Optimal control of visceral, cutaneous and post kala-azar leishmaniasis

Abstract

Abstract This article focuses on the eradication of different strains of leishmaniasis with the help of almost nonpharmaceutical interventions (NPIs). A comprehensive mathematical model of the disease is formulated incorporating three types of populations: sandflies, humans and dogs (reservoirs), and 3-types of strains: C l $C_{l}$ , cutaneous leishmaniasis; V l $V_{l}$ , kala-azar; and PKDL, post kala-azar. We find R 0 $R_{0}$ , the basic reproduction number of the infection. On the basis of sensitivity test of R 0 $R_{0}$ , the most active/sensitive parameters are investigated. These active parameters are controlled with the help of control variables. In some cases different parameters depend on the same single parameter, like ovigenesis and biting rate, both of which are linked to the blood source. Therefore we introduce three nonpharmaceutical control variables in the proposed model to control the biting rate of sandflies, density of seropositive dogs, and density of vector population. Nonpharmaceutical interventions include bed nets, eradiation of infectious dogs, and residual sprays, and thus extend the proposed model to an optimal control model. Using Lagrangian and Hamiltonian, we minimize the densities infected classes in human, sandfly and vector populations. Adopting optimality approach, we check the existence of the optimal control for the system. Using Matlab, we produce numerical simulations for the validation of results of control variables.

Keywords

• Leishmaniasis
• Basic reproduction number
• Mathematical model
• Sensitivity
• Pontryagin’s maximum principle
• Optimal control