Optimizing Control Measures for a Vector-Host Epidemic Model: A Mathematical Analysis

Abstract

Malaria, Rift Valley Fever, Dengue Fever and West Nile Fever are vector-host infections that are endemic among most of the populations mainly across the tropical regions. As much as these diseases are treatable and preventable they have proved to be among the greatest attributes to mortality among the populations of the world. This could be attributed to the fact that planning and measures that are proper and timely have not been put in place by the governments.To curb the effects of such diseases mathematical models can be used to study the dynamics of these infections and the effectiveness of various controls towards them including prevention and treatment. In this paper, we come up with an appropriate system of equations to represent the dynamics of Malaria, Dengue Fever, Rift Valley Fever and West Nile Fever infections. In particular, we have developed a mathematical model for the transmission and control of Dengue Fever, incorporating prevention and treatment as control parameters. We have further shown that our model has a unique disease-free equilirium point which is locally and globally asymptotically stable when R0 < 1. The model also have a unique endemic equilibrum point, which is locally and globally asymptotically stable when R0 > 1. We determined the parameters of the model, using Data from [1] while some of the data was approximated. To determine an optimal combination of prevention and treatment that effectively reduces the transmission, we formulated an optimal control problem, with an appropriate cost function. We then used Pontryagin’s Maximum Principle to determine the optimal control functions. Simulations of our model using various combinations of treatment and prevention indicate that infected vector and human populations can be drastically reduced, thus effectively controlling the transmission of Dengue Fever. In addition, effective treatment reduces the infected human population. The results also showed that when more resources are channeled towards the treatment and prevention of vector-host infections then these diseases can easily be eradicated within the shortest time possible. This implies the designing of proper policies and strategies to fully render a population disease-free.