Infectious Disease Modeling

Necessary mathematical definitions and concepts are presented in this chapter. Fundamental theory of ordinary differential equations is given first, followed by stability theory (including the notion of partial stability). The theory of delay differential equations, impulsive systems, and stochastic differential equations are also highlighted, with an emphasis placed upon stability notions and methods.

This chapter reviews the theory of switched systems.Governed by a combination of mode-dependent continuous/discrete dynamics and logic-based switching, and having a wide range of motivating applications, the qualitative behavior of switched systems is highlighted here. Stability theory is emphasized; topics of discussion include stability under arbitrary and constrained switching, as well as switching control.

The modeling of epidemics by hybrid and switched systems is introduced and analyzed. To begin, the classical SIR model is derived and its defining features are detailed. Motivated by variations in the contact rate between members of the population, a switched SIR model is formulated. The flexibility of the switched systems framework and its accompanying theory is highlighted by relaxing some of the population demographics and epidemiological assumptions. A switching incidence rate function form is considered to model abrupt changes in population behavior. The incorporation of stochastic perturbations into the model is also investigated. The findings here focus on the qualitative behavior of the models (i.e., stability theory). More specifically, global attractivity and partial stability are demonstrated, as well as persistence of the disease.

In this chapter, the methods developed thus far are applied to a variety of infectious disease models with different physiological and epidemiological assumptions. Many of the previous results are immediately applicable, thanks to the flexibility of the simple techniques used here. However, some complicating modeling assumptions lead to a need for different switched systems techniques not present in the previous chapter. First, the so-called SIS model is considered, followed by incorporation of media coverage, network epidemic models with interconnected cities (or patches), and diseases spread by vector agents (e.g., mosquitoes) which are modeled using time delays. Straightforward extensions of eradication results are given for models with vertical transmission, disease-induced mortality, waning immunity, passive immunity, and a model with general compartments.

This chapter is motivated by the application of control strategies to eradicate epidemics. In part, the previous switched epidemic models are reintroduced with continuous (e.g., vaccination of newborns continuously in time) or switching control (i.e., piecewise continuous application of vaccination or treatment schemes) for evaluation and optimization. As discussed earlier, infectious disease models are a crucial component in designing and implementing detection, prevention, and control programs (e.g., the World Health Organization’s program against smallpox, leading to its global eradication by 1977). The switched SIR model is first returned to for consideration and analysis of vaccination of the susceptible group (e.g., newborns or the entire cohort). Subsequently, the developed theoretical methods are applied to the switched SIR model with a treatment program in effect. Common Lyapunov functions are used to provide controlled eradication of diseases modeled by the so-called SEIR (Susceptible-Exposed-Infected-Recovered) model with seasonal variations captured by switching. A screening process, where traveling individuals are examined for infection, is proposed and studied for the switched multi-city model of the previous chapter. Switching control of diseases such as Dengue and Chikungunya which are spread via mosquito–human interactions, is investigated.

Building upon the previous chapter, impulsive control in epidemic models is formulated and analyzed in this part. Pulse vaccination, which is the control technique of applying vaccinations to a portion of the susceptible population in a relatively short time period (with respect to the dynamics of the disease) is considered. This is applied to the switched SIR model previously set forth in this monograph, along with pulse treatment strategies. Complications such as general switched incidence rates, vaccine failures, media coverage, and traveling individuals are considered. Conditions are found which guarantee eradication under the pulse schemes and an evaluation and comparison of control strategies (switching and impulsive) is performed in the context of a general vector-borne disease model.

This chapter is devoted to analyzing the spread of the Chikungunya virus, a vector-borne disease, modeled here according to interactions between human and mosquito populations. After introducing the full model, the remainder of this chapter focuses on studying the efficacy of different control strategies. Once the model is formulated and analyzed, a case study of the 2005–06 Chikungunya outbreak in Réunion is completed. Here, the mosquito birth rate is modeled as a time-varying switching parameter, to incorporate differences between the rainy season and dry season. Variations in the contact rate between mosquitoes and humans are also considered. Control strategies are analyzed and evaluated for comparison (e.g., destruction of breeding sites, reduced contact rates).

Concluding remarks are presented with a summary of contributions. Future work is also highlighted.