Solvability of Age-Structured Epidemiological Models with Intracohort Transmission
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The standard version of the epidemiological model with continuous age structure (Iannelli, Mathematical Theory of Age-Structured Population Dynamics, 1995) consists of the linear McKendrick model for the evolution of a disease-free population, coupled with one of the classical (SIS, SIR, etc.) models for the spread of the disease. A natural functional space in which the linear McKendrick model is well posed is the space of integrable functions. However, in the so-called intracohort models, the disease term contains pointwise products of the unknown functions; that is, of the age-specific densities of susceptibles, infectives and other classes (if applicable) which render the standard semilinear perturbation technique of proving the well-posedness of the full model not applicable in that space. This is due to the fact that the product of two integrable functions need not be integrable. Therefore, most works on the well-posedness of such problems have been done under additional assumption that the disease-free population is stable and the equilibrium has been reached (Busenberg et al. SIAM J Math Anal 22(4):1065–1080, 1991, Prüß, J Math Biol 11:65–84, 1981). This allowed for showing, after some algebraic manipulations, that the order interval [0, 1] in the space of integrable functions was invariant under the action of the linear McKendrick semigroup and thus the usual iteration technique could be applied in this interval, yielding a bounded solution. An additional advantage of adopting the stability assumption was that it eliminated from the model the death rate which, in all realistic cases, is unbounded and creates serious technical difficulties. The aim of this note is to show that a careful modification of the standard Picard iteration procedure allows for proving the same result without the stability assumption. Since in such a case we are not able to suppress the unbounded death rate, we will briefly present the necessary linear results in a way which simplifies and unifies some classical results existing in the literature (Inaba, Math Popul Stud 1:49–77, 1988, Prüß, J Math Biol 11:65–84, 1981, Webb, Theory of Nonlinear Age Dependent Population Dynamics, 1985).
Mathematics Subject ClassificationPrimary 92D25 47N20 Secondary 35Q80 47D06
KeywordsAge-structured population epidemiology McKendrick model semilinear equation
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