A STOCHASTIC DIFFERENTIAL EQUATION (SDE) BASED MODEL FOR THE SPREAD OF TUBERCULOSIS

Authors

Keywords:

Deterministic model, Stochastic model, Stochastic Differential Equation, Tuberculosis

Abstract

Understanding dynamics of an infectious disease helps in designing appropriate strategies for containing its spread in a population. In this work, a deterministic and stochastic model of the transmission dynamics of Tuberculosis is developed and analyzed. The models involve the Susceptible, Exposed, Infectious and Recovered individuals. We computed the basic reproduction number  and showed that for, the disease-free equilibrium is globally asymptotically stable. The resulting deterministic model was transformed into an equivalent stochastic model resulting in stochastic differential equation. The drift coefficient, the covariance matrix and the diffusion matrix were determined using the method proposed by Allen et al. (2008).

Dimensions

Adetunde, I.A. (2008). The mathematical models of the dynamical behaviour of tuberculosis disease in the upper region of the northern part of Ghana: a case study of Bawku. Current Research in Tuberculosis, 1(1): 1-6.

Allen E.J. Allen L.J.S., Arciniega A., and Greenwood P. E. (2008). Construction of Equivalent Stochastic Differential Equation Models. Stochastic analysis and applications, 26:274-297. DOI: 10.1080/07362990701857129

Allen, E. (2007) Modeling with Itô Stochastic Differential Equations. Vol. 22, Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5953-7

Allen, L.J.S. (2017). A primer on stochastic epidemic model formulation, numerical simulation, and analysis Infectious Disease Modelling. Dio: 10.1016/j.idm.2017.03.001.

Ibrahim, M.O., Ejieji, C.N. and Egbetade, S.A. (2013). A mathematical model for the epidemiology of tuberculosis with estimate of the basic reproduction number. IOSR Journal of Mathematics, 5(5): 46-52.

Kalu, A.U., and Inyama, S.C. (2012). Mathematical model of the role of vaccination and treatment on the transmission dynamics of tuberculosis. Gen. Math. Notes, 11(1): 10-23.

Kipruto, H., Mung’atu, J., Ogila, K.O., and Mwalili, S. (2015). Application of Deterministic and Stochastic processes to Model Evolving Epidemiology of Tuberculosis in Kenya. International Journal of Science and Research, vol.4 Issue 9.

Lamba, H., Mattingly, J.C., and Stuart, A.M. (2007). An adaptive Euler- Maruyama scheme for SDE: convergence and stability. IMA J Num Anal, 27:479-506.

Mbakoma, M., and Oukouomi, S.C. (2017). Mathematical Analysis of a Stochastic Tuberculosis Model. Journal of Analysis and Applications,15(1):21-50.

Olabode D, Cup J. and Fisher A. (2021), Deterministic and stochastic models for the epidemic dynamics of COVID-19 in Wuhan, China, Mathematical Biosciences & Engineering 18(1):950-967, DOI: 10.3934/mbe.2021050

Omame, A., Umana, R. A., and Inyama, S.C. (2015). Stochastic model and analysis of the dynamics of tuberculosis. Researchjournal’s Journal of Mathematics, vol.2 No.5.

Umana, R.A., Omame, A., and Inyama, S.C. (2016). Deterministic and stochastic models of the dynamics of Drug resistant Tuberculosis. Futo journal series, 2:173-194.

Waller, H.T., Geser, A., and Anderson, S. (1962). The use of mathematical models in the study of the epidemiology of tuberculosis. Am. J. Publ. Health, 52: 1002-1013.

Wang (2019), Deterministic and stochastic models for the epidemic dynamics of COVID-19 in Wuhan, China, Mathematical Biosciences and Engineering, AIMS Press, 2021, Volume 18 (1), 950-967. doi: 10.3934/mbe.2021050

World Health Organization (2007). Vaccine preventable diseases: Monitoring system, Global summary. www.who.org.

Published

27-10-2023

How to Cite

A STOCHASTIC DIFFERENTIAL EQUATION (SDE) BASED MODEL FOR THE SPREAD OF TUBERCULOSIS. (2023). FUDMA JOURNAL OF SCIENCES, 7(5), 9-17. https://doi.org/10.33003/fjs-2023-0705-1990

Issue

Vol. 7 No. 5 (2023): FUDMA Journal of Sciences - Vol. 7 No. 5

Section

Research Articles
Copyright & Licensing

How to Cite

A STOCHASTIC DIFFERENTIAL EQUATION (SDE) BASED MODEL FOR THE SPREAD OF TUBERCULOSIS. (2023). FUDMA JOURNAL OF SCIENCES, 7(5), 9-17. https://doi.org/10.33003/fjs-2023-0705-1990

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