MATHEMATICAL MODEL FOR CHOLERA DYNAMICS WITH VACCINATION AND TEMPERATURE DEPENDENT PARAMETER

Abstract

Cholera is an acute intestinal infection caused by ingesting Vibrio cholerae. It is severely contagious acute bacterial infection which is caused by colonization and multiplication of Vibrio cholerae inside the small intestine. When ingested, the binding sub units of the enterotoxin combine extraordinarily rapidly to the molecules in the cell wall of the intestine. The binding then becomes irreversible within minutes since they become incorporated into the cell membrane and eventu ally results into modification of the binding protein which leads to rapid excretion of electrolytes into the small intestine. Health providers have put several control measures into place such as provision of clean water, improved food safety, good sanitation and more so availability and access of vaccine which is administered to stimulate the immune system to produce antibodies for protection. Despite the use of the vaccine, cholera epidemics are still very frequent. Most of the cholera vaccine mathematical models so far done are between-host models. Mathematical modelling of the dynamics of Vibrio cholerae within the human host has largely remained unexplored yet it is the interaction between the bacteria and the cells of the small intestine that play a major role in the dynamics of the cholera disease. In this research, a within-host cholera mathematical model has been developed using a system of ODEs incorporating vaccine efficacy. The solutions of the model have been shown to be well-posed. The vaccine R0V has been done using the next gener ation matrix approach. Analysis of the model shows that IFE point is both locally and globally asymptotically stable when R0V < 1 and IE point is locally asymp totically stable when R0V > 1. Further analysis shows the existence of backward bifurcation. To highlight the relevance of vaccine efficacy, numerical simulation of the model with respect to vaccination is carried out and shows that when the vaccine efficacy γ is high, there will be lower infection rate of cells. Due to the ex istence of backward bifurcation in the within-host cholera model with vaccination, there is shedding out V ibrio cholerae to the environment whose multiplication is affected by changes in temperatures, it is for this reason that a between-host mathematical cholera model with temperature dependent parameter is formulated to investigate the effect of temperature change on the dynamics of cholera dis ease. Analysis of the model shows that DFE point is both locally and globally asymptotically stable when R0 < 1 and EE point is locally asymptotically stable when R0 > 1. Sensitivity analysis shows that increasing the temperature of the environment would help reduce the infection rate of the pathogen thus reduce the reproduction number R0. Numerical simulation of the model has been carried out with respect to temperature change and the analysis shows that cholera pathogens can multiply and spread faster under temperatures of 230C but between the tem perature range of 230C < T ≤ 430C their multiplication and spread is lowered. Finally this research adopts the use of Caputo fractional order time derivative to analyse the dynamics of between-host cholera model and the results compared to that obtained from the ordinary order derivatives and it shows that the fractional order simulated results gave a better understanding of how temperature of the environment is key to the control of the disease.