The Law of Mass Action in Epidemiology

Wilson, Edwin Bidwell; Worcester, Jane

doi:10.1073/pnas.31.1.24

Cited by 80 publications

(57 citation statements)

References 3 publications

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“…Let S(t) be the number of susceptible individuals at time t, I (t) be the number of infective individuals at time t. In 1945, Wilson and Worcester were the first to consider the nonlinear incidence rate βSI q [7,8]. In 1969, Severo considered a more general form βS p I q , where q < 1, but did not investigate behavior of the epidemic model in detail [9].…”

Section: Introductionmentioning

confidence: 99%

Analysis of stability and bifurcation for an SEIV epidemic model with vaccination and nonlinear incidence rate

Zhou

Cui

2010

Nonlinear Dyn

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In this paper, an SEIV epidemic model with vaccination and nonlinear incidence rate is formulated. The analysis of the model is presented in terms of the basic reproduction number R 0 . It is shown that the model has multiple equilibria and using the center manifold theory, the model exhibits the phenomenon of backward bifurcation where a stable diseasefree equilibrium coexists with a stable endemic equilibrium for a certain defined range of R 0 . We also discuss the global stability of the endemic equilibrium by using a generalization of the Poincaré-Bendixson criterion. Numerical simulations are presented to illustrate the results.

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Section: Introductionmentioning

confidence: 99%

Analysis of stability and bifurcation for an SEIV epidemic model with vaccination and nonlinear incidence rate

Zhou

Cui

2010

Nonlinear Dyn

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“…Researchers should therefore be careful in formulating discrete-time models with mass-action (see Diekmann & Heesterbeek 2000). proaches appeared together in a detailed analysis of their differences and similarities was in work by Burke (1942, 1943); they later appeared in the papers by Wilson and Worcester (1945). Unknown to all these authors was the work by En'ko in Russian in the late nineteenth century who published, also in connection to measles, another alternative mathematical model for discrete time (described earlier).…”

Section: A Science Taking Flightmentioning

confidence: 98%

The Law of Mass-Action in Epidemiology: A Historical Perspective

Heesterbeek¹

2005

Ecological Paradigms Lost

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The law of mass-action as a paradigm for describing the contact rate of individuals in a population has been in widespread and heavy use in ecology and epidemiology for almost 100 years. The law roughly states that the rate at which individuals of two types, X and Y, meet is proportional to the product of the (spatial) densities of the respective subpopulations: xy.The law originates in the theory and practice of chemical reaction kinetics. The analogy between individuals meeting when moving around in space and molecules meeting when moving around in a gas or solution is intuitively pleasing. Moreover, the simplicity of the interaction term widens the possibilities to analytically study the behaviour of the systems of differential equations incorporating this description of the contact process. Both of these factors have undoubtedly been major determinants in creating the success of mass action as a modelling concept.

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“…Wilson and Worcester [36,37] used λIS q -type incidence and found it to be best data fitting for q = 1. Severo [29] formulated models with incidence rates kI p S q (q < 1).…”

Section: Introductionmentioning

confidence: 99%

Dynamics of SVEIS epidemic model with distinct incidence

2015

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In this paper, we study the global dynamics of a SVEIS epidemic model with distinct incidence for exposed and infectives. The model is analyzed for stability and bifurcation behavior. To account for the realistic phenomenon of non-homogeneous mixing, the effect of diffusion on different population subclasses is considered. The diffusive model is analyzed using matrix stability theory and conditions for Turing bifurcation are derived. Numerical simulations support our analytical results on the dynamic behavior of the model. Int. J. Biomath. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 09/29/15. For personal use only. 1550076-2 Int. J. Biomath. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 09/29/15. For personal use only. Average length of infection of the infected individuals 1 σ Average time-span of infected individuals in exposed class α The relative measure of infectiousness for the asymptomatic class E 1550076-3 Int. J. Biomath. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 09/29/15. For personal use only. 1550076-4 Int. J. Biomath. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 09/29/15. For personal use only.

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The Law of Mass Action in Epidemiology

Cited by 80 publications

References 3 publications

Analysis of stability and bifurcation for an SEIV epidemic model with vaccination and nonlinear incidence rate

Analysis of stability and bifurcation for an SEIV epidemic model with vaccination and nonlinear incidence rate

The Law of Mass-Action in Epidemiology: A Historical Perspective

Dynamics of SVEIS epidemic model with distinct incidence

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