Age-structured Epidemic Model Research Articles

Hopf bifurcation of the age-structured SIRS model with the varying population sizes

<abstract><p>The purpose of this paper is to investigate the impact of the varying population sizes on the dynamic behavior of the age-structured epidemic model. A age-structured SIRS epidemic model with the varying population sizes is established and investigated to take into account time delay. The non-negativity of the solution, the existence and stability of the steady states, and the existence of the Hopf bifurcation are discussed. The numerical simulations show that the varying population sizes can cause the age-structured SIRS model to produce multiple stability switches.</p></abstract>

Structural identifiability analysis of age-structured PDE epidemic models

Computational and mathematical models rely heavily on estimated parameter values for model development. Identifiability analysis determines how well the parameters of a model can be estimated from experimental data. Identifiability analysis is crucial for interpreting and determining confidence in model parameter values and to provide biologically relevant predictions. Structural identifiability analysis, in which one assumes data to be noiseless and arbitrarily fine-grained, has been extensively studied in the context of ordinary differential equation (ODE) models, but has not yet been widely explored for age-structured partial differential equation (PDE) models. These models present additional difficulties due to increased number of variables and partial derivatives as well as the presence of boundary conditions. In this work, we establish a pipeline for structural identifiability analysis of age-structured PDE models using a differential algebra framework and derive identifiability results for specific age-structured models. We use epidemic models to demonstrate this framework because of their wide-spread use in many different diseases and for the corresponding parallel work previously done for ODEs. In our application of the identifiability analysis pipeline, we focus on a Susceptible-Exposed-Infected model for which we compare identifiability results for a PDE and corresponding ODE system and explore effects of age-dependent parameters on identifiability. We also show how practical identifiability analysis can be applied in this example.

Stability and Hopf Bifurcation in Delayed Age-Structured SVIR Epidemic Model with Vaccination and Incubation

Stability and Hopf Bifurcation in Delayed Age-Structured SVIR Epidemic Model with Vaccination and Incubation

Hopf Bifurcation of an Age-Structured Epidemic Model with Quarantine and Temporary Immunity Effects

In this paper, we investigate an epidemic model with quarantine and recovery-age effects. Reformulating the model as an abstract nondensely defined Cauchy problem, we discuss the existence and uniqueness of solutions to the model and study the stability of the steady state based on the basic reproduction number. After analyzing the distribution of roots to a fourth degree exponential polynomial characteristic equation, we also derive the conditions of Hopf bifurcation. Numerical simulations are performed to illustrate the results.

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.

Traveling wave solutions for a diffusive age-structured SIR epidemic model

The purpose of this paper is to study the existence and non-existence of traveling wave solutions for a diffusive age-structured SIR epidemic model. One of the main features of this model is that the susceptible, infected and removed individuals are all involved in this model, and the total population is not assumed a constant. In this paper, we show that the basic reproduction number R0 corresponding to the spatially homogeneous system is an important threshold parameter in determining the existence of the traveling wave solutions. More precisely, when R0>1 and the wave speed is greater than the minimal wave speed c*, we establish the existence of traveling wave solutions. On the other hand, when R0>1 and 0<c<c* or R0<1, we show the non-existence of traveling wave solutions. Our study provides some insights on how to deal with the high dimensional age-structured epidemic models.

An age-structured epidemic model with boosting and waning of immune status.

In this paper, we developed an age-structured epidemic model that takes into account boosting and waning of immune status of host individuals. For many infectious diseases, the immunity of recovered individuals may be waning as time evolves, so reinfection could occur, but also their immune status could be boosted if they have contact with infective agent. According to the idea of the Aron's malaria model, we incorporate a boosting mechanism expressed by reset of recovery-age (immunity clock) into the SIRS epidemic model. We established the mathematical well-posedness of our formulation and showed that the initial invasion condition and the endemicity can be characterized by the basic reproduction number $ R_0 $. Our focus is to investigate the condition to determine the direction of bifurcation of endemic steady states bifurcated from the disease-free steady state, because it is a crucial point for disease prevention strategy whether there exist subcritical endemic steady states. Based on a recent result by Martcheva and Inaba [1], we have determined the direction of bifurcation that endemic steady states bifurcate from the disease-free steady state when the basic reproduction number passes through the unity. Finally, we have given a necessary and sufficient condition for backward bifurcation to occur.

Threshold dynamics and optimal control on an age-structured SIRS epidemic model with vaccination.

