Mathematical analysis of an extended SIRS-SI malaria model incorporating seasonal vector recruitment with standard incidence

Alphablondy Achieng; Charles Wachira; Benjamin Kikwai

doi:10.51867/ajernet.maths.7.2.122

Auteurs

Alphablondy Achieng Department of Mathematics and Statistics, Machakos University, Kenya https://orcid.org/0009-0005-1978-8314
Charles Wachira Department of Mathematics and Statistics, Machakos University, Kenya https://orcid.org/0000-0003-1415-2905
Benjamin Kikwai Department of Mathematics and Statistics, Machakos University, Kenya https://orcid.org/0000-0002-6741-5011

DOI :

https://doi.org/10.51867/ajernet.maths.7.2.122

Mots-clés :

Basic Reproduction Number, Malaria Free Infection, Seasonality, Vaccine Reproduction Number

Résumé

In this study an extended deterministic SIRS-SI malaria model incorporating seasonality on vector recruitment with standard incidence and vaccination is developed and analyzed. Positivity and boundedness of model solutions are shown. The vaccine reproductive number R_v is obtained using next generation matrix approach. The Routh Hurwitz criterion is applied to analyze the local stability of Malaria-Free Equilibrium (MFE) points which is found to be locally asymptotically stable whenever R_v < 1. Gershgorin discs is applied to analyze the local stability of Malaria Persistence Equilibrium (MP E) points which is locally asymptotically stable when R_v > 1, implying that the disease would persist in the population. The Castillo-Chavez technique is applied to analyze the global stability of MF E. The analysis shows that the MF E point is globally asymptotically stable provided that R_v < 1. The numerical simulation show that an increase in vaccination rate (α) while using a vaccine of higher efficacy (ϵ) leads to reduction of malaria infection. It is shown that during rainy seasons, there is high mosquito recruitment hence there are expectation of malaria infectious periods. Therefore, the policy makers should educate on the use of mosquito treated bed nets, clearing of bushes around homestead, and spraying the insecticides during rainy seasons.

Téléchargements

Les données relatives au téléchargement ne sont pas encore disponibles.

Références

Anderson, R. M., & May, R. M. (1991). Infectious diseases of humans: Dynamics and control. Oxford University Press. https://doi.org/10.1093/oso/9780198545996.001.0001

Akowe, E., Ahman, Q. O., Agbata, B. C., Joseph, S. O., Senewo, E. O., Danjuma, A. Y., & Yahaya, D. J. (2025). A novel malaria mathematical model: Integrating vector and non-vector transmission pathways. BMC Infectious Diseases, 25(1), 322. https://doi.org/10.1186/s12879-025-10653-8

Ayalew, A., Molla, Y., & Woldegbreal, A. (2024). Modeling and stability analysis of the dynamics of malaria disease transmission with some control strategies. Abstract and Applied Analysis, 2024, Article 8837744. https://doi.org/10.1155/2024/8837744

Bejarano, D., Ibargüen-Mondragón, E., & Gómez-Hernández, E. A. (2018). A stability test for nonlinear systems of ordinary differential equations based on the Gershgorin circles. Contemporary Engineering Sciences, 11(91), 4541-4548. https://doi.org/10.12988/ces.2018.89504

Castillo-Chavez, C., Blower, S., van den Driessche, P., Kirschner, D., & Yakubu, A. A. (Eds.). (2002). Mathematical approaches for emerging and reemerging infectious diseases: Models, methods, and theory (Vol. 126). Springer.

https://doi.org/10.1007/978-1-4613-0065-6

Charles, W. M., George, L. O., & Malinzi, J. (2018). A spatiotemporal model on the transmission dynamics of Zika virus disease. Asian Research Journal of Mathematics, 10(4), 1-15. https://doi.org/10.9734/ARJOM/2018/43944

Duve, P., Charles, S., Munyakazi, J., Lühken, R., & Witbooi, P. (2024). A mathematical model for malaria disease dynamics with vaccination and infected immigrants. Mathematical Biosciences and Engineering, 21(1), 1082-1109. https://doi.org/10.3934/mbe.2024045

