Mathematical modeling and projections of a vector-borne disease with optimal control strategies: A case study of the Chikungunya in Chad

Abstract

In this work, we extend existing models of vector-borne diseases by including density-dependent rates and some existing control mechanisms to decrease the disease burden in the human population. We begin by analyzing the vector model dynamics and by determining the offspring reproductive number denoted by N as well as the trivial and nontrivial equilibria. Using theory of cooperative systems and the general theory of Lyapunov, we prove that, although there is a possibility that the trivial equilibrium coexists with a positive equilibrium, it remains globally asymptotically stable whenever N ≤ 1 . The fact that the non-trivial equilibrium is globally asymptotically stable permits us to reduce the study of the full model to the study of a reduced model whenever N > 1 . Thus, we analyze the reduced model by computing the basic reproduction number R 0 , equilibrium points as well as asymptotic stability of each equilibrium point. We also explore the nature of the bifurcation for the disease-free equilibrium from R 0 = 1 . By the application of the centre manifold theory, we prove that the backward bifurcation phenomenon can occur in our model, which means that the necessary condition R 0 < 1 is not sufficient to guarantee the final extinction of the disease in human populations. To calibrate our model, we estimate model parameters on clinical data from the last Chikungunya epidemic which occurred in Chad, using the non-linear least-square method. We find out that R 0 = 1.8519 , which means that we are in an endemic state since R 0 > 1 . To determine model parameters that are responsible for disease spread in the human community, we perform sensitivity analysis (SA) using a global method. It follows that the density-dependent death rate of mosquitoes and the average number of mosquito bites are key parameters in the disease dynamics. Following this, we thus formulate an optimal control model by including in the autonomous model, four time-dependent control functions to fight the disease spread. Pontryagin’s maximum principle is used to characterize our optimal controls. Numerical simulations, using parameter values of Chikungunya transmission dynamics, and efficiency analysis, are conducted to determine the better control strategy which guaranteed the final extinction of the disease in human populations.