We consider a vaccination control into a age-structured susceptible-infective-recovered-susceptible (SIRS) model and study the global stability of the endemic equilibrium by the iterative method. The basic reproduction number $ R_0 $ is obtained. It is shown that if $ R_0 < 1 $, then the disease-free equilibrium is globally asymptotically stable, if $ R_0 > 1 $, then the disease-free and endemic equilibrium coexist simultaneously, and the global asymptotic stability of endemic equilibrium is also shown. Additionally, the Hamilton-Jacobi-Bellman (HJB) equation is given by employing the Bellman's principle of optimality. Through proving the existence of viscosity solution for HJB equation, we obtain the optimal vaccination control strategy. Finally, numerical simulations are performed to illustrate the corresponding analytical results.

Optimal control strategies for a heroin epidemic model with age-dependent susceptibility and recovery-age

<abstract> In the present manuscript, an age-structured heroin epidemic model is formulated with the assumption that susceptibility and recovery are age-dependent. Keeping in view some control measures for heroin addiction, a control problem for further analysis is presented. The main results are the existence of control variables, sensitivities, adjoint system and the setting of an optimal control problem. We used the techniques of weak derivatives and a general principle of Pontryagin's type for obtaining the optimal control problem. To compare our results, we demonstrated sample simulations which show the effect of control on the entire population. </abstract>

An age-structured model for coupling within-host and between-host dynamics in environmentally-driven infectious diseases.

In this paper, an age-structured epidemic model for coupling within-host and between-host dynamics in environmentally-driven infectious diseases is investigated. The model is described by a mixed system of ordinary and partial differential equations which is constituted by the within-host virus infectious fast time ordinary system and the between-host disease transmission slow time age-structured system. The isolated fast system has been investigated in previous literatures, and the main results are introduced. For the isolated slow system, the basic reproduction number Rb0, the positivity and ultimate boundedness of solutions are obtained, the existence of equilibria, the local stability of equilibria, and the global stability of disease-free equilibrium are established. We see that when Rb0 ≤ 1 the system only has the disease-free equilibrium which is globally asymptotically stable, and when Rb0 > 1 the system has a unique endemic equilibrium which is local asymptotically stable. With regard to the coupled slow system, the basic reproduction number Rb, the positivity and boundedness of solutions and the existence of equilibria are firstly obtained. Particularly, the coupled slow system can exist two positive equilibria when Rb < 1 and a unique endemic equilibrium when Rb > 1. When Rb < 1 the disease-free equilibrium is local asymptotically stable, and when Rb > 1 and an additional condition is satisfied the unique endemic equilibrium is local asymptotically stable. When there exist two positive equilibria, under an additional condition the local asymptotic stability of a positive equilibrium and the instability of other positive equilibrium also are established. The numerical examples show that the additional condition may be removed. The research shows that the coupled slow age-structured system has more complex dynamical behavior than the corresponding isolated slow system.

Global threshold dynamics of an infection age-structured SIR epidemic model with diffusion under the Dirichlet boundary condition

In this paper, we are concerned with the global asymptotic behavior of an infection age-structured SIR epidemic model with diffusion in a general n-dimensional bounded spatial domain under the homogeneous Dirichlet boundary condition. By using the method of characteristics, we reformulate the model into a system of a reaction-diffusion equation and a Volterra integral equation. We define the basic reproduction number R0 by the spectral radius of a compact positive linear operator and show that if R0<1, then the disease-free steady state is globally attractive, whereas if R0>1, then a positive endemic steady state exists and the system is uniformly persistent. By numerical simulation for the 2-dimensional case, we show that R0 depends on the shape of the spatial domain. This result is in contrast with the case of the homogeneous Neumann boundary condition, in which R0 is independent of the spatial domain.

Dynamics and optimal control of an age-structured SIRVS epidemic model

Vaccination and treatment are both effective methods of preventing the spread of infectious diseases. In this paper, we propose an SIRVS epidemic model with ages of vaccination and recovery, involving vaccine-induced immunity and infection-induced immunity. The basic reproduction number of the epidemic model, R 0 , is first obtained. If R 0 < 1, the local and global stabilities of the disease-free steady state are strictly proved. If R 0 > 1, to control the disease, an optimal control problem is discussed by evaluating the cost of control strategies (vaccination and treatment) and using Pontryagin's maximum principle. We use specific parameter values related to influenza A to do some numerical simulations. Results show that the recovery age plays an important role in the optimal control, where precisely, the control with vaccination only has no effect when the acquired immunity period is not more than the vaccination immunity period; otherwise, it plays a significant role in disease control.

Periodic solutions of an age-structured epidemic model with periodic infection rate

In this paper we consider an age-structured epidemic model of the susceptible-exposed-infectious-recovered (SEIR) type. To characterize the seasonality of some infectious diseases such as measles, it is assumed that the infection rate is time periodic. After establishing the well-posedness of the initial-boundary value problem, we study existence of time periodic solutions of the model by using a fixed point theorem. Some numerical simulations are presented to illustrate the obtained results.

