Modelling and analysis of HIV‐TB co‐infection in a variable size population

Abstract

In this paper, a nonlinear mathematical model is proposed for the transmission dynamics of HIV and a curable TB pathogen within a population of varying size. In the model, we have divided the population into four sub classes of susceptibles, TB infectives, HIV infectives and that of AIDS patients. The model exhibits four equillibria namely, a disease free, HIV free, TB free and a co‐infection equilibrium. The model has been studied qualitatively using stability theory of nonlinear differential equations. It is shown that the positive co‐infection equilibrium is always locally stable but it may become globally stable under certain conditions showing that the disease becomes endemic due to constant migration of the population into the habitat. A numerical study of the model is also performed to investigate the influence of certain key parameters on the spread of the disease.

Šiame darbe pateikiamas netiesinis matematinis modelis, skirtas aprašyti ŽIV ir išgydomo TB patogeno plitimui kintamo dydžio populiacijoje. Modelyje populiacija dalinama i keturias klases – galintys užsikresti, TB infekuoti, ŽIV infekuoti ir AIDS pacientai. Modelis turi keturias pusiausvyros padetis: nesergantys, nesergantys ŽIV, nesergantys TB ir sergantys abiem ligom. Modelis analizuojamas kokybiniu požiūriu, naudojant netiesiniu diferencialiniu lygčiu stabilumo teorija. Irodyta, kad teigiama dvieju infekciju pusiausvyros padetis visada yra lokaliai stabili, be to, esant tam tikroms salygoms, ta padetis taip pat būna ir globaliai stabili. Tai reiškia, kad esant pastoviai migracijai i areala, liga tampa endemine. Atlikta modelio skaitine analize, skirta nagrineti kai kuriu svarbiausiu parametru itaka AIDS ir TB ligu plitimui.

Key words:

• HIV/AIDS epidemic
• TB pathogen
• stability
• region of attraction
• equilibrium points