Mathematical modeling and Analysis of Leptospirosis With Optimal Control

Abstract

Leptospirosis is an emerging zoonotic disease with high health and economic damage that is caused by Leptospiara bacteria. In this dissertation, we developed a deterministic mathe matical model that describes the dynamics of leptospirosis transmission in cattle herds, rats, and humans using both integer and non-integer systems of differential equations. The study examined the role of asymptomatic cattle and rats in the transmission of animal and human leptospirosis. For this, we formulated three models: cattle-environment, cattle-rat-environment, and cattle-human-environment models based on the epidemiological behavior of the Leptospira pathogen. For each model, we proved the well-posedness and also found the basic/effective re production numbers using the next-generation matrix approach. Analytically, we have shown the stability of Leptospira-free and Leptospira-persistent equilibrium points for the threshold value less than unity and greater than unity. Sensitivity analysis is also done to identify the most sensitive parameters in the proposed models. Accordingly, the contact rate between cattle, the contact rate between cattle and humans, the vaccination rate of cattle, the recovery rate of cattle, the recovery rate of humans, the rat mortality rate, and the environmental Leptospira decay rate are the most sensitive parameters. Furthermore, the first model is extended to an op timal control problem to optimally manage the spread of disease and determine the cost-efficient strategy. Numerical simulations have been conducted for each model using the Python language (odeint solver, scipy.optimize.fSolve, and GEKKO optimization packages). Accordingly, lower ing the rate of contact with asymptomatic infected cattle, promoting the rate of recovery for both asymptomatic infected cattle and humans, increasing the rate of vaccination among susceptible cattle, raising the rate of rat mortality, and maximizing the rate of Leptospira decay could all contribute to a significant drop in the prevalence of the disease. Findings of this dissertation underscore the remarkable importance of targeted interventions, such as maximizing prevention efforts, enhancing regular vaccination of cattle, pre-diagnosis treatment of cattle and humans using proper antibiotics, and implementing environmental sanitation programs, which are very useful to manage leptospirosis. In addition, the consideration of different fractional orders in the last model provides valuable insights into the model’s behavior, highlighting its variability and sensitivity to changes in the ordering.