SIR infection age models with a very general class of nonlinear incidence rates f(S,J) are investigated. We give a necessary and sufficient condition for global asymptotic stability of the ...free-equilibrium related to the basic reproduction number. Furthermore, additional conditions allow us to prove an exponential stability of this disease-free equilibrium. Finally, by using a Lyapunov functional, we show the global asymptotic stability of the endemic equilibrium whenever it exists.
This paper studies the global dynamics of an SEIR (Susceptible–Exposed–Infectious–Recovered) model with nonlocal diffusion. We show the model’s well-posedness, proving the solutions’ existence, ...uniqueness, and positivity, along with a disease-free equilibrium. Next, we prove that the model admits the global threshold dynamics in terms of the basic reproduction number R0, defined as the spectral radius of the next-generation operator. We show that the solution map has a global compact attractor, offering insights into long-term dynamics. In particular, the analysis shows that for R0<1, the disease-free equilibrium is globally stable. Using the persistence theory, we show that there is an endemic equilibrium point for R0>1. Moreover, by constructing an appropriate Lyapunov function, we establish the global stability of the unique endemic equilibrium in two distinct scenarios.
A hybrid inclusion combines a differential inclusion, a difference inclusion, and constraints on the resulting continuous-time and discrete-time dynamics. Under basic assumptions on the data of a ...hybrid inclusion, the paper proves a version of the Conley’s fundamental theorem, which states that a compact and invariant set for a hybrid inclusion decomposes into a part that is chain-recurrent and the complementary part where the dynamics is gradient-like. This is done by constructing a total Lyapunov function. The proof is inspired by the well-known approaches to the Conley’s fundamental theorem for continuous or for discrete dynamics, and has foundations in the previously established asymptotic stability theory for hybrid inclusions.
Prostate cancer is a serious disease that endangers men's health. The genetic mechanism and treatment of prostate cancer have attracted the attention of scientists. In this paper, we focus on the ...nonlinear mixed reaction diffusion dynamics model of neuroendocrine transdifferentiation of prostate cancer cells with time delays, and reveal the evolutionary mechanism of cancer cells mathematically. By applying operator semigroup theory and the comparison principle of parabolic equation, we study the global existence, uniqueness and boundedness of the positive solution for the model. Additionally, the global invariant set and compact attractor of the positive solution are obtained by Kuratowski's measure of noncompactness. Finally, we use the Pdepe toolbox of MATLAB to carry out numerical calculations and simulations on an example to check the correctness and effectiveness of our main results. Our results show that the delay has no effect on the existence, uniqueness, boundedness and invariant set of the solution, but will affect the attractor.
•We provide alternative descriptions of those generalized possibly infinite iterated function systems whose constitutive functions are affine contractions which are topologically contracting ...generalized iterated function systems.•We establish conditions under which the attractor of a contractive affine generalized possibly infinite iterated function system is compact.•We are dealing with one way to generalize the notion of IFS that was introduced by theauthors under the name of generalized iterated function system
In this paper we provide alternative descriptions of those generalized possibly infinite iterated function systems whose constitutive functions are affine contractions which are topologically contracting generalized iterated function systems. As a by-product, we establish conditions under which the attractor of a contractive affine generalized possibly infinite iterated function system is compact.
The global stability of the endemic equilibrium is shown for an endemic model with infinite-dimensional population structure using a Volterra like Lyapunov function and the Krein–Rutman theorem.
Spatially distributed populations with two sexes may face the problem that males and females concentrate in different parts of the habitat and mating and reproduction does not happen sufficiently ...often for the population to persist. For simplicity, to explore the impact of sex-dependent dispersal on population survival, we consider a discrete-time model for a semelparous population where individuals reproduce only once in their life-time, during a very short reproduction season. The dispersal of females and males is modeled by Feller kernels and the mating by a homogeneous pair formation function. The spectral radius of a homogeneous operator is established as basic reproduction number of the population,
R
0
. If
R
0
<
1
, the extinction state is locally stable, and if
R
0
>
1
the population shows various degrees of persistence that depend on the irreducibility properties of the dispersal kernels. Special cases exhibit how sex-biased dispersal affects the persistence of the population.
In this paper, we investigate an SIRS epidemic model with chronological age structure in a demographic steady state. Although the age-structured SIRS model is a simple extension of the well-known ...age-structured SIR epidemic model, we have to develop new technique to deal with problems due to the reversion of susceptibility for recovered individuals. First we give a standard proof for the well-posedness of the normalized age-structured SIRS model. Next we examine existence of endemic steady states by fixed point arguments and bifurcation method, where we introduce the next generation operator and the basic reproduction number R
to formulate endemic threshold results. Thirdly we investigate stability of steady states by the bifurcation calculation and the comparison method, and we show existence of a compact attractor and discuss the global behavior based on the population persistence theory. Finally we give some numerical examples and discuss the effect of mass-vaccination on R
and the critical coverage of immunization based on the reinfection threshold.
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