Epidemic Dynamics and Adaptive Vaccination Strategy: Renewal Equation Approach
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Epidemic Dynamics and Adaptive Vaccination Strategy: Renewal Equation Approach Aubain Nzokem1
· Neal Madras1
Received: 1 April 2020 / Accepted: 1 September 2020 © Society for Mathematical Biology 2020
Abstract We use analytical methods to investigate a continuous vaccination strategy’s effects on the infectious disease dynamics in a closed population and a demographically open population. The methodology and key assumptions are based on Breda et al. (J Biol Dyn 6(Sup2):103–117, 2012). We show that the cumulative force of infection for the closed population and the endemic force of infection in the demographically open population can be reduced significantly by combining two factors: the vaccine effectiveness and the vaccination rate. The impact of these factors on the force of infection can transform an endemic steady state into a disease-free state. Keyword Force of infection, Cumulative force of infection, Scalar-renewal equation, Per capita death rate, Lambert function, Adaptive vaccination strategy
1 Introduction The paper of Kermack and McKendrick (1927) is one of the best known contributions to the mathematical theory of epidemic modeling. The paper provides the condition of outbreak and the final size equation in a closed population setting. One of the key features of Kermack and McKendrick (1927) was to introduce an age of infection model. In such a model, the general infectivity function ( A(τ )) of an individual is considered and depends on the time (τ ) elapsed since the infection took place. Kermack and McKendrick’s framework encompasses a wide family of epidemic models; Breda et al. (2012) have illustrated the generalization by providing the following age infection functions for standard SIR and SEIR models.
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Aubain Nzokem [email protected] Neal Madras [email protected]
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Department of Mathematics and Statistics, York University, Toronto, ON, Canada 0123456789().: V,-vol
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A. Nzokem, N. Madras
Fig. 1 The transfer diagram of the model: the force of infection function (the probability per unit of time that a susceptible becomes infected) t → F(t); the rate of vaccination function t → φ(t); S (unvaccinated susceptible); V (vaccinated susceptible); θ (vaccine parameter (0 ≤ θ ≤ 1)); I (infected population); B (constant birth rate); μ (constant per capita death rate)
A(τ ) = βe−ατ ⇐⇒ SIR γ (e−ατ − e−γ τ ) ⇐⇒ SEIR A(τ ) = β γ −α
(1.1)
The paper of Breda et al. (2012) “On the formulation of epidemic models (an appraisal of Kermack and McKendrick)” revised Kermack and McKendrick’s paper and produced the same results, but the method used was different. In fact, Breda et al. (2012) considered the force of infection as a result of a nonlinear scalar-renewal equation, and they analyzed the cumulative force of infection or the simple force of infection at the disease-free equilibrium and the endemic equilibrium. For related work of interest, see (Kermack and McKendrick (1932); Diekmann and Heesterbeek (2013) Meehan et al. (2017); Andersson and Britton (2000); Huppert and Katri
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