Asymptotic perturbation and sensitivity analysis of malaria transmission in Nigeria: a mathematical model

Sunday N. Aloke; Henry O. Adagba; Uchenna E. Michael; Okorie Nwite; Theresa E. Efor; Aloysius N. Ezaka; Chika Agha

doi:10.61298/rans.2026.4.1.203

Authors

Sunday N. Aloke
[email protected]

Department of Industrial Mathematics and Health Statistics, David Umahi Federal University of Health Scioences, Uburu, Nigeria
Henry O. Adagba
Department of Industrial Mathematics and Applied Statistics, Ebonyi State University, Abakaliki, Nigeria
Uchenna E. Michael
Department of Mathematics/Statistics, Alex Ekwueme Federal University Ndufu Alike, Ebonyi State, Nigeria
Okorie Nwite
Department of Industrial Mathematics and Applied Statistics, Ebonyi State University, Abakaliki, Nigeria
Theresa E. Efor
Department of Industrial Mathematics and Applied Statistics, Ebonyi State University, Abakaliki, Nigeria
Aloysius N. Ezaka
Department of Industrial Mathematics and Applied Statistics, Ebonyi State University, Abakaliki, Nigeria
Chika Agha
Department of Industrial Mathematics and Applied Statistics, Ebonyi State University, Abakaliki, Nigeria

Keywords:

Malaria transmission, Perturbation method, Sensitivity analysis, Basic reproduction number

Abstract

This article analyzes malaria dynamics among high-risk groups in Nigeria, with the human population divided into five compartments and the mosquito population divided into two compartments. The resulting system of nonlinear ordinary differential equations is examined by asymptotic perturbation and sensitivity-analysis methods. The perturbation analysis describes the behavior of the system at the malaria-free equilibrium (MFE) and the effect of a small perturbation around that state. The first-order perturbation solution indicates exponential growth in disease prevalence, showing that the MFE is unstable when infection is introduced. The sensitivity analysis shows that preventing mosquitoes from surviving long enough to become infectious is the most effective way to reduce the malaria transmission cycle. The infected adult male compartment, I_M, also acts as an important long-term reservoir and is the second most influential factor in the model. The estimated basic reproduction number satisfies R₀> 1, indicating a high potential for malaria spread and the need for effective control measures.

Dimensions

REFERENCES

[1] S. S. Nundu, R. Culleton, S. V. Simpson, H. Arima, J.-J. Muyembe, T. Mita, S. Ahuka & T. Yamamoto, ‘‘Malaria parasite species composition of Plasmodium infections among asymptomatic and symptomatic schoolage children in rural and urban areas of Kinshasa, Democratic Republic of Congo’’, Malaria Journal 20 (2021) 389. https://doi.org/10.1186/s12936-021-03919-4.

[2] R. W. Snow, ‘‘Global malaria eradication and the importance of Plasmodium falciparum epidemiology in Africa’’, BMC Medicine 13 (2015) 23. https://doi.org/10.1186/s12916-014-0254-7.

[3] L. S. Garcia, ‘‘Malaria’’, Clinics in Laboratory Medicine 30 (2010) 93. https://doi.org/10.1016/j.cll.2009.10.001.

[4] World Health Organization, ‘‘World malaria report 2019’’, Geneva, Switzerland: World Health Organization (2019). https://www.who.int/publications/i/item/9789241565721.

[5] N. Chitnis, J. M. Cushing & J. M. Hyman, ‘‘Bifurcation analysis of a mathematical model for malaria transmission’’, SIAM Journal on Applied Mathematics 67 (2006) 24. https://doi.org/10.1137/050638941.

[6] A. Ducrot, S. B. Sirima, B. Somé & P. Zongo, ‘‘A mathematical model for malaria involving differential susceptibility, exposedness and infectivity of human host’’, Journal of Biological Dynamics 3 (2009) 574. https://dx.doi.org/10.1080/17513750902829393.

