Computation of the basic reproduction numbers for reaction-diffusion epidemic models

We consider a class of $k$-dimensional reaction-diffusion epidemic models ($k = 1, 2, \cdots$) that are developed from autonomous ODE systems. We present a computational approach for the calculation and analysis of their basic reproduction numbers. Particularly, we apply matrix theory to study the relationship between the basic reproduction numbers of the PDE models and those of their underlying ODE models. We show that the basic reproduction numbers are the same for these PDE models and their associated ODE models in several important scenarios. We additionally provide two numerical examples to verify our analytical results

A Mathematical Model for the Spatial Spread of HIV in a Heterogeneous Population

An important factor in the dynamic transmission of HIV is the spatio-temporal mobility of the host population. One key challenge in HIV epidemiology therefore, is determining how the spatial structure of the host population influences disease transmission. The aim of this paper is to study how the movement of individuals impacts the spatial spread of HIV. We constructed a deterministic reaction-diffusion equation model for the spread of HIV in a heterosexually mobile population, under the assumption of a varying population size to study the dynamics of HIV spread in a spatially structured population and obtained the minimal wave speed. Then we considered the existence of traveling waves and the influences of parameters on HIV prevalence and the minimal wave speed. Numerical simulation showed that a stationary labyrinthine pattern emerges in the distribution of the infection population density in the two high-risk groups as a result of diffusion. Keywords: Spatial distribution, HIV, reaction-diffusion, epidemiology

Selected topics on reaction-diffusion-advection models from spatial ecology

We discuss the effects of movement and spatial heterogeneity on population dynamics via reaction-diffusion-advection models, focusing on the persistence, competition, and evolution of organisms in spatially heterogeneous environments. Topics include Lokta-Volterra competition models, river models, evolution of biased movement, phytoplankton growth, and spatial spread of epidemic disease. Open problems and conjectures are presented

On a Vector-host Epidemic Model with Spatial Structure

In this paper, we study a reaction-diffusion vector-host epidemic model. We define the basic reproduction number $R_0$ and show that $R_0$ is a threshold parameter: if $R_0\le 1$ the disease free steady state is globally stable; if $R_0>1$ the model has a unique globally stable positive steady state. We then write $R_0$ as the spectral radius of the product of one multiplicative operator $R(x)$ and two compact operators with spectral radius equalling one. Here $R(x)$ corresponds to the basic reproduction number of the model without diffusion and is thus called local basic reproduction number. We study the relationship between $R_0$ and $R(x)$ as the diffusion rates vary