Analysis of an Age-structured SIL model with demographics process and vertical transmission
Ramses Djidjou Demasse (1) (2), Jean Jules Tewa (3), Samuel Bowong (4) (5)
(1) Département de Mathématiques Université de Yaoundé 1 = Department of Mathematics [Yaoundé, Cameroon]
(2) Département de Mathématiques [Yaoundé I] = Department of Mathematics [Yaoundé, Cameroon]
(3) Ecole Nationale Supérieure Polytechnique de Yaoundé
(4) Laboratoire International de Recherche en Informatique et Mathématiques Appliquées
(5) Faculté des Sciences [Douala]
Abstract
We consider a mathematical SIL model for the spread of a directly transmitted infectious disease in an age-structured population; taking into account the demographic process and the vertical transmission of the disease. First we establish the mathematical well-posedness of the time evolution problem by using the semigroup approach. Next we prove that the basic reproduction ratio R0 is given as the spectral radius of a positive operator, and an endemic state exist if and only if the basic reproduction ratio R0 is greater than unity, while the disease-free equilibrium is locally asymptotically stable if R0<1. We also show that the endemic steady states are forwardly bifurcated from the disease-free steady state when R0 cross the unity. Finally we examine the conditions for the local stability of the endemic steady states.
