Tuberculosis (TB) spreads through contact between a susceptible person and smear positive pulmonary TB case (TPM+). The spread of TB is highly dependent on people migration between cities or regions that may have different contact rates and different environmental parameters, leading to different disease spread speed in the population. In this work, a metapopulation model, i.e., networks of populations connected by migratory flows, which overcomes the assumption of homogeneous mixing between different regions was constructed. The TB model was combined to a simple demographic structure for the population living in a multi-patch environment (cities, towns, regions or countries). The model consist of a system of differential equations coupling TB epidemic at different strength and mobility between the patches. Constant recruitment rate, slow and fast progression to the disease, effective chemoprophylaxis, diagnostic and treatment are taken into account to make the model including the reality of people in the sub-Saharan African countries. The basic reproduction number (\(\mathcal {R}_0\)) was computed and it was demonstrated that the disease-free equilibrium is globally asymptotically stable if \(\mathcal {R}_0 < 1\). When \(\mathcal {R}_0 > 1\), the disease-free equilibrium is unstable and there exists one endemic equilibrium. Moreover, the impact of increasing migration rate between patches on the TB spread was quantified using numerical implementation of the model. Using an example on 15 inter-connected patches on the same road, we demonstrated that most people was most likely to get infected if the disease starts in a patch in the middle than in border patches.

A deterministic model of tuberculosis (TB) in sub-Saharan Africa including undetected and lost-sight cases is presented and analyzed. The model is shown to exhibit the phenomenon of backward bifurcation, when a stable disease-free equilibrium co-exists with one or more stable endemic equilibrium points when the associated basic reproduction number (R-0) is less than unity. Analyzing the model obviously reveals that exogenous reinfection plays a key role on the existence of backward bifurcation. However, an analysis of the ranges of exogenous reinfection suggested that backward bifurcation occurs only for very high and unrealistic ranges of the exogenous reinfection rate. Random perturbation of reinfection rates was performed to gain insight into the role of this latter on the stability of the disease free equilibrium. (C) 2016 Published by Elsevier B.V.

Tuberculosis (TB) is a common lethal infectious disease usually caused by Mycobacterium tuberculosis. According to the WHO, TB to date, claims the second largest number of victims due to a single infectious agent right after HIV/AIDS. Although a widespread implementation of control measures focus on case finding and short-course chemotherapy, the global burden of TB has increased over the past two decades. A deterministic TB model with a demographic structure for the population living in a multi-patch environment (cities, towns, regions or countries, etc.) is formulated. The basic reproduction number R-0 is computed. A sensitivity analysis of the model parameters is performed to estimate the most influential parameters in TB dynamics. Using the iterative Gauss-Newton method to solve the inverse problem of parameter identification, estimability of parameters have been studied, and estimable unknown parameters have been computed using real data of TB in Cameroon, subdivided into four regions. Numerical simulations showed the model to reproduce the TB dynamics in Cameroon and predict a short term increase in the number of TB active cases over next years.

This paper considers the optimal control of tuberculosis through education, diagnosis campaign and chemoprophylaxis of latently infected. A mathematical model which includes important components such as undiagnosed infectious, diagnosed infectious, latently infected and lost-sight infectious is formulated. The model combines a frequency dependent and a density dependent force of infection for TB transmission. Through optimal control theory and numerical simulations, a cost-effective balance of two different intervention methods is obtained. Seeking to minimize the amount of money the government spends when tuberculosis remain endemic in the Cameroonian population, Pontryagin's maximum principle is used to characterize the optimal control. The optimality system is derived and solved numerically using the forward-backward sweep method (FBSM). Results provide a framework for designing cost-effective strategies for diseases with multiple intervention methods. It comes out that combining chemoprophylaxis and education, the burden of TB can be reduced by 80% in 10 years. (C) 2014 Elsevier B.V. All rights reserved.

Drinking alcohol cause the liver to become even more damaged in people infected with the hepatitis B virus (HBV). We present a deterministic model for HBV in a community in order to determine the effects of alcohol in the spread of the disease. The important mathematical features of the HBV model are thoroughly investigated. The epidemic threshold known as the basic reproduction number and equilibria for the model are determined and stabilities analyzed. The model is numerically analyzed to assess the effects of alcohol drinking on the transmission dynamics of HBV. *Presene address: UMI209IRD/UPMC UMMISCO, Bondy ,Projet MASAIE INRIA Grand Est, France and Projet GRIMCAPE, LIRIMA, Cameroon. Correspondence/Reprint request: Dr. S. Bowong, Laboratory of Applied Mathematics, Department of Mathematics and Computer Science, Faculty of Science, University of Douala, PO Box 24157, Douala Cameroon. E-mail: sbowong@gmail.com

Tuberculosis (TB) remains a major global health problem. A possible risk factor for TB is diabetes (DM), which is predicted to increase dramatically over the next two decades, particularly in low and middle income countries, where TB is widespread. This study aimed to assess the strength of the association between TB and DM. We present a deterministic model for TB in a community in order to determine the impact of DM in the spread of the disease. The important mathematical features of the TB model are thoroughly investigated. The epidemic threshold known as the basic reproduction number and equilibria for the model are determined and stabilities analyzed. The model is numerically analyzed to assess the impact of DM on the transmission dynamics of TB. We perform sensitivity analysis on the key parameters that drive the disease dynamics in order to determine their relative importance to disease transmission and prevalence. Numerical simulations suggest that DM enhances the TB transmission and progression to active TB in a community. The results suggest that there is a need for increased attention to intervention strategies such as the chemoprophylaxis of TB latent individuals and treatment of active TB in people with DM, which may include testing for suspected diabetes, improved glucose control, and increased clinical and therapeutic monitoring in order to reduce the burden of the disease.

Infection with the hepatitis C virus (HCV) is the most common coinfection in people with the human immunodeficiency virus (HIV), and hepatitis C is categorized as an HIV-related illness. The study of the joint dynamics of HIV and HCV present formidable mathematical challenges in spite the fact that they share similar routes of transmission. A deterministic model for the co-interaction of HCV and HIV in a community is presented and rigorously analyzed. The disease-free equilibrium is shown to be locally asymptotically stable when the associated epidemic threshold known as the basic reproduction number for the model is less than the unity. The Centre Manifold theory is used to show that the HCV only and HIV/AIDS only endemic equilibria are locally asymptotically stable when their associated reproduction numbers are greater than the unity. We compute two coexistence thresholds for the stability of boundary equilibria. Numerical results are presented to validate analytical results.