbf{I}\_{\{\widehat{U}\_{n,k}(T) > v\_0(T) > 1.644853\}} $$

where *U<sup>n</sup>*,1(*T*),..., *U<sup>n</sup>*,*M*(*T*) are independent and identically distributed random variables, which have the same distribution as *U<sup>n</sup>*(*T*) under HΔ. Even, it can prove that the convergence of *<sup>π</sup><sup>n</sup>* to *<sup>π</sup>n*, as *<sup>M</sup>* goes to <sup>∞</sup>, is uniform on compact subsets of **<sup>R</sup>**+, I.e., for each *<sup>T</sup>* <sup>&</sup>gt; 0,

$$\lim\_{M \to \infty} \sup\_{0 \le \Delta \le T} |\widehat{\pi}\_{\mathfrak{n}}(\Delta) - \pi\_{\mathfrak{n}}(\Delta)| = 0, \quad a.s.$$

In Table 4 below, *<sup>M</sup>* <sup>=</sup> 1, 000 simulations of *<sup>π</sup><sup>n</sup>*(Δ) are carried out for a population size equals 300 and for 150 values of Δ between 0 and 14.9, with step sizes of equal length Δ = 0.1. Since


**Table 4.** Frequency of times *U<sup>n</sup>*(*T*)/*v*0(*T*) > 1.644853 for different values of Δ.

for <sup>Δ</sup> <sup>=</sup> 0 the power of the test should be close to 0.05, the random function *<sup>π</sup><sup>n</sup>* cannot be considered a good estimator for *πn*, at least for *n* = 300. However, the asymptotic power function *π* seems to give a better approximation for *πn*. The values of *π*(Δ) are given in Table 5 below for 150 values of Δ between 0 and 14.9, with step sizes of equal length Δ = 0.1. In


**Table 5.** Values of *π*(Δ) for different values of Δ.

22 Will-be-set-by-IN-TECH

frequency

0.0 0.1 0.2 0.3 0.4 0.5

**Histogram associated to the Basic Reproduction Number**

standard normal density

**7.2. The power function of the test for the basic reproduction number**

centered estimator of the Basic Reproduction Number for n=300

According to Proposition 6.2, the power function *πn*(Δ) converges to the asymptotic power function (apf) *π*(Δ) = Φ(Δ/*v*0(*T*) − *tα*). The level of significance of the test is assumed to be *<sup>α</sup>* <sup>=</sup> 0.05 and consequently, *<sup>t</sup><sup>α</sup>* <sup>=</sup> 1.644853. Notice *<sup>π</sup>n*(Δ) can be estimated by *<sup>π</sup><sup>n</sup>*(Δ), the frequency H0 is rejected (rf). Hence, *<sup>π</sup><sup>n</sup>*(Δ) is the frequency of times that *<sup>U</sup><sup>n</sup>*(*T*)/*v*0(*T*) <sup>&</sup>gt;

where *U<sup>n</sup>*,1(*T*),..., *U<sup>n</sup>*,*M*(*T*) are independent and identically distributed random variables, which have the same distribution as *U<sup>n</sup>*(*T*) under HΔ. Even, it can prove that the convergence of *<sup>π</sup><sup>n</sup>* to *<sup>π</sup>n*, as *<sup>M</sup>* goes to <sup>∞</sup>, is uniform on compact subsets of **<sup>R</sup>**+, I.e., for each *<sup>T</sup>* <sup>&</sup>gt; 0,

−4 −2 0 2 4

**Histogram associated to the removal rate**

standard normal density

n=300 centered estimator of the removal rate

√*n*. By the Law of Large Number, a strong consistent

{*<sup>U</sup>n*,*k*(*T*)/*v*0(*T*)>1.644853},

<sup>|</sup>*<sup>π</sup><sup>n</sup>*(Δ) <sup>−</sup> *<sup>π</sup>n*(Δ)<sup>|</sup> <sup>=</sup> 0, *<sup>a</sup>*.*s*.

−4 −2 0 2 4

**Histogram associated to the infection rate**

standard normal density

frequency

0.0 0.1 0.2 0.3 0.4 0.5

> n=300 centered estimator of the infection rate

> > frequency

0.0 0.1 0.2 0.3 0.4 0.5

**Figure 5.** Histograms of *Xn*,3(10) for a population size equals 300.

*<sup>π</sup><sup>n</sup>*(Δ) = <sup>1</sup>

lim *<sup>M</sup>*→<sup>∞</sup> sup 0≤Δ≤*T*

*M*

*M* ∑ *k*=1 I

1.644853, under H<sup>Δ</sup> : *R*<sup>0</sup> = 1 + Δ/

estimator of *πn*(Δ) is given by

**Figure 4.** Histograms of *Xn*,1(10) and *Xn*,2(10) for a population size equals 300.

−4 −2 0 2 4

Figure 6 below, a graphical comparison between *<sup>π</sup>* and *<sup>π</sup><sup>n</sup>* is given for a population size equals *n* = 300.

