Dynamics of a host-pathogen model with constant mortality rate

Qamar Din

doi:10.15388/na.2017.2.3

Abstract

In this paper, we propose a discrete-time host-pathogen model and study its qualitative behavior. The model is for the spread of an infectious disease with constant mortality rate of hosts. Moreover, the time-step is equal to the duration of the infectious phase and the host mortality is taken at some constant rate d >0. This two-dimensional discrete-time epidemic model has complex dynamical behavior. More precisely, we investigate the existence and uniqueness of positive equilibrium point, boundedness character, local and global asymptotic stability of unique positive equilibrium point, and the rate of convergence of positive solutions that converge to unique positive equilibrium point. Numerical simulations are provided to illustrate our theoretical results.

Highlights

Differential and difference equations are used to study a wide range of population models
When the population remains small over a number of generations or remains essentially constant over a generation, it would seem that the dynamics of the population is best described by a discrete-time model [17]
We want to investigate stability analysis of the case where there is host mortality at some constant rate d > 0 and the susceptible dynamics become Sn+1 = (1 − d)Sn + β − In, where 0 < d < 1

Summary

Introduction

Differential and difference equations are used to study a wide range of population models. Most of the SI -type models consist of the mass action principle, i.e., the assumption that new cases arise in a simple proportion to the product of the number of individuals which are susceptible and the number of individuals which are infectious This principle has limited validity, and in discrete models, this principle leads to biologically irrelevant results unless some restrictions are suggested for the parameters. We want to investigate stability analysis of the case where there is host mortality at some constant rate d > 0 and the susceptible dynamics become Sn+1 = (1 − d)Sn + β − In, where 0 < d < 1. For some interesting results related to the qualitative behavior of difference equations, we refer the reader to [3,4,5,6,7,8,9,10,11,12,13,14]

Boundedness

Existence and local stability of positive equilibrium

Global stability analysis

Rate of convergence

Numerical simulations and discussion