Modelling the co-infection of malaria and zika virus disease

Emmanuel Chidiebere Duru; Godwin Christopher Ezike Mbah; Michael Chimezie Anyanwu; Nicholas Topman Nnamani

doi:10.46481/jnsps.2024.1938

Authors

Emmanuel Chidiebere Duru
[email protected]

Department of Mathematics, Michael Okpara University of Agriculture, P.M.B. 7267 Umudike, Nigeria
Godwin Christopher Ezike Mbah
Department of Mathematics, University of Nigeria, P.M.B. 3147 Nsukka, Nigeria
Michael Chimezie Anyanwu
Department of Mathematics, Michael Okpara University of Agriculture, P.M.B. 7267 Umudike, Nigeria
Nnamani Nicholas Topman
Department of Mathematics, University of Nigeria, P.M.B. 3147 Nsukka, Nigeria

Keywords:

Malaria , Zika virus, Vaccination, Sterile-insect technology

Abstract

In this work, a new model for the co-infection of malaria and zika virus disease incorporating vaccination, treatment and vector control using sterile-insect technology (SIT) is formulated. The importance of this study is to highlight the possibility of the co-infection of humans with malaria and zika virus disease in any environment where both diseases co-circulate. Also, to suggest a new and comprehensive method for controlling the individual diseases and their co-infection. Through stability analysis, we showed that the disease-free equilibrium, (DFE) point of the co-infection model is locally asymptotically stable when the basic reproduction numbers, Rmz is less than one, and unstable otherwise. But, the DFE failed to be globally stable when Rmz < 1 which is an indication of existence of backward bifurcation in the model. This shows that bringing down the reproduction number, Rmz to less than one is not enough to eradicate the co-infection of the two diseases. Furthermore, it is shown that the two diseases have positive impact on the spread of each other, which could be attributed to misdiagnoses of one disease as the other. We also showed that effective treatment of infectious humans, increasing the rate of vaccination and employing sterile-insect technique to control the vectors significantly helped to control the individual diseases as well as the co-infection. From the results obtained in the study, it can be concluded that effective control of malaria and zika virus disease requires measures that will control their spread in both human and mosquito populations.

Dimensions

REFERENCES

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

S. I. Onah, O. C. Collins, C. Okoye, & G. C. E. Mbah, “Dynamics and control measures for malaria using a mathematical epidemiological model”, Electronic Journal of Mathematical Analysis and Applications 7 (2019) 65. https://fcag-egypt.com/Journals/EJMAA/.

World Health Organization, “World Health Organization fact sheet on Malaria”, [Online]. https://www.who.int/news-room/fact-sheets/details/malaria.

O. M. Ogunmiloro, “Mathematical modeling of the co-infection dynamics of malaria-toxoplasmosis in the tropics”, Biometrical Letters 56 (2019) 139. https://doi.org/10.2478/bile-2019-0013.

D. Aldila, “Dynamical analysis on a malaria model with relapse preventive treatment and saturated fumigation”, Computational and Mathematical Methods in Medicine 2022 (2022) 1135452. https://doi.org/10.1155/2022/1135452.

V. S. Moorthy & F. Binka, “R21/matrix-m: a second malaria vaccine?,? The Lancet 397 (2021) 1782. https://doi.org/10.1016/s0140-6736(21)01065-5.

J. Lamwong & P. Pongsumpun, “Age Structural model of zika virus”, International Journal of Modelling and Optimization 8 (2018) 17. https://doi.org/10.7763/IJMO.2018.V8.618.

M. C. Anyanwu, G. C. E. Mbah & E. C. Duru, “On mathematical model for zika virus disease control with wolbachia-infected mosquitoes”, Abacus Mathematics Science Series 47 (2020) 35. https://www.man-nigeria.org.ng/ABA-SCI-2020-4-1.

R. Rakkiyappan, V. P. Latha & F. A. Rihan, “A fractional-order model for zika virus infection with multiple delays”, Wiley Hindawi Complexity 20 (2019) 1. https://doi.org/10.1155/2019/4178073.

