Stability and optimal control of a disease model with vertical transmission and saturated incidence

Abstract

This paper studies a nonlinear SI-model featuring vertical transmission and saturated incidence terms. The model is analyzed by utilizing some dynamical system tools to assess the dynamic behavior of solutions to a system of first order nonlinear ordinary differential equations arising from the spread of a vertically transmitted disease in a population. The mathematical well-posedness of the governing model is proven through positivity and boundedness properties of solutions. Lyapunov functions with LaSalle’s invariance principle are used to establish the global asymptotic dynamics of the system about the disease-free and endemic steady states. Additionally, sensitivity analysis is conducted to show how the epidemiological threshold (basic reproduction number) of the model is affected by variations in parameters. The study is extended to explore the significance of a time-dependent optimal control for blocking vertical transmission route using Pontryagin’s maximum principle of the optimal control theory. The obtained qualitative results are supplemented by numerical simulations. It is shown that incorporation of the optimal control helps in preventing vertical transmission of infection, and in turn diminishes the spread of the disease in the population.