Existence, uniqueness and stability change of endemic states for the age-structured S-I-R epidemic model
Abstract
We consider an age structured S-I-R epidemic model with inter-cohort form for the force of infection. The population is structured into three disjoint classes according to disease status: susceptibles, infecteds, and removeds. The basic parameters for this model are the age-specific birth rate, death rate, removal rate, contact rate of disease, infectiousness and probability of vertical transmission of disease. In the existing literature a threshold condition was found for the existence of a unique endemic equilibrium and its stability under several restrictions. It was assumed that there is no vertical transmission that the removal rate is constant, and that the force of infection is of the general inter-cohort form. Here we shall generalize these results in the following way, we assume that the probability of vertical transmission is arbitrary and the removal rate is an arbitrary function of age. However, for the force of infection we shall use a separable inter-cohort form. This is a less general form but it is not too restrictive. Indeed, dealing with S-I-R models and inter-cohort transmission many of the existing papers have used an even simpler form of the force of infection. Under the above general assumptions, we can find an explicitly computable threshold condition for the existence of an endemic equilibrium. The existing paper also gave a threshold condition but in that case the threshold was the spectral radius of an integral operator, which was not explicitly computable. We can also prove that the endemic equilibrium is unique in most cases. For example, if the removal rate is non-decreasing, we can prove that there is a unique endemic state provided that the threshold parameter exceeds unity. We also have proved stability results. If the threshold parameter is less than one, we can show that the trivial equilibrium is always locally asymptotically stable. Conversely, if the threshold parameter is larger than one, the trivial equilibrium is always unstable. If there is no vertical transmission, we have the same stability result for the endemic states as the existing paper had under similar assumptions. In the general case, we find an explicit condition which determines the stability change of the endemic states.
Degree
Ph.D.
Advisors
Milner, Purdue University.
Subject Area
Public health|Mathematics
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