World Health Organization, “The history of Zika virus”. (2016). [Online]. https://www.who.int/news-room/feature-stories/detail/the-history-of-zika-virus .

CDC Responds to ZIKA, “Zika Virus: Information for Clinicians”. [Online]. https://www.cdc.gov/zika/pdfs/clinicianppt.

T. A. Perkins, A. S. Siraj, C. W. Ruktanonchai, M. U. G. Kraemer & A. J. Tatem, “Model based projections of Zika virus infections in child bearing women in the Americas”, Nature Microbiology 1 (2016) 16126. https://doi.org/10.1038/nmicrobiol.2016.126.

G. Calvet, R. S. Aguiar, A. S. O. Melo, et al., “Detection and sequencing of zika virus from amniotic fluid of fetuses with microcephaly in Brazil: a case study?, Lancet Infectious Diseases 16 (2016) 653. https://doi.org/10.1016/S1473-3099(16)00095-5.

S. B. Boret, R. Escalante & M. Villasana, “Mathematical modelling of Zika virus in Brazil”, Journal of LaTeX Templates 14 (2021) 1. https://hal.archives-ouvertes.fr/hal-03505850.

M. J. B. Vreysen, A. S. Robinson & J. Hendrichs, Area-wide control of insect pests, from research to field implementation, Springer, Dordrecht, the Netherlands, 2007. https://doi.org/10.1007/978-1-4020-6059-5.

V. A. Dyck, J. Hendrichs & A. S. Robbinson, “Sterile insect technique: principles and practice in area-wide integrated pest management”, 2nd Edition, Boca Raton, FL, CRC Press, 2021, pp. 1–799. https://doi.org/10.1201/9781003035572.

W. Atokolo & G. C. E. Mbah, “Modeling the control of Zika virus vector population using the sterile insect technology”, Journal of Applied Mathematics 20 (2020) 1. https://doi.org/10.1155/2020/6350134.

R. Anguelov, Y. Dumont & J. Lubuma, “Mathematical modeling of sterile insect Technology for control of Anopheles mosquito?, Computers and Mathematics with Applications 64 (2012) 374389. https://doi.org/10.1016/j.camwa.2012.02.068.

UN FAO, “Tsetse fly eradicated on the Island of Zanzibar”. [Online]. https://www.fao.org/in-action-Tsetse-fly-eradicated-on-the-Island-of-Zanzibar/

UN FAO, “Senegal celebrates first victory against tsetse fly eradication”. [Online]. https://www.fao.org/in-action-Senegal-celebrates-first-victory-against-tsetse-fly/eradication/en.

W. Klassen, “Introduction: development of the sterile insect technique for african malaria vectors”, Malaria Journal 8 (2009) 11. https://doi.org/10.1186/1475-2875-8-S2-I1.

K. Dietz, L. Molineux & A. Thomas, “A malaria model tested in the African Savannah”, Bull. World Health Organization 50 (1974) 347. https://www.researchgate.net/publication/18551729-A-Malaria-Model-Tested-in-the-African-Savannah.

D. Omale, J. P. Omale & W. Atokolo, “Mathematical modeling on the transmission dynamics and control of malaria with treatment within a population”, Academic Journal of Statistics and Mathematics (AJSM) 6 (2020) 1. https://www.researchgate.net/publication/346021061 .

Fatmawati, F. F. Herdicho, Windarto, W. Chukwu & H. Tasman, “An optimal control of malaria transmission model with mosquito seasonal factor”, Results in Physics 25 (2021) 1. https://doi.org/10.1016/j.rinp.2021.104238.

N. Chitnis, A. Schapira, C. Schindler, M. A. Penny & T. A. Smith, “Mathematical analysis to prioritise strategies for malaria elimination”, Journal of Theoretical Biology 455 (2018) 118. https://doi.org/10.1016/j.jtbi.2018.07.007.

A. Nwankwo & D. Okuonghae, “Mathematical assessment of the impact of different microclimate conditions on malaria transmission dynamics”, Mathematical Biosciences and Engineering 16 (2019) 1414. https://doi.org/10.3934/mbe.2019069.

