Mathematical Analysis Of Control Strategies For Elimination Of Leishmaniasis

Abstract

In this study, we investigate mathematical models of cutaneous and visceral strains of leishmaniasis. The models help in analyzing and understanding di erent dy- namics of leishmaniasis. The study focuses both zoonotic and anthroponotic syn- dromes. While formulating di erent compartments of the models we used medical research, carried out on di erent stages of leishmaniasis. First, we design a mathematical model of Zoonotic Cutaneous leishmaniasis (CL) taking into account humans, vectors and reservoirs. The total population is as- sumed as homogenously mixed. The results show that the disease can be con- trolled if sand y bitting rate is controlled. However for disease eradication from the community, vector control must be followed by human treatment and reservoir control. We have formulated mathematical model of Visceral leishmaniasis. The model incorporates both Visceral leishmaniasis and its further complication; Post Kala- Azar Dermal Leishmaniasis. The disease can be eradicated in short time if e ective medicines are used and vector control is focussed. Mathematical model for optimal control of Anthroponotic Visceral leishmaniasis is presented. The model incorporates rst line and second line treatment. On the basis of sensitivity analysis of the reproduction number, we propose four control strategies. For quanti cation of prevalence period of the disease, we perform nu- merical simulations. The results show that the disease can be eradicated however the prolonged prevalence of PKDL needs special attention. Mathematical model to describe sensitivity analysis and optimal control of An- throponotic Cutaneous Leishmania is designed. with the help of sensitivity indices we nd the relative importance of the role of di erent parameters, in the transmis- sion of ACL. We address three key parameters; sand y biting rate, mortality rate of sand ies, and healing (recovery) rate of infectious humans. For this we intro- duce three control variables (strategies), in the optimal control problem, each for reducing sand y biting rate, increasing mortality rate of sand y and increasing recovery rate of humans. These control strategies causes decrease in initial trans- mission rate R0. Since the control strategies are always e ected by economics constraints, therefore, we take into account the constraints imposed by limited resources, in our objective functional. The results obtained from numerical simu- lations show that the control strategies are very e ective, if implemented, on the same time, in the same area.