[7] R. B. Mbewe, J. B. Keven, C. Mangani, M. L. Wilson, T. Mzilahowa, D. P. Mathanga, C. Valim, M. K. Laufer, E. D. Walker & L. M. Cohee, ‘‘Genotyping of Anopheles mosquito blood meals reveals nonrandom human host selection: implications for human-to-mosquito Plasmodium falciparum transmission’’, Malaria Journal 22 (2023) 115. https://doi.org/10.1186/s12936-023-04541-2.

[8] W. Takken, D. Charlwood & S. W. Lindsay, ‘‘The behaviour of adult Anopheles gambiae, sub-Saharan Africa’s principal malaria vector, and its relevance to malaria control: a review’’, Malaria Journal 23 (2024) 161. https://doi.org/10.1186/s12936-024-04982-3.

[9] J. H. Ellwanger, J. D. C. Cardoso & J. A. B. Chies, ‘‘Variability in human attractiveness to mosquitoes’’, Current Research in Parasitology and Vector-Borne Diseases 1 (2021) 100058. https://doi.org/10.1016/j.crpvd.2021.100058.

[10] J. E. Ayodele, C. T. Omisakin, E. T. Oyedele & F. Kolawole, ‘‘Evaluation of severity of malaria infection and effect of anti-malaria drugs on gender differences using blood cell lines parameters’’, American Journal of Medical Sciences and Medicine 2 (2014) 89. http://pubs.sciepub.com/ajmsm/2/5/2.

[11] O. M. Akanbi, J. A. Badaki, O. Y. Adeniran & O. O. Olotu, ‘‘Effect of blood group and demographic characteristics on malaria infection, oxidative stress and haemoglobin levels in South Western Nigeria’’, African Journal of Microbiology Research 4 (2010) 877. https://academicjournals.org/article/article1380211377_Akanbi%20et%20al.pdf

[12] A. Cernetich, L. Garver & S. Jedlicka, ‘‘Involvement of gonadal steroids and gamma interferon in sex differences in responses to blood stage malaria infection’’, Infection and Immunity 74 (2006) 222. https://doi.org/10.1128/IAI.00008-06.

[13] J. Krucken, M. A. Dkhil & J. V. Braun, ‘‘Testosterone suppresses protective responses of the liver to blood stage malaria’’, Infection and Immunity 73 (2005) 436. https://doi.org/10.1128/IAI.73.1.436-443.2005.

[14] M. Zuk & A. K. McKean, ‘‘Sex differences in parasite infections: Patterns and processes’’, International Journal for Parasitology 26 (1996) 1009. https://doi.org/10.1016/S0020-7519(96)80001-4.

[15] P. Duve, S. Charles, J. Munyakazi, R. Lühken & P. Witbooi, ‘‘A mathematical model for malaria disease dynamics with vaccination and infected immgrants", Mathematical Biosciences and Engineering 21(2024) 1082. https://doi.org/10.3934/mbe.2024045.

[16] Dorsey, ‘‘Gender difference in the incidence of malaria diagnosed at public Health Organization, Geneva, Switzerland, 2019’’, Tanzania Journal of Science 47 (2021) 953. https://doi.org/10.4314/tjs.v47i3.7.

[17] A. Kalula, E. Mureithi, T. Marijani & I. Mbalawata, ‘‘Optimal control and cost effectiveness analysis of age-structured malaria model with asymptomatic carrier and temperature variability’’, Journal of Biological Dynamics 17 (2023) 2199766. https://doi.org/10.1080/17513758.2023.2199766.

[18] G. T. Azu-Tungmah, F. T. Oduro & G. A. Okyere, ‘‘Analysis of an agestructuredmalariamodelincorporatinginfantsandpregnantwomen’’, Journal of Advances in Mathematics and Computer Science 30 (2019) 1. https://doi.org/10.9734/JAMCS/2019/46649.

