ABSTRACT
This chapter describes the transmission dynamics of infectious diseases in communities, regions, and countries that can help us in designing control programs for these diseases. In this chapter, basic terminology that is often used in epidemiology (basic reproduction number, nonlinear incidence, next-generation operator method, and so on) is introduced. The formulation of spatial epidemiology models and the study of the spatial and temporal dynamics of the epidemic compartment models are discussed. Five different types of epidemic models – Susceptible-Infected (SI), Susceptible-Infected-Susceptible (SIS), Susceptible-Infected-Removed (SIR), Susceptible-Infected-Removed-Susceptible (SIRS), and Susceptible-Exposed-Infected-Recovered (SEIR) models – are analyzed in detail. The SI model with nonlinear incidence rate, self-diffusion, and cross-diffusion is analyzed. The influenza epidemic is studied as an SI model. The spatial SIR model with vital dynamics and SIR model with different treatment rates are analyzed with a bilinear incidence rate. With a limited capacity constant treatment, the model exhibits various bifurcations including saddle-node bifurcation, subcritical Hopf bifurcation, and homoclinic bifurcation. The existence of traveling waves in a delayed SIR epidemic model with a nonlinear incidence rate has been discussed. The spatiotemporal complexity of a spatial epidemic model with a nonlinear incidence rate and a distributed contact spatial analog of the basic SIRS endemic model are investigated. A diffusive SEIR epidemic model with varying infectivity and distributed and infinite delay in a bounded domain is investigated for the asymptotic behavior of the positive solution. A diffusive epidemic model (SEIR type) for H1N1 influenza with a variable transmission coefficient is presented. The system supports the existence of sustained and damped oscillations depending on the initial population distributions.
