Existence and stability results in a fractional optimal control model for dengue and two-strains of salmonella typhi

Abstract

Co-infection with dengue and salmonella typhi could lead to devastating consequences, and sometimes even result in deaths. This could lead to tremendous hazards not only to country’s economy but also overloading health-care centers. In this article, a fractional co-infection model for dengue, and two-strains (drug-sensitive and drug-resistant) of salmonella typhi is developed by implementing Caputo fractional derivative. Existence, uniqueness and stability of the model are proved by implementing Arzela Ascoli’s theorem, Banach fixed point theorem and Hyers-Ulam stability criteria, respectively. To control the diseases, control measures namely prevention control against dengue, $u_1\left(t\right)$, prevention control against drug-sensitive salmonella typhi, $u_2\left(t\right)$, and prevention control against drug-resistant salmonella typhi, $u_3\left(t\right)$, are introduced into the considered model. The optimality system for corresponding fractional optimal control problem is illustrated by employing Pontryagin’s maximum principle. The simulations of the model are performed by employing fractional Euler scheme to see the impact of control measures and fractional order on the respective diseases.