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# Numerical Optimal Control of HIV Transmission in Octave/MATLAB

by 1,†, 2,*,† and
1
Department of Mathematics, Polytechnic of Leiria, 2411-901 Leiria, Portugal
2
Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Math. Comput. Appl. 2020, 25(1), 1; https://doi.org/10.3390/mca25010001
Received: 1 October 2019 / Revised: 7 December 2019 / Accepted: 17 December 2019 / Published: 19 December 2019
(This article belongs to the Special Issue Numerical and Symbolic Computation: Developments and Applications)
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. View Full-Text
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Campos, C.; Silva, C.J.; Torres, D.F.M. Numerical Optimal Control of HIV Transmission in Octave/MATLAB. Math. Comput. Appl. 2020, 25, 1.