Optimal control analysis of malaria and typhoid fever co-dynamics

Tchoumi, S. Y.; Chukwu, C. W.; Diagne, M. L.; Rwezaura, H.; Tchuenche, J. M.

doi:10.47974/JIM-2063

Open Access Research Article

Optimal control analysis of malaria and typhoid fever co-dynamics

^*S. Y. TchoumiCorresponding authorsytchoumi83@gmail.comDepartment of Mathematics and Computer Sciences ENSAI, University of Ngaoundere P. O. Box 455 ; Department of Mathematics and Applied Mathematics University of PretoriaDepartment of Mathematics and Computer Sciences ENSAI, University of Ngaoundere P. O. Box 455 Ngaoundere, CameroonView full profile → , C. W. Chukwuwiliam.chukwu@gmail.comDepartment of Mathematical Sciences Georgia Southern University 65 Georgia Ave. P.O. Box 8093, StatesboroGeorgia, 30460, U.S.A.View full profile → , M. L. Diagnecouragelamine@gmail.comDepartement de Mathematiques UFR des Sciences et Technologies Universite de ThiesThies, SenegalView full profile → , H. Rwezauraheriethrwezaula@gmail.comDepartment of Mathematics University of Dar es Salaam P. O. Box 35062Dar es Salaam, TanzaniaView full profile → , J. M. Tchuenchejmtchuenche@gmail.comSchool of Computer Science and Applied Mathematics University of the Witwatersrand Private Bag 3, Wits 2050; School of Computational and Communication Sciences and Engineering Nelson Mandela African Institution of Science and Technology P. O. Box 447School of Computer Science and Applied Mathematics University of the Witwatersrand Private Bag 3, Wits 2050Johannesburg, South AfricaView full profile →

* Corresponding author · click or hover a name for details

pp. 2567–2600Vol. 28Issue 7October 2025DOI: 10.47974/JIM-2063XML

Received:: 13 Mar 2024
Published Online:: 19 Aug 2025
Article type:: Research Article
Language:: EN
Article no.:: JIM-2063
Pages:: 2567–2600

Abstract

Malaria is an infectious vector-borne disease spread by infected mosquitoes, while typhoid fever is contracted by either drinking water or eating food contaminated with the Salmonella typhoid bacteria. Both diseases affect millions of individuals every year, causing a great deal of morbidity and mortality. Previous mathematical models of the co-dynamics of these two diseases have not considered the interplay between symptomatic and asymptomatic individuals infected with typhoid. To fill this gap, we formulate a malaria and typhoid co-infection model explicitly including both of these classes and use standard theory of dynamical systems to analyze the model. The sub-models reproduction numbers are derived. Theoretical results show that the disease-free and endemic equilibria could co-exist (backward bifurcation) for both the typhoid only and malaria only sub-models when the respective reproduction number is less than unity. The potential impact of malaria on typhoid reveals that the increase in the number of cases due to malaria could lead to a decrease of the number of typhoid fever cases. To mitigate the spread of both malaria and typhoid fever, the model is extended to include three control measures: malaria prevention, typhoid vaccination and treatment. Numerical simulations are carried out and graphically depicted, and it is noted that reduction in the spread of typhoid greatly impacts the decrease in the number of malaria infectious individuals. Also, as expected, the most effective combination control strategy is the simultaneous implementation of malaria prevention, typhoid treatment and vaccination.

Keywords

Malaria Typhoid fever Co-infection Optimal control Mathematical model Basic reproduction number

