Optimal Control of Mathematical Model of Diphtheria Spreading

Putri Amalia(1*), Syamsuddin Toaha(2),

(1) Department Mathematics, FMIPA Universitas Hasanuddin, Indonesia
(2) Department Mathematics, FMIPA Universitas Hasanuddin, Indonesia
(*) Corresponding Author

DOI: https://doi.org/10.26858/jdm.v10i2.35776


This article examines optimal control model for the spread of diphtheria disease. Diphtheria is an infectious disease caused by the bacterium Corynebacterium diphtheriae. This model is divided into six compartments, namely population of susceptibles (𝑆), population of latent (L), population of infected with symptoms (Is), population of infected without symptoms (Ia), population of recovered with full immunity (Rf) and population of recovered with partial immunity (Rp). Two optimal controls are applied in the model, namely vaccination and treatment. The problem of optimal control is solved by using Pontryagin's minimal principle, which consists in solving a set of necessary conditions that must be satisfied by the optimal control and its associated state. The numerical method used to solve the optimal control problem is the forward-backward sweep method. Based on the results of numerical simulations, both controls should be administered in large numbers and continuously since the beginning of observation in order to reduce the number of diphtheria infected population and to control the spread of diphtheria.


Mathematical model of diphtheria spread, optimal control problem, state, forward-backward sweep.

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