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Acquired immunity dependent upon exposure in an SIRS epidemic model. (English) Zbl 0637.92007
Summary: The dynamics of immunity boosted by reexposure to infection are incorporated into an SIRS (Susceptible/Infected/Recovered (immune)/Susceptible) epidemic model. The basic reproduction rate, $$R_ 0$$, is derived. If $$R_ 0$$ is less than unity, there exists a unique, stable equilibrium which is disease-free. If $$R_ 0$$ is greater than unity, the disease-free equilibrium is unstable and there exists a unique stable endemic equilibrium. The qualitative picture is unchanged if immune individuals also contribute to transmission, as long as the degree of infectivity of immune individuals does not exceed that of infected individuals who are ill.
Analysis of this simple model illustrates how the phenomenon of boosted immunity complicates disease control. Partially effective disease control may be harmful because reduction of transmission may lead to increased prevalence of illness. Furthermore, predictions concerning the effect of control are sensitive to uncertainty about the relative contributions of symptomatic infected individuals and asymptomatic immune carriers to the reservoir of infection.

##### MSC:
 92D25 Population dynamics (general) 34D05 Asymptotic properties of solutions to ordinary differential equations 34D99 Stability theory for ordinary differential equations
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##### References:
 [1] Hethcote, H.W.; Stech, H.W.; van den Driessche, P., Periodicity and stability in epidemic models: A survey, (), 65-82 [2] Aron, J.L., Dynamics of acquired immunity boosted by exposure to infection, Math. biosci., 64, 249-259, (1983) · Zbl 0515.92026 [3] Nedelman, J., Introductory review. some new thoughts about some old malaria models, Math. biosci., 73, 159-182, (1985) · Zbl 0567.92020 [4] Anderson, R.M.; May, R.M., Directly transmitted infectious diseases: control by vaccination, Science, 215, 1053-1060, (1982) · Zbl 1225.37099 [5] Dutertre, J., Etude d’un modele epidemiologique applique au paludisme, Ann. soc. belge medecine tropicale, 56, 127-141, (1976) [6] Elderkin, R.H.; Berkowitz, D.P.; Farris, F.A.; Gunn, C.F.; Hickernell, F.J.; Kass, S.N.; Mansfield, F.I.; Taranto, R.G., On the steady state of an age-dependent model for malaria, (), 491-512
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