Integrating Disease Dynamics and Environmental Sanitation: A Mathematical Model of Tuberculosis Spread in Communities with Compromised Water and Waste Management

Department of Matematics, Faculty of Science and Mathematics, Diponegoro University, Semarang, Indonesia

^* Corresponding author: widowati@lecturer.undip.ac.id

Abstract

Tuberculosis (TB) remains a global health challenge, particularly in areas with inadequate environmental sanitation. Poor water quality and ineffective waste management can indirectly exacerbate TB transmission by weakening public health resilience, increasing vulnerability to infection, and complicating recovery. This study develops a compartmental mathematical model that integrates TB disease dynamics with environmental sanitation factors, specifically focusing on communities facing disrupted water supply and waste disposal systems. TB is a disease caused by infection with the bacterium Mycobacterium tuberculosis. The developed mathematical model of TB spread is a system of nonlinear differential equations with six variables: susceptible, vaccinated, latent, infected, drug-resistant, and cured, while incorporating sanitation-related parameters that influence susceptibility and recovery levels. The basic reproduction number (R0) is then derived as a function of epidemiological and sanitation parameters, using a next-generation matrix. Improved sanitation results in a smaller R0. Furthermore, local stability analysis around non-endemic and endemic equilibrium points is performed using the Routh-Hurwitz method. If R0 is less than one, the non-endemic equilibrium is asymptotically stable, indicating that TB will disappear from the community over time. If R0 is greater than one, the endemic equilibrium is asymptotically stable, indicating that TB will spread throughout the community over time. Simulation results show that if parameters related to sanitation are improved, the number of individuals recovering will increase. These results highlight the link between public health and the environment, which policymakers can use to evaluate interventions to reduce TB transmission in the community.