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

  • Divine WandukuEmail author
  • C. Newman
  • O. Jegede
  • B. Oluyede
Part of the Forum for Interdisciplinary Mathematics book series (FFIM)


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.


Influenza epidemics Chain binomial model MLE method EM algorithm Basic reproduction number 



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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Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  • Divine Wanduku
    • 1
    Email author
  • C. Newman
    • 1
  • O. Jegede
    • 1
  • B. Oluyede
    • 1
  1. 1.Department of Mathematical SciencesGeorgia Southern UniversityStatesboroUSA

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