Subject Classifications

34D2034D2349K1049M0592B0592D30

References

[1] Mayo Clinic Staff, “Malaria – Symptoms and causes,” Mayo Clinic, [Online]. Available: https://www.mayoclinic.org/diseases-conditions/malaria/symptoms-causes/syc-20351184. [Accessed: Mar. 5, 2024].
[2] World Health Organization, “Typhoid,” WHO Fact Sheets, [Online]. Available: https://www.who.int/news-room/fact-sheets/detail/typhoid. [Accessed: Mar. 6, 2024].
[3] National Institute for Communicable Diseases, “Update on enteric fever in South Africa – 18 Feb (2022),” [Online]. Available: https://www.nicd.ac.za/update-on-enteric-fever-in-south-africa-18-feb-2022/. [Accessed: Mar. 7, 2024].
[4] Hamadou Abboubakar and Reinhard Racke, Mathematical Modelling and Optimal Control of Typhoid Fever (2019).
[5] Akeem Abdulfatai, Ismaila M. Ali, Blessing David, and Daniel J. Washachi, “Mathematical modelling of malaria and typhoid co-infection incorporating vector and loss of immunity,” unpublished manuscript.
[6] Folashade B. Agusto, “Optimal isolation control strategies and cost-effectiveness analysis of a two-strain avian influenza model,” Biosystems, vol. 113, no. 3, pp. 155–164 (2013).
[7] Ouma C. Akinyi, John Y. T. Mugisha, Andrew Manyonge, Charles Ouma, and Kennedy Maseno, “A model on the impact of treating typhoid with antimalarial: Dynamics of malaria concurrent and co-infection with typhoid,” International Journal of Mathematical Analysis, vol. 9, no. 9–12, pp. 541–551 (2015).
[8] Adolf Ammah, Theresa Nkuo-Akenji, Roland Ndip, and J. E. Deas, “An update on concurrent malaria and typhoid fever in Cameroon,” Transactions of the Royal Society of Tropical Medicine and Hygiene, vol. 93, no. 2, pp. 127–129 (1999).
[9] Roumen Anguelov, Yves Dumont, Jean M.-S. Lubuma, and Esau Mureithi, “Stability analysis and dynamics-preserving nonstandard finite difference schemes for a malaria model,” Mathematical Population Studies, vol. 20, no. 2, pp. 101–122 (2013).
[10] Ehsan Anwar, Eliyahu Goldberg, Adam Fraser, Claudia J. Acosta, Mical Paul, and Leonard Leibovici, “Vaccines for preventing typhoid fever,” Cochrane Database of Systematic Reviews, no. 1 (2014).
[11] Bruno Buonomo, “Analysis of a malaria model with mosquito host choice and bed-net control,” International Journal of Biomathematics, vol. 8, no. 6, article 1550077 (2015).
[12] Ib Christian Bygbjerg, Claus Lanng, et al., “Septicaemia as a complication of falciparum malaria,” Transactions of the Royal Society of Tropical Medicine and Hygiene, vol. 76, no. 5 (1982).
[13] Bill Bynum, “Typhomalarial fever,” The Lancet, vol. 360, no. 9342, p. 1339 (2002).
[14] Jack Carr, Applications of Center Manifold Theory, New York, NY, USA: Springer-Verlag (1981).
[15] Nakul Chitnis, James M. Hyman, and James M. Cushing, “Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model,” Bulletin of Mathematical Biology, vol. 70, no. 5, pp. 1272–1296 (2008).
[16] C. W. Chukwu, M. L. Juga, Z. Chazuka, and J. Mushanyu, Mathematical analysis and sensitivity assessment of HIV/AIDS-listeriosis co-infection dynamics. International Journal of Applied and Computational Mathematics, 8(5), p.251 (2022).
[17] Mamadou L. Diagne, Folashade B. Agusto, Herieth Rwezaura, Jean M. Tchuenche, and Suzanne Lenhart, “Optimal control of an epidemic model with treatment in the presence of media coverage,” Scientific African, vol. 24, article e02138 (2024).
[18] F. F. Herdicho, C. W. Chukwu, and H. Tasman, “An optimal control of malaria transmission model with mosquito seasonal factor,” Results in Physics, vol. 25, article 104238 (2021).
[19] David Kirschner, Suzanne Lenhart, and Steve Serbin, “Optimal control of the chemotherapy of HIV,” Journal of Mathematical Biology, vol. 35, no. 7, pp. 775–792 (1997).
[20] Suzanne Lenhart and John T. Workman, Optimal Control Applied to Biological Models. Boca Raton, FL, USA: Chapman and Hall/CRC (2007).
[21] Edmore Mtisi, Herieth Rwezaura, and Jean M. Tchuenche, “A mathematical analysis of malaria and tuberculosis codynamics,” Discrete and Continuous Dynamical Systems - Series B, vol. 12, no. 4, pp. 827–847 (2009).