Bifurcation analysis of an age-structured SIRI epidemic model.

In this paper, an SIRI epidemic model with age of infection and the proliferation of susceptible individuals with logistic growth is investigated. By using the theory of integral semigroup and Hopf bifurcation theory for semilinear equations with non-dense domain, it is shown that if the threshold parameter is greater than unity, sufficient condition is derived for the occurrence of the Hopf bifurcation. Numerical simulations are carried out to illustrate the theoretical results.

Strong convergence of the partially truncated Euler–Maruyama scheme for a stochastic age-structured SIR epidemic model

we study in this paper strong convergence of the partially truncated Euler-Maruyama scheme for an age-structured Susceptible-Infected-Removed (SIR) epidemic model with environmental noise. Using the semigroup theory, the existence and uniqueness of global positive solution for the model is first proved. We then define a truncated function and develop a partially truncated EM numerical solutions to the stochastic age-structured SIR epidemic model. We present the pth moment boundedness of the partially truncated EM numerical approximate solutions under appropriate conditions. Furthermore, the strong Lq convergence is established for the condition of 2 ≤ q < p of the partially truncated EM scheme. Finally, numerical simulations and examples are provided to demonstrate theoretical results and to illustrate validity of the partially truncated EM scheme.

The basic reproduction number [Formula: see text] in time-heterogeneous environments.

In the previous paper (Inaba in J Math Biol 65:309-348, 2012), we proposed a new (most biologically natural) definition of the basic reproduction number [Formula: see text] for structured population in general time-heterogeneous environments based on the generation evolution operator. Using the mathematical definition for cone spectral radius, we show that our [Formula: see text] is given by the spectral radius of the generation evolution operator in the time-state space. Then as far as we consider linear population dynamics, our [Formula: see text] is a threshold value for population extinction and persistence in time-heterogeneous environments. Next we prove that even for nonlinear systems, our [Formula: see text] plays a role of a threshold value for population extinction in time-heterogeneous environments. For periodic systems, we can show that supercritical condition [Formula: see text] implies existence of positive periodic solution. Finally using the idea of [Formula: see text] in time-heterogeneous environment, we examine existence and stability of periodic solution in the age-structured SIS epidemic model with time-periodic parameters.

Mathematical Analysis for an Age-Structured Heroin Epidemic Model

In this paper, an age-structured heroin epidemic model, where the susceptibility of individuals and the relapse of heroin users in treatment are described by two age-dependent variables, is formulated and analyzed. The basic reproduction ratio of the model is derived and proved to be a threshold condition, which completely determines the global behaviors of the model. The asymptotic smoothness of the semiflow generated by the family of solutions, uniform persistence and existence of an interior global attractor have been presented for establishing and defining a Lyapunov functional on this attractor. Some control strategies of heroin and two special cases of the model formulation are addressed.

Hopf bifurcation in an age-structured SIR epidemic model

In this paper, we study the occurrence of a sustained periodic solution via the Hopf bifurcation in an age-structured SIR epidemic model. Under the assumption that the transmission rate depends on the age of infective individuals and the product of the transmission rate and the population age distribution is concentrated in a specific age, we reformulate the model into an integral equation of Fredholm type. We then define the basic reproduction number R0 and show that the unique positive endemic equilibrium of the integral equation exists if and only if R0>1. We derive a characteristic equation for the endemic equilibrium, and regarding the specific age as a bifurcation parameter, we obtain a sufficient condition for the occurrence of the Hopf bifurcation. Finally, we provide a numerical example that supports our theoretical result.

Global stability of an age-structured epidemic model with general Lyapunov functional.

In this paper, we focus on the study of the dynamics of a certain age structured epidemic model. Our aim is to investigate the proposed model, which is based on the classical SIR epidemic model, with a general class of nonlinear incidence rate with some other generalization. We are interested to the asymptotic behavior of the system. For this, we have introduced the basic reproduction number R₀ of model and we prove that this threshold shows completely the stability of each steady state. Our approach is the use of general constructed Lyapunov functional with some results on the persistence theory. The conclusion is that the system has a trivial disease-free equilibrium which is globally asymptotically stable for R₀ < 1 and that the system has only a unique positive endemic equilibrium which is globally asymptotically stable whenever R₀ > 1. Several numerical simulations are given to illustrate our results.

Mathematical analysis for an age-structured SIRS epidemic model.

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 R0 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 R0 and the critical coverage of immunization based on the reinfection threshold.