Differential susceptibility and infectivity epidemic models
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1.
Center for Nonlinear Studies (MS B284), Los Alamos National Laboratory, Los Alamos, NM 87545
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2.
Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899
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Received:
01 January 2005
Accepted:
29 June 2018
Published:
01 November 2005
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MSC :
34D20, 34D23, 92D30.
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We formulate differential susceptibility and differential infectivity
models for disease transmission in this paper. The susceptibles are divided into
n groups based on their susceptibilities, and the infectives are divided into m
groups according to their infectivities. Both the standard incidence and the
bilinear incidence are considered for different diseases. We obtain explicit
formulas for the reproductive number. We define the reproductive number
for each subgroup. Then the reproductive number for the entire population
is a weighted average of those reproductive numbers for the subgroups. The
formulas for the reproductive number are derived from the local stability of
the infection-free equilibrium. We show that the infection-free equilibrium is
globally stable as the reproductive number is less than one for the models with
the bilinear incidence or with the standard incidence but no disease-induced
death. We then show that if the reproductive number is greater than one,
there exists a unique endemic equilibrium for these models. For the general
cases of the models with the standard incidence and death, conditions are
derived to ensure the uniqueness of the endemic equilibrium. We also provide
numerical examples to demonstrate that the unique endemic equilibrium is
asymptotically stable if it exists.
Citation: James M. Hyman, Jia Li. Differential susceptibility and infectivity epidemic models[J]. Mathematical Biosciences and Engineering, 2006, 3(1): 89-100. doi: 10.3934/mbe.2006.3.89
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Abstract
We formulate differential susceptibility and differential infectivity
models for disease transmission in this paper. The susceptibles are divided into
n groups based on their susceptibilities, and the infectives are divided into m
groups according to their infectivities. Both the standard incidence and the
bilinear incidence are considered for different diseases. We obtain explicit
formulas for the reproductive number. We define the reproductive number
for each subgroup. Then the reproductive number for the entire population
is a weighted average of those reproductive numbers for the subgroups. The
formulas for the reproductive number are derived from the local stability of
the infection-free equilibrium. We show that the infection-free equilibrium is
globally stable as the reproductive number is less than one for the models with
the bilinear incidence or with the standard incidence but no disease-induced
death. We then show that if the reproductive number is greater than one,
there exists a unique endemic equilibrium for these models. For the general
cases of the models with the standard incidence and death, conditions are
derived to ensure the uniqueness of the endemic equilibrium. We also provide
numerical examples to demonstrate that the unique endemic equilibrium is
asymptotically stable if it exists.
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