**Asymptotic and Estimated Power Function**

In order to develop statistical inference on the parameters of the model, martingale estimators are proposed. This fact allows to obtain closed form for the estimators. Even, such estimators can be obtained when the distribution of the process governing the epidemic is not completely

Discrete-Time Stochastic Epidemic Models and Their Statistical Inference 255

All the models belong to the class that we are presenting in this chapter assume a stochastic latent period, and consequently, the Reed-Frost model are not included here. As a matter of fact, none of these models could be considered an extension or a particular case of the Reed-Frost model. Moreover, the modeling presented in this chapter need not be Markovian,

This work was partially supported by Instituto de Matemática of the Pontificia Universidad

*Instituto de Matemática, Pontificia Universidad Católica de Valparaíso, Centro de Investigación y*

[1] Andersson, H. & Britton, T. [2000]. *Stochastic Epidemic Models and Their Statistical*

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*Mathematics and Computers in Simulation*, Camberra, Australia, pp. 1767–1773. [10] Buckley, F. & Pollet, P. [2010]. Limit theorems for discrete-time metapopulation models,

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even though some Markovian epidemic models are included in this setting.

Católica de Valparaíso and by FONDECYT under project 1120879.

*Modelamiento de Fenómenos Aleatorios, Universidad de Valparaíso, Chile*

*Analysis. Lecture Notes in Statistics 151*, Springer, New York.

by aids modelling, *Journal of Applied Probability* Vol. 25: 39–62.

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known.

**Acknowledgements**

**Author details**

**9. References**

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*Probability* Vol. 9(No. 3): 519–541.

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New York.

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Raúl Fierro

**Figure 6.** Grafics of *<sup>π</sup>*(Δ) and *<sup>π</sup><sup>n</sup>*(Δ) for a population size equals *<sup>n</sup>* <sup>=</sup> 300.

## **8. Conclusions**

A wide class of discrete-time stochastic epidemic models was introduced and analyzed in this work, the convergence of the stochastic model to the deterministic one as the population increases was proved. Moreover, it was proved the convergence in distribution of a process, depending on the fluctuations between the stochastic and deterministic models, to the solution to a stochastic differential equation. The statistical analysis was focused in proving asymptotic normality of natural martingale estimators, for the parameters of the model, and these results were applied to hypothesis tests for the General Epidemic Model. As a consequence, a hypothesis test for the reproduction number was stated and numerical simulations validated the size populations under which the results are useful.

The stochastic models considered here represent an alternative modeling to those using counting processes and having transitions occurring at random times. However, both are asymptotically consistent with their deterministic counterparts. The advantage of considering stochastic models at discrete times is that it is not necessary to observe the epidemic over a long period of time; however, in order to attain the asymptotical consistency mentioned, our model requires frequent observation when dealing with a large population. Another important conclusion is obtained from Subsection 4.2, where numerical simulations was carried out for the SIS epidemic model. Indeed, the simulation showed large fluctuations of the SIS stochastic epidemic model, for *n* = 10, regarding the deterministic one. However, by simulating the process for *n* = 1, 000 it could be appreciated the trajectories of the deterministic and stochastic models were quite closed. Moreover, Theorem 2 provides confidence bounds which give an insight of the fluctuations of the stochastic model regarding the deterministic one. The simulation in Subsection 4.2 for the SIS epidemic model shows the coherence between these bounds and the simulated process (Figure 3). As a conclusion, the deterministic and stochastic model are quite different for small populations, while for large populations both models perform similarly. However, it is worth noting, although the stochastic and deterministic trajectories are similar for large size populations, an attractor state for the deterministic model is not necessarily an attractor for the stochastic model. As shown in Subsection 4.2, this is the case for the SIS epidemic model.

In order to develop statistical inference on the parameters of the model, martingale estimators are proposed. This fact allows to obtain closed form for the estimators. Even, such estimators can be obtained when the distribution of the process governing the epidemic is not completely known.

All the models belong to the class that we are presenting in this chapter assume a stochastic latent period, and consequently, the Reed-Frost model are not included here. As a matter of fact, none of these models could be considered an extension or a particular case of the Reed-Frost model. Moreover, the modeling presented in this chapter need not be Markovian, even though some Markovian epidemic models are included in this setting.
