Numerical Optimal Control of HIV Transmission in Octave/MATLAB

TL;DR

This paper presents accessible GNU Octave/MATLAB code for simulating HIV transmission models and solving optimal control problems to identify effective prevention strategies using Pontryagin's maximum principle.

Contribution

It introduces a clear implementation of HIV transmission modeling and optimal control solutions in Octave/MATLAB, including algorithms for both uncontrolled and controlled scenarios.

Findings

Numerical solutions for HIV transmission dynamics using ode45 and Runge-Kutta methods.

Optimal control strategies that maximize uninfected individuals with minimal infections.

Algorithms compatible with MATLAB and GNU Octave for practical application.

Abstract

We provide easy and readable GNU Octave/MATLAB code for the simulation of mathematical models described by ordinary differential equations and for the solution of optimal control problems through Pontryagin's maximum principle. For that, we consider a normalized HIV/AIDS transmission dynamics model based on the one proposed in our recent contribution (Silva, C.J.; Torres, D.F.M. A SICA compartmental model in epidemiology with application to HIV/AIDS in Cape Verde. Ecol. Complex. 2017, 30, 70--75), given by a system of four ordinary differential equations. An HIV initial value problem is solved numerically using the ode45 GNU Octave function and three standard methods implemented by us in Octave/MATLAB: Euler method and second-order and fourth-order Runge-Kutta methods. Afterwards, a control function is introduced into the normalized HIV model and an optimal control problem is formulated, where the goal is to find the optimal HIV prevention strategy that maximizes the fraction of uninfected HIV individuals with the least HIV new infections and cost associated with the control measures. The optimal control problem is characterized analytically using the Pontryagin Maximum Principle, and the extremals are computed numerically by implementing a forward-backward fourth-order Runge-Kutta method. Complete algorithms, for both uncontrolled initial value and optimal control problems, developed under the free GNU Octave software and compatible with MATLAB are provided along the article.