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Zbl 0665.92013
Greenhalgh, David
Threshold and stability results for an epidemic model with an age- structured meeting rate.
(English)
[J] IMA J. Math. Appl. Med. Biol. 5, No.2, 81-100 (1988). ISSN 0265-0746

The paper extends the analysis of stability results of an epidemic model of the same author [ibid. 4, 109-144 (1987)] to the case where the meeting rate $\beta$ depends on age. The results are similar to those for the model where the meeting rate is constant. There is a threshold value that corresponds to the spectral radius of a Fredholm operator with kernel $\phi$ equal to one. \par In particular case when $\beta =\beta (a,a')$ factorizes as f(a)g(a') then there are two possible equilibria, one with no disease present and one with disease present. The second equilibrium is possible only if the threshold is exceeded. If the threshold is exceeded then the equilibrium with no disease present is locally unstable, if the threshold is not exceeded then this equilibrium is locally stable. If the threshold is exceeded there is an additional possible equilibrium with disease present, but it is not possible to determine if it is stable or not. \par In the general case it is supposed that if $\phi$ exceeds one then a nontrivial equilibrium exists and is unique. These conjectures are true when $\beta =f(a)g(a')$ and also when $\beta$ is expressed as a finite sum of products of positive functions of a and a'. This includes the case where the age range is divided into n age groups and the meeting rate is represented by an $n\times n$ matrix.
[S.Totaro]
MSC 2000:
*92D25 Population dynamics
45K05 Integro-partial differential equations
47A53 (Semi-)Fredholm operators; index theories
47A10 Spectrum and resolvent of linear operators

Keywords: epidemiology; age structure; threshold value; spectral radius; equilibria; meeting rate

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