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Modeling the Stochastic Dynamics of Influenza Epidemics with Vaccination Control, and the Maximum Likelihood Estimation of Model Parameters

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Mathematical Modelling in Health, Social and Applied Sciences

Part of the book series: Forum for Interdisciplinary Mathematics ((FFIM))

Abstract

This study presents a family of stochastic models for the dynamics of influenza in a closed human population. We consider treatment for the disease in the form of vaccination and incorporate the periods of effectiveness of the vaccine and infectiousness for the individuals in the population. Our model is a SVIR model, with trinomial transition probabilities, where all individuals who recover from the disease acquire permanent natural immunity against the strain of the disease. A special SVIR model in the stochastic family based on correlated vaccination and infection probabilities at any instant is presented. The methods of maximum likelihood and expectation–maximization algorithm are applied to find estimates for the parameters of the chain. Moreover, estimators for some special epidemiological control parameters, such as the basic reproduction number, are computed. A numerical simulation example is presented to find the MLE of the parameters of the model.

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Notes

  1. 1.

    The cumulative sums in (5.48)–(5.49) were calculated using “for-loops” in R.

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Acknowledgements

This work was completed during the graduate studies of Cameron Newman, Omotomilola Jegede and Mymuna Monem in the department of Mathematical Sciences of Georgia Southern University (GSU) in 2018–2019 academic year, supervised by Dr. Wanduku. Mymuna Monem was part of the general group discussions.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Georgia Southern University, 65 Georgia Ave., Room 3042, Statesboro, GA, 30460, USA

    Divine Wanduku, C. Newman, O. Jegede & B. Oluyede

Authors
  1. Divine Wanduku
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  2. C. Newman
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  3. O. Jegede
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  4. B. Oluyede
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Corresponding author

Correspondence to Divine Wanduku .

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Editors and Affiliations

  1. Department of Mathematics, Gauhati University, Guwahati, Assam, India

    Hemen Dutta

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Wanduku, D., Newman, C., Jegede, O., Oluyede, B. (2020). Modeling the Stochastic Dynamics of Influenza Epidemics with Vaccination Control, and the Maximum Likelihood Estimation of Model Parameters. In: Dutta, H. (eds) Mathematical Modelling in Health, Social and Applied Sciences. Forum for Interdisciplinary Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-15-2286-4_2

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