Abstract
This paper is concerned with the existence of traveling waves for a delayed SIRS epidemic diffusion model with saturation incidence rate. By using the cross-iteration method and Schauder’s fixed point theorem, we reduce the existence of traveling waves to the existence of a pair of upper-lower solutions. By careful analyzsis, we derive the existence of traveling waves connecting the disease-free steady state and the endemic steady state through the establishment of the suitable upper-lower solutions.
Highlights
Since Kermack and Mckendrick [1] proposed an ordinary differential system to study epidemiology in 1927, various models have been used to describe various kinds of epidemics, and the dynamics of these systems have been investigated
Ṙ (t) = γI (t) − (δ + d) R (t), where the parameters A, d, β, δ, γ, μ are positive constants and A is the recruitment rate of the population, d is the natural death rate of the population, β is the transmission rate, δ is the rate at which recovered individuals lose immunity and return to the susceptible class, γ is the recovery rate of the infective individuals, and μ is the death rate of the infective individuals due to disease
The SIRS model assumes that the recovered individuals have only temporary immunity, which is reasonable in the study of some communicable diseases
Summary
Since Kermack and Mckendrick [1] proposed an ordinary differential system to study epidemiology in 1927, various models have been used to describe various kinds of epidemics, and the dynamics of these systems have been investigated. Mena-Lorca and Hethcote [2] considered the following SIRS epidemic model:. The SIRS model assumes that the recovered individuals have only temporary immunity, which is reasonable in the study of some communicable diseases. Gan et al [7] considered the following delayed SIRS epidemic model with spatial diffusion:. As the number of susceptible individuals is large, it is reasonable to consider the saturation incidence rate (see [8]) instead of the bilinear incidence rate. Motivated by the works mentioned above, we will consider the following delayed SIRS epidemic diffusion model with nonlinear saturation rate. (δ d) R (x, t) and study its traveling wave solutions. We point out that the nonlinear terms in (3) do not satisfy the common various (exponential) monotonicity conditions such as in [10,11,12]; the main difficulty is the construction and verification of the upperlower solutions
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