Optimal control of a heroin epidemic mathematical model

A heroin epidemic mathematical model with prevention information and treatment, as control interventions, is analyzed, assuming that an individual's behavioral response depends on the spreading of information about the effects of heroin. Such information creates awareness, which helps individuals to participate in preventive education and self-protective schemes with additional efforts. We prove that the basic reproduction number is the threshold of local stability of a drug-free and endemic equilibrium. Then, we formulate an optimal control problem to minimize the total number of drug users and the cost associated with prevention education measures and treatment. We prove existence of an optimal control and derive its characterization through Pontryagin's maximum principle. The resulting optimality system is solved numerically. We observe that among all possible strategies, the most effective and cost-less is to implement both control policies.publishe

Mathematical and numerical analysis of an acid-mediated cancer invasion model with nonlinear diffusion. ETNA - Electronic Transactions on Numerical Analysis

In this paper, we study the existence of weak solutions of the nonlinear cancer invasion parabolic system with density-dependent diffusion operators. To establish the existence result, we use regularization, the Faedo-Galerkin approximation method, some a priori estimates, and compactness arguments. Furthermore in this paper, we present results of numerical simulations for the considered invasion system with various nonlinear density-dependent diffusion operators. A standard Galerkin finite element method with the backward Euler algorithm in time is used as a numerical tool to discretize the given cancer invasion parabolic system. The theoretical results are validated by numerical examples

Weak solution for time-fractional strongly coupled three species cooperating model

This article investigates the solvability of the time-fractional strongly coupled three-species cooperating system of equations with the Dirichlet boundary conditions. This model expresses the interactions of cooperating species. First, a suitable approximation problem is considered to overcome the strong degeneracy of the original model. Then, we establish the existence of a weak solution for the proposed system using the Faedo–Galerkin method and some compactness arguments

A time-fractional HIV infection model with nonlinear diffusion

[EN] This paper deals with a set of three partial differential equations involving time-fractional derivatives and nonlinear diffusion operators. This model helps us to understand the HIV spread and transmission into the patient. First, we prove the existence and uniqueness of weak solutions to the mathematical model. Then, the Galerkin finite element scheme is implemented to approximate the solution of the model. Further, a-priori error bounds and convergence estimates for the fully-discrete problem are derived. The second order convergence for the proposed scheme is also proved. Numerical tests are shown to validate the theoretical studies.Manimaran, J.; Shangerganesh, L.; Debbouche, A.; Cortés, J. (2021). A time-fractional HIV infection model with nonlinear diffusion. Results in Physics. 25:1-13. https://doi.org/10.1016/j.rinp.2021.104293S1132