[22] Zindoga Mukandavire, Abba B. Gumel, Witness Garira, and Jean M. Tchuenche, “Mathematical analysis of a model for HIV-malaria co-infection,” Mathematical Biosciences and Engineering, vol. 6, no. 2, pp. 333–362 (2009).
[23] Shelton Mushayabasa, Impact of vaccines on controlling typhoid fever in Kassana-Nankana district of Upper East Region of Ghana: Insight from a mathematical model (2011). [Unpublished or institutional report].
[24] Shelton Mushayabasa, Courage P. Bhunu, and N. A. Mhlanga, “Modeling the transmission dynamics of typhoid in malaria-endemic settings,” Applications and Applied Mathematics: An International Journal (AAM), vol. 9, no. 1, article 9, pp. 49–65 (2014).
[25] Joseph M. Mutua, Feng-Bin Wang, and Nitin K. Vaidya, “Modeling malaria and typhoid fever co-infection dynamics,” Mathematical Biosciences, vol. 264, pp. 128–144 (2015).
[26] Emmanuel Njeuhmeli, Melissa Schnure, Andrea Vazzano, Elizabeth Gold, Peter Stegman, Katharine Kripke, Michel Tchuenche, Lori Bollinger, Steven Forsythe, and Catherine Hankins, “Using mathematical modeling to inform health policy: A case study from voluntary medical male circumcision scale-up in Eastern and Southern Africa and proposed framework for success,” PLoS ONE, vol. 14, no. 3, article e0213605 (2019).
[27] Oluwadare M. Ogunmiloro, “Mathematical modeling of the co-infection dynamics of malaria-toxoplasmosis in the tropics,” Biometrical Letters, vol. 56, no. 2, pp. 139–163 (2019).
[28] Bimala Pantha, Folashade Agusto, and Ibrahim Elmojtaba, “Optimal control applied to a visceral leishmaniasis model,” (2020). [Unpublished or conference proceedings if no journal is specified].
[29] Virginia E. Pitzer, Catherine C. Bowles, Stephen Baker, Gagandeep Kang, Venugopal Balaji, Jeremy J. Farrar, and Bryan T. Grenfell, “Predicting the impact of vaccination on the transmission dynamics of typhoid in South Asia: A mathematical modeling study,” PLoS Neglected Tropical Diseases, vol. 8, no. 1, article e2642 (2014).
[30] Lev S. Pontryagin, Vladimir G. Boltyanskii, Revaz V. Gamkrelidze, and Evgenii F. Mishchenko, The Mathematical Theory of Optimal Processes, vol. 4. New York, NY, USA: Interscience Publishers (1986).
[31] Israel Simon-Oke and Michael Akinbote, “Prevalence of malaria and typhoid co-infection in relation to haematological profile of university students in Akure, Nigeria,” Journal of Infectious Diseases and Epidemiology, vol. 6, article 166 (2020).
[32] David C. Smith, “The rise and fall of typhomalarial fever: I. Origins,” Journal of the History of Medicine and Allied Sciences, vol. 37, no. 2, pp. 182–220 (1982).
[33] H. Tasman, H., D. Aldila, P. A. Dumbela, M. Z. Ndii, Fatmawati, F. F. Herdicho, and C. W. Chukwu, “Assessing the impact of relapse, reinfection, and recrudescence on malaria eradication policy: A bifurcation and optimal control analysis,” Tropical Medicine and Infectious Disease, vol. 7, no. 10, article 263 (2022).
[34] Stephane Y. Tchoumi, Chidozie W. Chukwu, Mamadou L. Diagne, Herieth Rwezaura, Mary L. Juga, and Jean M. Tchuenche, “Optimal control of a two-group malaria transmission model with vaccination,” Network Modeling Analysis in Health Informatics and Bioinformatics, vol. 12, no. 1, article 7 (2022).
[35] Stephane Y. Tchoumi, Herieth Rwezaura, and Jean M. Tchuenche, “A mathematical model with numerical simulations for malaria transmission dynamics with differential susceptibility and partial immunity,” Healthcare Analytics, vol. 3, article 100165 (2023).
[36] Getachew Teshome Tilahun and Haileyesus Tessema Alemneh, “Mathematical modeling and optimal control analysis of COVID-19 in Ethiopia,” Journal of Interdisciplinary Mathematics, vol. 24, no. 8, pp. 2101–2120 (2021).
[37] Pauline Van den Driessche and James Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical Biosciences, vol. 180, no. 1–2, pp. 29–48 (2002).
[38] World Health Organization, World Malaria Report 2021. [Online]. Available: https://www.who.int/publications/i/item/9789240040496. [Accessed: Feb. 5, 2024].
[39] Zemene Amare Workie and Poomani Raj Koya, “Mathematical modelling of the co-infection dynamics of typhoid fever with Plasmodium vivax and Plasmodium falciparum with treatment,”.

Views: 110Downloads: 10Citations: 0