Analysis of SIS-SI Stochastic Model with CTMC on the Spread of Malaria Disease
DOI:
https://doi.org/10.5614/j.math.fund.sci.2021.53.2.1Keywords:
basic reproduction number, expected time to reach disease-free equilibrium, malaria disease, outbreak probability, quasi-stationary distributionAbstract
Various mathematical models have been developed to describe the transmission of malaria disease. The purpose of this study was to modify an existing mathematical model of malaria disease by using a CTMC stochastic model. The investigation focused on the transition probability, the basic reproduction number (R0), the outbreak probability, the expected time required to reach a disease-free equilibrium, and the quasi-stationary probability distribution. The population system will experience disease outbreak if R0>1, whereas an outbreak will not occur in the population system if R0?1. The probability that a mosquito bites an infectious human is denoted as k, while ?is associated with human immunity. Based on the numerical analysis conducted, k and ?have high a contribution to the distribution of malaria disease. This conclusion is based on their impact on the outbreak probability and the expected time required to reach a disease-free equilibrium.
References
WHO, World Malaria Report 2017, World Health Organization, https://www.who.int/publications/i/item/9789241565523, Nov. 2017.
WHO, World Malaria Report 2018, World Health Organization, https://www.who.int/publications/i/item/9789241565653, Nov. 2018.
Mandal, S., Sakar, R.R. & Sinha, S., Mathematical Models of Malaria, Malaria Journal, 10(202), pp. 1-19, 2011.
Nuraini, N., Tasman, H., Soewono, E. & Sidarto, K.A., A With-in Host Dengue Infection Model with Immune Response, Mathematical and Computer Modelling, 49, pp. 1148-1155, 2008.
Nuraini, N., Soewono, E. & Sidarto, K.A., A Mathematical Model of Dengue Internal Transmission Process, J. Indones. Math. Soc, 13(1), pp. 123-132, 2006.
Bakary, T., Boureima, S. & Sado, T., A Mathematical Model of Malaria Transmission in A Periodic Environment, Journal of Biological Dynamics, 12(1), pp. 400-432, 2018.
Mbogo, R.W., Luboobi, L.S. & Odhiambo, J.W., A Stochastic Model for Malaria Transmission Dynamics, Journal of Applied Mathematics, 1-13, 2018.
Gahungu, P., Wahid, B.K.A., Oumarou, A.M., & Bisso, S., Stochastic Age-structured Malaria Transmission Model, Journal of Applied Mathematics & Bioinformatics, 7(2), pp. 29-50, 2017.
Wanduku, D. The Stochastic Extinction and Stability Conditions for Nonlinear Malaria Epidemics, Mathematical Biosciences and Engineering, 16(5), pp. 3771-3806, 2019.
Ross, R., An Application of the Theory of Probabilities to the Study of A Priori Pathometry Part I, Proc R Soc, 92(638), pp. 204-230, Feb.1916.
Macdonald, G., The Analysis of Equilibrium in Malaria, Trop. Dis. Bull, 49(9), pp. 813-829, 1952.
Kingsolver, J.G., Mosquito Host Choice and the Epidemiology of Malaria, Am. Nat, 130(6), pp. 811-827, 1987.
Lacroix, R., Mukabana, W.R., Gouagna, L.C. & Koella, J.C., Malaria Infection Increases Attractiveness of Humans to Mosquitos, Biology, 3(9), pp. 1590-1593, 2005.
Chamchod, F. & Britton, N.F., Analysis of a Vector-bias Model of Malaria Transmission, Mathematical Biology, 73(1), pp. 639-657, 2010.
Mading, M. & Yunarko, M., Immune Response Against Malaria Parasites Infection, Vektor Penyakit, 8(2), pp. 45-52, 2014. (Text in Indonesian and Abstract in English).
Anderson, R.M. & May, R.M., Infectious Disease of Humans: Dynamics and Control, 1st ed., England (UK): Oxford University Press, 1992.
Filipe, J.A.N., Riley, E.M., Drakeley, C., Sutherland, C.J. & Ghani, A.C., Determination of the Processes Driving the Acquisition of Immunity to Malaria Using a Mathematical Transmission. PLOS Computational Biology, 3(12), pp. 225, 2007.
Gupta, S., Swinton, J. & Anderson, R.M., Theoretical Studies of the Effect of Heterogeneity in the Parasite Population on the Transmission Dynamics of Malaria. Biological Sciences, 256, pp. 231-238, 1994.
Allen, L.J.S. & Lahodny, G.E., Extinction Thresholds in Deterministic and Stochastic Epidemic Models, J Biological Dynamics, 6(2), pp. 590-611, 2016.
Nasell, I., On the Quasi-stationary Distribution of the Ross Malaria Model, Mathematical Biosciences, 107(1), pp. 187-207, 1991.