Elmojtaba, I. M., Mugisha, J. Y. T., & Hashim, M. H. (2010). Mathematical analysis of the dynamics of visceral leishmaniasis in the Sudan. Applied Mathematics and Computation, 217(6), 2567-2578. https://doi.org/10.1016/j.amc.2010.07.069

Fikadu, M., & Ashenafi, E. (2023). Malaria: An overview. Infection and Drug Resistance, 16, 3339-3347. https://doi.org/10.2147/IDR.S405668

https://doi.org/10.2147/IDR.S405668

Gatore Sinigirira, K. J., Ogana, W., & Chirove, F. (2025). Mathematical modelling of malaria integrating temperature, rainfall, and vegetation index. Acta Applicandae Mathematicae, 199(1), 5. https://doi.org/10.1007/s10440-025-00740-y

Guo, S. B., He, M., & Cui, J. A. (2023). Global stability of a time-delayed malaria model with standard incidence rate. Acta Mathematicae Applicatae Sinica, English Series, 39(2), 211-221. https://doi.org/10.1007/s10255-023-1042-y

Habtamu, K., Petros, B., & Yan, G. (2022). Plasmodium vivax: The potential obstacles it presents to malaria elimination and eradication. Tropical Diseases, Travel Medicine and Vaccines, 8(1), 27. https://doi.org/10.1186/s40794-022-00185-3

Lotfi, E. M., Maziane, M., Hattaf, K., & Yousfi, N. (2014). Partial differential equations of an epidemic model with spatial diffusion. International Journal of Partial Differential Equations, 2014, Article 186437. https://doi.org/10.1155/2014/186437

Ochieng, F. O. (2024). SEIRS model for malaria transmission dynamics incorporating seasonality and awareness campaign. Infectious Disease Modelling, 9(1), 84-102. https://doi.org/10.1016/j.idm.2023.11.010

Ochwach, J. O., Mark, O. O., & Murwayi, A. L. M. (2023). On basic reproduction number (R₀): Derivation and application. Journal of Engineering and Applied Sciences Technology, 5, 173, 2-14. https://doi.org/10.47363/JEAST/2023(5)173

Osman, M. A., Ebenezer, A., Hassan, N. A., & Yang, C. (2019). Sensitivity analysis of a mathematical model for malaria transmission with saturated incidence rate. Journal of Scientific Research and Reports, 22(3), 1-10. https://doi.org/10.9734/JSRR/2019/46131

Okombo, J., & Fidock, D. A. (2025). Towards next-generation treatment options to combat Plasmodium falciparum malaria. Nature Reviews Microbiology, 23(3), 178-191. https://doi.org/10.1038/s41579-024-01099-x

Qu, Z., Patterson, D., Zhao, L., Ponce, J., Edholm, C. J., Prosper Feldman, O. F., & Childs, L. M. (2025). Mathematical modeling of malaria vaccination with seasonality and immune feedback. PLOS Computational Biology, 21(5). https://doi.org/10.1371/journal.pcbi.1012988

Rajnarayanan, A., Kumar, M., & Tridane, A. (2025). Analysis of a mathematical model for malaria using a data-driven approach. Scientific Reports, 15(1), 27272. https://doi.org/10.1038/s41598-025-12078-4

Siagian, F. E. (2020). Malaria epidemiology: Specific vulnerable groups in the population. International Journal of Research and Reports in Hematology, 3(2), 41-52.

Tchoumi, S. Y., Chukwu, C. W., Diagne, M. L., Rwezaura, H., Juga, M. L., & Tchuenche, J. M. (2022). Optimal control of a two-group malaria transmission model with vaccination. Network Modeling Analysis in Health Informatics and Bioinformatics, 12(1), 7. https://doi.org/10.1007/s13721-022-00403-0

Walldorf, J. A., Mayers, G. F., Amponsa-Achiano, K., Jalang'o, R., Chisema, M., Khalayi, L., & Okine, R. N. (2025). Post-introduction evaluation (PIE) of the malaria vaccine introduced in three pilot countries (Ghana, Kenya, and Malawi) in 2021. Malaria Journal, 24(1), 333. https://doi.org/10.1186/s12936-025-05590-5

World Health Organization. (2024). Global malaria programme operational strategy. World Health Organization.