**1. Introduction**

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Throughout human history, infectious diseases have caused debilitation and premature death to large portions of the human population, leading to serious social-economic concerns. Many factors have contributed to the persistence and increase in the occurrence of infectious disease (demographic factors, political, social and economic changes, environmental change, public health care and infrastructure, microbial adaptation, etc.), which according to the World Health Organization (WHO), are the second leading cause of death globally (≈ 23 % of deaths) after cardiovascular diseases (WHO, 2010).

Research on basic and applied aspects of host, pathogen, and environmental factors that influence disease emergence, transmission and spread have been supported so far, and the development of diagnostics, vaccines, and therapeutics has been greatly increased. In recent years, mathematical modeling became an interesting tool for the understanding of infectious diseases epidemiology and dynamics, leading to great advances in providing tools for identifying possible approaches to control, including vaccination programs, and for assessing the potential impact of different intervention measures.

Epidemiological models are a formal framework to convey ideas about the components of a host-parasite interaction and can act as a tool to predict, understand and develop strategies to control the spread of infectious diseases by helping to understand the behaviour of the system under various conditions. They can also aid data collection, data and interpretation and parameter estimation. The purpose of epidemiological models is to take different aspects of the disease as inputs and to make predictions about the numbers of infected and susceptible people over time as output.

In the early 20*th* century, mathematical models were introduced into infectious disease epidemiology, and a series of deterministic compartment models such as SI (susceptible-infected), SIS (susceptible-infected-susceptible), and e.g SIR (susceptibleinfected-recovered) have been proposed based on the flow patterns between compartments of hosts. In our days, most of the models developed try to incorporate other factors focusing on several different aspects of the disease, which can imply rich dynamic behaviour even in the most basic dynamical models. Factors that can go into the models include the duration

susceptibles *S*, infected *I* and recovered *R*,

again after waning immunity rate *α*.

(ODE) reads,

*<sup>S</sup>* <sup>+</sup> *<sup>I</sup> <sup>β</sup>*

*<sup>S</sup>*˙ <sup>=</sup> <sup>−</sup> *<sup>β</sup>*

individuals in a given population *N*, when solving the above ODE system.

interpretation for the class abundances in percentages.

calculated setting the rates of change *S*˙ and ˙

the solution is given by

S,I,R (t)

˙ *<sup>I</sup>* <sup>=</sup> *<sup>β</sup>*

*<sup>I</sup> <sup>γ</sup>* −→ *R <sup>R</sup> <sup>α</sup>* −→ *<sup>S</sup>*

Modeling Infectious Diseases Dynamics: Dengue Fever, a Case Study 231

for a host population of *N* individuals, with contact and infection rate *β*, recovery rate *γ* and waning immunity rate *α*. The dynamic model in terms of ordinary differential equations

where we use the time derivative *S*˙ = *dS*/*dt* with time *t* for a constant population size of *N* = *S* + *I* + *R* individuals. The solution of *R*(*t*) is given by *R*(*t*) = *N* − *I*(*t*) − *S*(*t*) which can be calculated using the solution of the ODEs. The susceptible individuals become infected with infection rate *β*, recover from the infection with recovery rate *γ* and become susceptible

In Fig. 1 we show the dynamical behavior of the susceptible, infected and recovered

0 10 20 30 40 50

t

Fig. 1. Time dependent solution simulation for the SIR epidemic model. With a population *N* = 100, and starting values *I* = 40, *S* = 60 and *R* = 0, we fixed *β* = 2.5, *α* = 0.1, and *γ* = 1. In green the dynamics for the susceptibles *S*(*t*), in pink the dynamics for the infected *I*(*t*)

The basic SIR model has only fixed points as possible stationary solutions, that can be

*I* ∗

*S*∗

and in blue the dynamics of the recovered *R*(*t*). Note that *N* = 100 allows for the

−→ *I* + *I*

*<sup>N</sup> IS* <sup>+</sup> *<sup>α</sup>*(*<sup>N</sup>* <sup>−</sup> *<sup>S</sup>* <sup>−</sup> *<sup>I</sup>*) (1)

*<sup>N</sup> IS* <sup>−</sup> *<sup>γ</sup><sup>I</sup>* , (2)

*I* to zero. For the disease free equilibrium state,

<sup>1</sup> = 0 (3)

<sup>1</sup> = *N* (4)

 infected susceptible recovered

of disease, the duration of infectivity, the infection rate, the waning immunity, and so forth. In such a way, differential equation models are a simplified representation of reality, which are designed to facilitate prediction and calculation of rates of change as functions of the conditions or the components of the system.

There are two common approaches in modeling, the deterministic and the stochastic one. In the first case, the model is one in which the variable states are uniquely determined by parameters in the model and by sets of previous states of these variables. In mathematics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. In a stochastic model, randomness is present, and variable states are not described by unique values, but rather by probability distributions. Stochastic epidemic models are appropriate stochastic processes that can be used to model disease propagation. Disease propagation is an inherently stochastic phenomenon and there are a number of reasons why one should use stochastic models to capture the transmission process. Real life epidemics, in the absence of intervention from outside, can either go extinct with a limited number of individuals getting ultimately infected, or end up with a significant proportion of the population having contracted the disease in question. It is only stochastic, as opposed to deterministic, models that can capture this behavior and the probability of each event taking place.

Only few stochastic processes can be solved explicitly. The simplest and most thoroughly studied stochastic model of epidemics are based on the assumption of homogeneous mixing, i.e. individuals interact randomly at a certain rate. The mean field approximation is a good approximation to be used in order to understand better the behavior of the stochastic systems in certain parameter regions, where the dynamics of the mean quantities are approximated by neglecting correlations, giving closed ordinary differential equations (ODE) systems, hence mathematically deterministic systems which are easier to analyze.

In the following section of this chapter we present the properties of the basic SIR epidemic model for infectious diseases with a summary of the analysis of the dynamics, identifying the thresholds and equilibrium points. The goal is to introduce notation, terminology, and results that will be generalized in later sections on more advanced models motivated by dengue fever epidemiology as an example of multi-strain systems.
