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Optimal Control of Aquatic Diseases: A Case Study of Yemen’s Cholera Outbreak

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Abstract

We propose a mathematical model for the transmission dynamics of some strains of the bacterium Vibrio cholerae, responsible for the cholera disease in humans. We prove that, when the basic reproduction number is equal to one, a transcritical bifurcation occurs for which the endemic equilibrium emanates from the disease-free point. A control function is introduced into the model, representing the distribution of chlorine water tablets for water purification. An optimal control problem is then proposed and analyzed, where the goal is to determine the fraction of susceptible individuals who should have access to chlorine water tablets in order to minimize the total number of new infections plus the total cost associated with the distribution of chlorine water tablets, over the considered period of time. Finally, we consider real data of the cholera outbreak in Yemen, from April 27, 2017 to April 15, 2018, choosing the values of the parameters of the uncontrolled model that fit the real data. Using our optimal control results, we show, numerically, that the distribution of chlorine water tablets could have stopped, in a fast way, the worst cholera outbreak that ever occurred in human history. Due to the critical situation of Yemen, we also simulate the case where only a small percentage of susceptible individuals has access to chlorine water tablets and obtain an optimal control solution that decreases, substantially, the maximum number of infective individuals affected by the outbreak.

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Acknowledgements

This research was supported by the Portuguese Foundation for Science and Technology (FCT) within Projects UIDB/04106/2020 and UIDP/04106/2020 (CIDMA) and PTDC/EEI-AUT/2933/2014 (TOCCATA), funded by Project 3599 – Promover a Produção Científica e Desenvolvimento Tecnológico e a Constituição de Redes Temáticas and FEDER funds through COMPETE 2020, Programa Operacional Competitividade e Internacionalização (POCI). Lemos-Paião is also supported by the Ph.D. fellowship PD/BD/114184/2016; Silva by national funds (OE), through FCT, I.P., in the scope of the framework contract foreseen in the numbers 4, 5, and 6 of the article 23, of the Decree-Law 57/2016, of August 29, changed by Law 57/2017, of July 19. The research of Ezio Venturino has been partially supported by the project “Metodi numerici e computazionali per le scienze applicate” of the Dipartimento di Matematica “Giuseppe Peano”. The authors are very grateful to two anonymous referees for several constructive remarks and questions that helped them to improve the quality of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Center for Research and Development in Mathematics and Applications (CIDMA), University of Aveiro, 3810-193, Aveiro, Portugal

    Ana P. Lemos-Paião, Cristiana J. Silva & Delfim F. M. Torres

  2. Department of Mathematics “Giuseppe Peano”, Università di Torino, Via Carlo Alberto 10, 10123, Turin, Italy

    Ezio Venturino

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  1. Ana P. Lemos-Paião
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  2. Cristiana J. Silva
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  3. Delfim F. M. Torres
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  4. Ezio Venturino
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Correspondence to Delfim F. M. Torres.

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Communicated by Alberto d’Onofrio.

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Ezio Venturino is a member of the INdAM reseach group GNCS.

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Lemos-Paião, A.P., Silva, C.J., Torres, D.F.M. et al. Optimal Control of Aquatic Diseases: A Case Study of Yemen’s Cholera Outbreak. J Optim Theory Appl 185, 1008–1030 (2020). https://doi.org/10.1007/s10957-020-01668-z

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