Mathematical analysis of reinfection and relapse in malaria dynamics Abstract   Recurrent malaria constitutes one of the greatest setbacks to the realization of malaria disease elimination from the population. In this paper, a deterministic model governed by a system of nonlinear differential equations is developed to assess the effects of recurrent malaria xe2x80x93 reinfection and relapse on the transmission dynamics of the disease. The model is distributed into autonomous and nonautonomous systems. Analysis of the autonomous model shows that reinfection, which is the recurrence of malaria symptoms due to new parasites infection, has the potential to trigger the existence of two endemic equilibrium points when the basic reproduction number is below unity. Consequently without reinfection, global asymptotic dynamics of the autonomous model is established in the presence of relapse with the aid of carefully constructed Lyapunov functions for both the diseasefree and endemic equilibria. The nonautonomous model with timedependent control strategies is analyzed using Pontryaginxe2x80x99s Maximum Principle to find the optimal solutions to the malaria control problem. Costeffectiveness analysis is conducted to buttress the results of the optimal control problem by using the average costeffectiveness ratio (ACER) and incremental costeffectiveness ratio (ICER) methods. Finally, numerical simulations are demonstrated to enhance the theoretical results.

Computational modelling and optimal control of measles epidemic in human population Abstract   Measles is an awfully contagious acute viral infection. It can be fatal, causing cough, red eyes, followed by a fever and skin rash with signs of respiratory infection. In this paper, we propose and analyze a model describing the transmission dynamics of a measles epidemic in the human population using the stability theory of differential equations. The model proposed undergoes a backward bifurcation for some parameter values. Sensitivity analysis is carried out on the model parameters in order to determine their impact on the disease dynamics. We extend the model to an optimal control problem by including timedependent control variables: prevention, treatment of infected people and vaccination of the susceptible humans. In an attempt to minimize the infected people and the cost applied we design the cost functional. Next, we show that optimal control exists for the system, and the Pontryagin maximum principle is employed to characterize the continuous controls. Numerical simulation is performed to justify the analytical results and discussed quantitatively.