Stability Change of the Endemic Equilibrium in Age-Structured Models for the Spread of S—I—R Type Infectious Diseases

Hopf bifurcation in a chronological age-structured SIR epidemic model with age-dependent infectivity

In this paper, we examine the stability of an endemic equilibrium in a chronological age-structured SIR (susceptible, infectious, removed) epidemic model with age-dependent infectivity. Under the assumption that the transmission rate is a shifted exponential function, we perform a Hopf bifurcation analysis for the endemic equilibrium, which uniquely exists if the basic reproduction number is greater than 1. We show that if the force of infection in the endemic equilibrium is equal to the removal rate, then there always exists a critical value such that a Hopf bifurcation occurs when the bifurcation parameter reaches the critical value. Moreover, even in the case where the force of infection in the endemic equilibrium is not equal to the removal rate, we show that if the distance between them is sufficiently small, then a similar Hopf bifurcation can occur. By numerical simulation, we confirm a special case where the stability switch of the endemic equilibrium occurs more than once.

Structure of epidemic models: toward further applications in economics

In this paper, we review the structure of various epidemic models in mathematical epidemiology for the future applications in economics. The heterogeneity of population and the generalization of nonlinear terms play important roles in making more elaborate and realistic models. The basic, effective, control and type reproduction numbers have been used to estimate the intensity of epidemic, to evaluate the effectiveness of interventions and to design appropriate interventions. The advanced epidemic models includes the age structure, seasonality, spatial diffusion, mutation and reinfection, and the theory of reproduction numbers has been generalized to them. In particular, the existence of sustained periodic solutions has attracted much interest because they can explain the recurrent waves of epidemic. Although the theory of epidemic models has been developed in decades and the development has been accelerated through COVID-19, it is still difficult to completely answer the uncertainty problem of epidemic models. We would have to mind that there is no single model that can solve all questions and build a scientific attitude to comprehensively understand the results obtained by various researchers from different backgrounds.

Mathematical analysis on an age-structured SIS epidemic model with nonlocal diffusion

In this paper we propose an age-structured susceptible-infectious-susceptible epidemic model with nonlocal (convolution) diffusion to describe the geographic spread of infectious diseases via long-distance travel. We analyze the well-posedness of the model, investigate the existence and uniqueness of the nontrivial steady state corresponding to an endemic state, and study the local and global stability of this nontrivial steady state. Moreover, we discuss the asymptotic properties of the principal eigenvalue and nontrivial steady state with respect to the nonlocal diffusion rate. The analysis is carried out by using the theory of semigroups and the method of monotone and positive operators. The spectral radius of a positive linear operator associated to the solution flow of the model is identified as a threshold.

Theoretical and numerical results for an age-structured SIVS model with a general nonlinear incidence rate

In this paper, we propose an SIVS epidemic model with continuous age structures in both infected and vaccinated classes and with a general nonlinear incidence. Firstly, we provide some basic properties of the system including the existence, uniqueness and positivity of solutions. Furthermore, we show that the solution semiflow is asymptotic smooth. Secondly, we calculate the basic reproduction number [Formula: see text] by employing the classical renewal process, which determines whether the disease persists or not. In the main part, we investigate the global stability of the equilibria by the approach of Lyanpunov functionals. Some numerical simulations are conducted to illustrate the theoretical results and to show the effect of the transmission rate and immunity waning rate on the disease prevalence.

On the formulation of epidemic models (an appraisal of Kermack and McKendrick)

The aim of this paper is to show that a large class of epidemic models, with both demography and non-permanent immunity incorporated in a rather general manner, can be mathematically formulated as a scalar renewal equation for the force of infection.

Age-structured homogeneous epidemic systems with application to the MSEIR epidemic model

In this paper, we develop a new approach to deal with asymptotic behavior of the age-structured homogeneous epidemic systems and discuss its application to the MSEIR epidemic model. For the homogeneous system, there is no attracting nontrivial equilibrium, instead we have to examine existence and stability of persistent solutions. Assuming that the host population dynamics can be described by the stable population model, we rewrite the basic system into the system of ratio age distribution, which is the age profile divided by the stable age profile. If the host population has the stable age profile, the ratio age distribution system is reduced to the normalized system. Then we prove the stability principle that the local stability or instability of steady states of the normalized system implies that of the corresponding persistent solutions of the original homogeneous system. In the latter half of this paper, we prove the threshold and stability results for the normalized system of the age-structured MSEIR epidemic model.

The Impact of the Allee Effect in Dispersal and Patch-Occupancy Age on the Dynamics of Metapopulations

In this paper, we introduce a Levins-type metapopulation model with empty and occupied patches, and dispersing population. We structure the proportion of occupied patches according to the patch-occupancy age. We observe that patch-occupancy age may destabilize the metapopulation, leading to persistent oscillations. We also allow for the dispersal rate to vary with the proportion of empty patches in a monotone or unimodal way. The unimodal dependence leads to multiple non-trivial equilibria and bistability when the reproduction number of the metapopulation R < 1 but greater than a lower critical value R(*). We show that the metapopulation will persist independently of its initial status if R > 1.

Existence and uniqueness of endemic states for the age-structured S-I-R epidemic model

The existence and uniqueness of positive steady states for the age structured S-I-R epidemic model with intercohort transmission is considered. Threshold results for the existence of endemic states are established for most cases. Uniqueness is shown in each case. Threshold used are explicitly computable in terms of demographic and epidemiological parameters of the model.

The dynamics of smallpox epidemics in Britain, 1550–1800

Time-series analysis, a valuable tool in studying population dynamics, has been used to determine the periodicity of smallpox epidemics during the seventeenth and eighteenth centuries in two contrasting representative situations: 1) London, a large city where smallpox was endemic, and 2) Penrith, a small rural town. The interepidemic period was found to be two years in London and five years in Penrith. Equations governing the dynamics of epidemics predict 1) a two-year periodicity and 2) that oscillatory epidemics die out quickly. It is suggested that epidemics were maintained by a periodic variation in susceptibility linked either to a five-year cycle of malnutrition or to an annual cycle. Computer modeling shows how the very different patterns of epidemics are related to population size and to the magnitude of the oscillation in susceptibility.