Abstract
In this paper, we investigate an SIRS epidemic model with chronological age structure in a demographic steady state. Although the age-structured SIRS model is a simple extension of the well-known age-structured SIR epidemic model, we have to develop new technique to deal with problems due to the reversion of susceptibility for recovered individuals. First we give a standard proof for the well-posedness of the normalized age-structured SIRS model. Next we examine existence of endemic steady states by fixed point arguments and bifurcation method, where we introduce the next generation operator and the basic reproduction number R0 to formulate endemic threshold results. Thirdly we investigate stability of steady states by the bifurcation calculation and the comparison method, and we show existence of a compact attractor and discuss the global behavior based on the population persistence theory. Finally we give some numerical examples and discuss the effect of mass-vaccination on R0 and the critical coverage of immunization based on the reinfection threshold.
Highlights
In a seminal series of papers published during the 1920s and the 1930s, Kermack and McKendrick proposed infection–age structured epidemic models that take into account demography of the host population, the waning immunity and reinfection of recovered individuals ([1, 2])
The Kermack–McKendrick infection-age dependent reinfection model has been reinvestigated by several authors ([3, 4, 5, 6, 7]), and it was shown that a backward bifurcation of endemic steady states is possible to occur
SIRS model that the disease will be naturally eradicated if R0 < 1, while it is strongly persistent and endemic steady states exists if R0 > 1
Summary
In a seminal series of papers published during the 1920s and the 1930s, Kermack and McKendrick proposed infection–age structured epidemic models that take into account demography of the host population, the waning immunity and reinfection of recovered individuals ([1, 2]). In their models, the total population is decomposed into three compartments, the susceptibles, the infectious and the recovered populations (SIR model), and it is assumed that reinfection occurs for the recovered population depending on the time since recovery (recovery-age). We give some numerical examples and discuss the effect of mass-vaccination
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