Formulation of a Vector SIS Malaria Model in a Patchy Environment with Two Age Classes

Josephine Wairimu,Wandera Ogana,Sallet Gauthier

doi:10.4236/am.2014.510147

Abstract

We formulate an SIS model describing transmission of highland malaria in Western Kenya. The host population is classified as children, age 1- 5 years and adults, above 5 years. The susceptibility and infectivity of an individual depend on age class and residence. The large scale system with 6n equations is reduced into a compact form of 3n equations by a change of variables. Then 3n equations are vectorialized using the matrix theory to get a one dimension, compact form of the system, equation in . Using Vidyasagar theorem [1], the graph of the reduced system is shown to be strongly connected and the system is a monotone dynamical system. This means that circulation of malaria parasites among the species and among the patches is strongly connected, hence transmission is sustained. We show that for then-dimensional age structured system the positive orthant is positively invariant for all positive values of the variables.

Highlights

In Kenya, malaria is the leading cause of morbidity and mortality
Motivated by the work of Auger [10] and Pongsumpun [8], we formulate an age structured model of malaria with susceptibility and infectivity depending on residence patch
We formulated an infinite model representing the spread of malaria in a heterogenous population classified

Summary

Introduction

In Kenya, malaria is the leading cause of morbidity and mortality. It accounts for 30% of all outpatient attendances and 19% of all admissions to health facilities. The U-shaped valleys are broad and with slow moving rivers with poor drainage This favor mosquitoes and high malaria infection and incidence. Children between 1 and 5 years of age are infected [2] They are not bitten in same way as adults, [3]. Auger [10] modified Ross [11] model to n patches without vectors migration. They model assumed that susceptibility and infectivity are similar in all patches. Motivated by the work of Auger [10] and Pongsumpun [8], we formulate an age structured model of malaria with susceptibility and infectivity depending on residence patch. Mosquitoes fly between patches less than 2 km apart, the approximate distance a mosquito can travel, Lutambi [12] and Kelly [13]

The Model

Reduced System

Basic Properties of the Model

Findings

Conclusions