**1. Introduction**

There is a wide range of models, both stochastic and deterministic, for the spread of an epidemic. Usually, when the population is constituted of a large number of individuals, a deterministic model is useful as a first approximation, and random variations can be neglected. As an alternative, a stochastic model could be more appropriate for describing the epidemic, but it is less tractable and its mathematical analysis is usually possible only when the population size is very small. However, most populations are not large enough to neglect the effect of statistical fluctuations, nor are they small enough to avoid cumbersome mathematical calculations in the stochastic model. In these cases, it uses to be convenient to take into account both types of models and their relationship. The interplay between ordinary differential equations and Markovian counting processes has been widely investigated in the literature. Major references on this subject can be found in [11, 20–22]. Concerning the deterministic epidemic models, those using ordinary differential equations in their formulation have received special attention and a great number of epidemics is modeled by means of Markovian counting processes. For example, some epidemic models known as SIR, SI, SIS, and others derived from these ones, use differential equations and Markovian counting processes in their formulations. Furthermore, stochastic models based on Markovian counting processes and differential equations are mainly used to carry out the statistical analysis of the model parameters. The Mathematical Theory of Infectious Diseases by Bailey [2] represents a classical reference containing a presentation and analysis of these models. However a more recent book by Andersson and Britton [1] entitled Stochastic Epidemic Models and their Statistical Analysis is a more appropriate reference according to the point of view of this chapter. The spread of these epidemics is developed in a closed population, which is divided into three individual compartments, i.e. susceptible, infective and removed cases; different types of transitions can occur among these three groups of individuals. These models include the stochastic and deterministic versions of the Kermack and McKendrick model [19] and the SIS epidemic model, among others. Moreover, a number of variations of these models has been widely studied. Modeling of epidemics by continuous-time Markov chains has a long history; thus, it seems pertinent to cite the works by [4–6, 18, 24, 28].

©2012 Fierro, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Fierro, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 2 Will-be-set-by-IN-TECH 232 Stochastic Modeling and Control Discrete-Time Stochastic Epidemic Models and their Statistical Inference <sup>3</sup>

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, could be carried out.

Results of Section 5 are applied here to some hypothesis tests. Section 7 is devoted to some

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

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

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

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

*<sup>m</sup>*(*t*))� and *A* is a 3 × *m*-incidence matrix.

*n*

*<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

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

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

*n*

*<sup>k</sup>* )) generated by *<sup>Z</sup>n*(*<sup>t</sup>*

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

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

*<sup>n</sup>*(*t*), *ρn*(*t*))�, *σn*(*t*) = *Sn*(*t*)/*n*, *ι*

*<sup>m</sup>*, and the number of individuals in each compartment is obtained, for

*<sup>m</sup>* take values in the set of non-negative integer numbers, have

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

*<sup>k</sup>*−1)), (*<sup>i</sup>* ∈ {1, . . . , *<sup>m</sup>*}),

*<sup>k</sup>* = *k*Δ/*n*, (Δ > 0), i.e. each time subinterval has length

Discrete-Time Stochastic Epidemic Models and Their Statistical Inference 233

*Xn*(*t*) = *Xn*(0) + *AZn*(*t*), (1)

<sup>1</sup> (0) = ··· <sup>=</sup> *<sup>Z</sup><sup>n</sup>*

*n*

*<sup>m</sup>*(0) = 0. Let <sup>F</sup>*<sup>n</sup>*

<sup>3</sup> : *<sup>u</sup>* <sup>+</sup> *<sup>v</sup>* <sup>+</sup> *<sup>w</sup>* <sup>=</sup> <sup>1</sup>} and it

*<sup>k</sup>* ). The stochastic processes

*<sup>k</sup>*−1−conditionally independent and satisfy

*<sup>n</sup>*(*t*) = *In*(*t*)/*n* and *ρn*(*t*) = *Rn*(*t*)/*n*.

*<sup>k</sup>* be

numerical simulations. Finally, Section 8 contains some conclusions.

*n*

**2. Modeling and preliminaries**

*<sup>k</sup>* }*k*∈**N**, where for each *<sup>k</sup>* ∈ **<sup>N</sup>**, *<sup>t</sup>*

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

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

*<sup>k</sup>*−1).

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

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

right-continuous trajectories and start at zero, i.e. *Z<sup>n</sup>*

1 (*t n*

IE(Δ*Z<sup>n</sup> <sup>i</sup>* (*t n <sup>k</sup>* )|F*<sup>n</sup>*

*n*

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>k</sup>* ),..., <sup>Δ</sup>*Z<sup>n</sup>*

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

is assumed the following condition holds:

*n*

**2.1. Discrete-time modeling**

individuals.

{*tn*

Δ*Z*(*t<sup>n</sup>*

*Zn* <sup>1</sup> ,..., *<sup>Z</sup><sup>n</sup>*

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

each *t* ≥ 0, by means of

where *Zn*(*t*)=(*Z<sup>n</sup>*

It is assumed *Z<sup>n</sup>*

the *σ*-field *σ*(*Zn*(*t*

(C) For each *<sup>k</sup>* <sup>∈</sup> **<sup>N</sup>**, <sup>Δ</sup>*Z<sup>n</sup>*

where *χn*(*t*)=(*σn*(*t*), *ι*

our setting.

processes *Z<sup>n</sup>*

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 normality is proved.

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 be obtained in a closed form.

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. Results of Section 5 are applied here to some hypothesis tests. Section 7 is devoted to some numerical simulations. Finally, Section 8 contains some conclusions.