S. Oke, M. Adeniyi, M. M. Ojo & M. B. Matadi, “Mathematical modelling of malaria, disease with control strategy”, Communications in Mathematical Biology and Neuroscience 43 (2020) 1. https://doi.org/10.28919/cmbn/4513.

O. C. Collins & K. J. Duffy, “A mathematical model for the dynamics and control of malaria in Nigeria”, Infectious Disease Modelling 7 (2022) 728. https://doi.org/10.1016/j.idm.2022.10.005.

P. J. Witbooi, S. M. Vyambwera, G. J. Van Schalkwyk & G. E. Muller, “Stability and control in a stochastic model of malaria population dynamics”, Advances in Continuous and Discrete Models 2023 (2023) 45. https://doi.org/10.1186/s13662-023-03791-3.

E. Bonyah & K. O. Okosun, “Mathematical modeling of Zika virus. Asian Pacific Journal of Tropical Disease”, Asian Pacific Journal of Tropical Disease 6 (2016) 673. https://doi.org/10.1016/S2222-1808(16) 61108-8.

H. Gwalani, S. M. Alshammari & A. R. Mikler, “An interactive model for vector borne diseases: A simulation for Zika in French polynesia”, (2016) [Online]. https://digital.library.unt.edu/ark:/67531/metadc919588 .

H. Nishiura, R. Kinoshita, K. Mizumoto, Y. Yasuda & K. Nah, “Transmission potential of Zika virus infection in the South pacific”, International Journal of Infectious Diseases 45 (2016) 95. https://doi.org/10.1016/j.ijid.2016.02.017.

C. Oleson & M. Artzrouni, “A patch model for the transmission dynamics of Zika Virus from Rio de Janeiro to miami during carnival and the Olympics”, SIAM Undergraduate Research Online 9 (2016) 1. https://www.siam.org/students/siuro/vol9/S01542.pdf.

N. K. Goswami, A. K. Srivastav, M. Ghosh & B. Shanmukha, “Mathematical modeling of Zika virus disease with nonlinear incidence and optimal control”, IOP Conf. Series: Journal of Physics Conference Series 1000 (2018) 1. https://doi.org/10.1088/1742-6596/1000/1/012114.

S. Olaniyi, “Dynamics of Zika virus model with nonlinear incidence and optimal control strategies”, International Journal of Applied Mathematics and Information Sciences 12 (2018) 969. http://dx.doi.org/10.18576/amis/120510.

P. Andayani, L.R. Sari, A. Suryanto & I. Darti, “Numerical study for Zika virus transmission with beddington-deangelis incidence rate”, Far East Journal of Mathematical Sciences, FJMS, Prayagraj India 111 (2019) 145. http://dx.doi.org/10.17654/MS111010145.

G. Gonzalez-Parra & T. Benincasa, “Mathematical modeling and numerical simulations of zika in colombia considering mutation”, Journal of Mathematical Computation and Simulation 163 (2019) 1. https://doi.org/10.3390/computation8030076.

J. Amaoh-Mensah, I. K. Dontwi & E. Bonyah, “Stability analysis of Zika ? malaria co-infection model for malaria endemic region”, Journal of Advances in Mathematics and Computer Science 26 (2018) 1. https://doi.org/10.9734/JAMCS/2018/37229.

A. A. Otu, U. A. Udoh, O. I. Ita, J. P. Hicks, I. Ukpeh & J. Walley, “Prevalence of Zika and malaria in patients with fever in secondary healthcare facilities in south-eastern Nigeria”, Tropical Doctor 50 (2020) 22. https://doi.org/10.1177/0049475519872580.

P. A. Mac, A. Kroeger, T. Daehne, C. Anyaike, R. Velayudhan & M. Panning, “Zika, flavivirus and malaria antibody cocirculation in Nigeria”, Tropical Medicine and Infectious Disease 8 (2023) 171. https://doi.org/10.3390/tropicalmed8030171.

