Solvability of Age-Structured Epidemiological Models with Intracohort Transmission
- 365 Downloads
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
- 1.Arendt, W., Grabosch, A., Greiner, G., Groh, U., Lotz, H. P., Moustakas, U., Nagel, R., Neubrander, F. and Schlotterbeck, U.: One-Parameter Semigroup of Positive Operators. Springer-Verlag, Berlin Heidelberg (1986)Google Scholar
- 3.Banasiak, J.: Kinetic models in natural sciences. In: Banasiak, J. and Mokhtar-Kharroubi, M. (eds.) Evolutionary Equations with Applications in Natural Sciences, Lecture Notes in Mathematics 2126. Springer, Heidelberg (2015)Google Scholar
- 5.Belleni-Morante, A., McBride, A.C.: Applied Nonlinear Semigroups: An Introduction. John Wiley & Sons. Inc., New York (1998)Google Scholar
- 8.Capasso V.: Mathematical Structures of Epidemic Systems, LNB 97, 2nd edn. Springer, Berlin (2008)Google Scholar
- 9.Iannelli, M.: Mathematical Theory of Age-Structured Population Dynamics. Consiglio Nazionale delle Ricerche (C. N. R.), Giardini (1995)Google Scholar
- 12.Li, J., Brauer, F.: Continuous-time age-structured models in population dynamics and epidemiology. In: Brauer, F., van den Driessche, P., Wu, J. (eds.) Mathematical Epidemiology, LNM 1945. Springer, Berlin (2008)Google Scholar
- 13.Martin, R.H.: Nonlinear Operators and Differential Equations in Banach Spaces. John Wiley & Sons. Inc., New York (1976)Google Scholar
- 14.M’pika Massoukou, R.Y.: Age structured models of mathematical epidemiology. PhD thesis, UKZN (2013)Google Scholar
Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.