In this paper, we present a study on a network-based susceptible-infected-recovered (SIR) epidemic model with a saturated treatment function. It is well known that treatment can have a specific effect on the spread of epidemics, and due to the limited resources of treatment, the number of patients during severe disease outbreaks who need to be treated may exceed the treatment capacity. Consequently, the number of patients who receive treatment will reach a saturation level. Thus, we incorporated a saturated treatment function into the model to characterize such a phenomenon. The dynamics of the present model is discussed in this paper. We first obtained a threshold value R0, which determines the stability of a disease-free equilibrium. Furthermore, we investigated the bifurcation behavior at R0=1. More specifically, we derived a condition that determines the direction of bifurcation at R0=1. If the direction is backward, then a stable disease-free equilibrium concurrently exists with a stable endemic equilibrium even though R0<1. Therefore, in this case, R0<1 is not sufficient to eradicate the disease from the population. However, if the direction is forward, we find that for a range of parameters, multiple equilibria could exist to the left and right of R0=1. In this case, the initial infectious invasion must be controlled to a lower level so that the disease dies out or approaches a lower endemic steady state.
This work investigates the diversity of traveling wave solutions for a class of delayed cellular neural networks on the one-dimensional integer lattice ℤ1. The dynamics of a given cell is characterized by instantaneous self-feedback and neighborhood interaction with distributed delay due to, for example, finite switching speed and finite velocity of signal transmission. Applying the monotone iteration scheme, we can deduce the existence of monotonic traveling wave solutions provided the templates satisfy the so-called quasi-monotonicity condition. We then consider two special cases of the delayed cellular neural network in which each cell interacts only with either the nearest m left neighbors or the nearest m right neighbors. For the former case, we can directly figure out the analytic solution in an explicit form by the method of step with the help of the characteristic function and then prove that, in addition to the existence of monotonic traveling wave solutions, for certain templates there exist nonmonotonic traveling wave solutions such as camel-like waves with many critical points. For the latter case, employing the comparison arguments repeatedly, we can clarify the deformation of traveling wave solutions with respect to the wave speed. More specifically, we can describe the transition of profiles from monotonicity, damped oscillation, periodicity, unboundedness and back to monotonicity as the wave speed is varied. Some numerical results are also given to demonstrate the theoretical analysis.
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