[19] P. K. Robert, J. I. Irunde & A. P. Mtunya, ‘‘Modelling malaria dynamics in children under five years, pregnant women, and the influence of temperature’’, Tanzania Journal of Science 50 (2024) 835. https://doi.org/10.4314/tjs.v50i4.10.

[20] P. van den Driessche & J. Watmough, ‘‘Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission’’, Mathematical Biosciences 180 (2002) 29. https://doi.org/10.1016/S0025-5564(02)00108-6.

[21] G. H. Golub & C. F. Van Loan, ‘‘Matrix computations’’, 4th ed. Baltimore, MD, USA: Johns Hopkins University Press (2013).

[22] J. P. Keener, ‘‘Principles of Applied Mathematics: Transformation and Approximation’’, Boca Raton, FL, USA: CRC Press (2000).

[23] C. M. Bender & S. A. Orszag, ‘‘Advanced mathematical methods for scientists and engineers’’, New York, NY, USA: McGraw-Hill (1978).

[24] S. Olaniyi & O. S. Obabiyi, ‘‘Mathematical Model for Malaria Transmission Dynamics in Human and Mosquito Population with Nonlinear Forces of Infection’’, International Journal of Pure and Applied Mathematics 88 (2013) 125. https://doi.org/10.12732/ijpam.v88i1.10.

[25] A. T. Haringo, L. L. Obsu & F. K. Bushu, ‘‘Impact of asymptomatic infections on malaria transmission dynamics’’, Infectious Disease Modelling 10 (2025) 1456. https://doi.org/10.1016/j.idm.2025.07.012.

[26] World Health Organization, ‘‘World Health Statistics 2023: Monitoring Health for the SDGs’’, Geneva, Switzerland: World Health Organization (2023). http://who.int/publications/i/item/9789240074323.

[27] National Population Commission (NPC) and ICF, ‘‘Nigeria Demographic and Health Survey 2018’’, NPC and ICF (2019). https://www.dhsprogram.com/pubs/pdf/fr359/fr359.pdf.

[28] National Malaria Elimination Programme [Nigeria] and ICF International, ‘‘Nigeria Malaria Indicator Survey 2015’’, Abuja, Nigeria, and Rockville, MD, USA: NMEP and ICF International (2016). https://dhsprogram.com/pubs/pdf/MIS20/MIS20.pdf.

[29] D. L. Smith, J. Dushoff & F. E. McKenzie, ‘‘The risk of a mosquito-borne infection in a heterogeneous environment’’, PLoS Biology 2 (2004) e368. https://doi.org/10.1371/journal.pbio.0020368.

[30] World Health Organization, ‘‘WHO guidelines for malaria’’, Geneva, Switzerland: World Health Organization (2019). https://www.who.int/publications/i/item/guidelines-for-malaria. Accessed March 1, 2026.

[31] J. Okiring, A. Epstein, J. F. Namuganga, E. V. Kamya, I. Nabende, M. Nassali, A. Sserwanga, S. Gonahasa, M. Muwema, S. M. Kiwuwa, S. G. Staedke, M. R. Kamya, J. I. Nankabirwa, J. Briggs & P. Jagannathan, ‘‘Gender difference in the incidence of malaria diagnosed at public health facilities in Uganda’’, Malaria Journal 21 (2022) 1. https://doi.org/10.1186/s12936-022-04046-4.

[32] J. C. Nmor, ‘‘Health care seeking behavior for malaria among laboratory confirmed outpatients in a rural community, Southern Nigeria’’, International Journal of Tropical Disease & Health 6 (2015) 8. https://doi.org/10.9734/IJTDH/2015/11814.

[33] National Malaria Elimination Programme [Nigeria], ‘‘National Malaria Strategic Plan 2021-2025: A path to malaria pre-elimination in Nigeria’’, (2020). https://mesamalaria.org/wp-content/uploads/2024/07/NATIONAL-MALARIA-STRATEGIC-PLAN-Nigeria-2021-2025-Final.pdf