A. Sow, C. Loucoubar, D. Diallo, O. Faye, Y. Ndiaye, C. S. Senghor, A. T. Dia, O. Faye, S. C. Weaver, M. Diallo, D. Malvy & A. Alpha Sall, “Concurrent malaria and arbovirus infections in Kedougou, southeastern Senegal ”, Malaria Journal 15 (2016) 47. https://doi.org/10.1186/s12936-016-1100-5 .

M. M. Baba, A. Ahmed, S. Y. Jackson & B. S. Oderinde, “Cryptic Zika virus infections unmasked from suspected malaria cases in Northeastern Nigeria”, PLoS ONE 18 (2023) e0292350. https://doi.org/10.1371/journal.pone.0292350.

Medical Park Hospital. (2022). disease-and-treatment/malaria#.[Online].

Medical News Today. (2022). where-is-it-most-common.https://www.medicalnewstoday.com/articles/150670# [Online]. https://www.medparkhospital.com/en-US/

D. Borghino, Malaria vaccine candidate shown to prevent thousands of cases. (2022). [Online]. http://www.gizmag.com/malaria-vaccine-candidate-trial/37205.

B. Bolaji, U. B. Odionyenma, B. I. Omede, P. B. Ojih & A. A. Ibrahim, “Modelling the transmission dynamics of Omicron variant of COVID-19 in densely populated city of Lagos in Nigeria”, Journal of the Nigerian Society of Physical Sciences 5 (2023) 1055. https://doi.org/10.46481/jnsps.2023.1055 .

R. Musa, O. J. Peter & F. A. Oguntolu, “A non-linear differential equation model of COVID-19 and seasonal influenza co-infection dynamics under vaccination strategy and immunity waning”, Healthcare Analytics 4 (2023) 100240. https://doi.org/10.1016/j.health.2023.100240.

B. I. Omede, O. J. Peter, W. Atokolo, B. Bolaji & T. A. Ayoola, “A mathematical analysis of the two-strain tuberculosis model dynamics with exogenous re-infection”, Healthcare Analytics 4 (2023) 1. https://doi.org/10.1016/j.health.2023.100266.

S. Ajao, I. Olopade, T. Akinwumi, S. Adewale & A Adesanya, “Understanding the transmission dynamics and control of HIV infection: A Mathematical Model Approach”, Journal of the Nigerian Society of Physical Sciences 5 (2023) 1389. https://doi.org/10.46481/jnsps.2023.1389.

P. Van Den Driessche & J. Watmough, “Reproduction Numbers and Sub-threshold Endemic Equilibria for Compartmental Models of Disease Transmission”, Mathematical Biosciences 180 (2002) 29. https://doi.org/10.S0025-5564(02)00108-6 .

O. Diekmann, J. A. Hesterbeek & M. G. Roberts, “Construction of next generation matrices for compartmental models in epidemics”, Journal of the Royal Society of Biology, Interface 7 (2010) 875. https://doi.org/10.1098/rsif.2009.0386 .

K. A. Tijani, C. E. Madubueze & R. I. Gweryina, “Typhoid fever dynamical model with cost-effective optimal control”, Journal of the Nigerian Society of Physical Sciences 5 (2023) 1579. https://doi.org/10.46481/jnsps.2023.1579.

J. H. Kim & Y. J. Song, “On stability of a polynomial”, Journal of Applied Mathematics and Informatics 36 (2018) 231. https://doi.org/10.14317/jami.2018.231.

C. Castillo-Chavez, Z. Feng & W. Huang, On the computation of R0 and its role on Global stability, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, Cornell University, Ithaca, NY 14853, USA, 2001. https//www.researchgate.net/publication/286207230-On-the-computation-of-$R_0$-and-its-role-on-Global-stability .

E. Dantas, M. Tosin & A. Cunha, Jr, “Calibration of a SEIR epidemic model to describe Zika Virus outbreak in Brazil”. (2017) [Online]. https://hal.archives-ouvertes.fr/hal-01456776v2.