Age-structured Epidemic Model Research Articles

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.

Dynamical analysis of an age-structured multi-group SIVS epidemic model.

Host heterogeneities such as space, gender, and age etc are intrinsic characters for investigating diseases mechanisms and transmission routes. First, we incorporate inter-group, intra-group and age structure to propose a multi-group SIVS epidemic model. Then we obtain the basic reproduction number of the system which is the spectral radius of the next generation operator by the renewal equation. Based on some assumptions for parameters, we obtain the existence and uniqueness of endemic equilibrium. By means of integral semigroup theory and Lyapunov methods, we show that the threshold dynamics of the system is completely determined by the basic.

Global dynamics of a discrete age-structured SIR epidemic model with applications to measles vaccination strategies

We investigate an SIR epidemic model with discrete age groups to understand the transmission dynamics of an infectious disease in a host population with an age structure. We derive the basic reproduction number R0 and show that it is a sharp threshold parameter. If R0≤1, the disease-free equilibrium E0 is globally stable. If R0>1,E0 is unstable, the model is uniformly persistent, and an endemic equilibrium exists. The global stability of the endemic equilibrium when R0>1 is established under a sufficient condition. The model is then used to analyze the measles data in India and evaluate the effectiveness of several vaccination strategies for the control of measles epidemics in India.

Analysis of an age-structured multi-group heroin epidemic model

This paper is concerned with the mathematical analysis of an age-structured multi-group heroin epidemic model, which can be used to describe the spread of heroin habituation and addiction in heterogeneous environment. Under general assumptions on the different level of susceptibility and the relapse to frequent heroin use, we establish sharp criteria for heroin spreading and vanishing. We rigorously investigate the well-posedness of the model, the existence of equilibria, the asymptotic smoothness of solution orbits, and the global stability of equilibria. Specifically, we rigorously show that the drug-free equilibrium is globally asymptotically stable if a threshold value ℜ0 is less than one, and the unique drug-endemic equilibrium is globally attractive if ℜ0 is greater than one. In the proofs of global stability of equilibria, we construct suitable Lyapunov functions by using a graph-theoretic method.

Correction: Kuniya, T. Stability Analysis of an Age-Structured SIR Epidemic Model with a Reduction Method to ODEs. Mathematics 2018, 6, 147

I have found an error in Equation (17) in my paper [1] [...]

Persistence and extinction for an age-structured stochastic SVIR epidemic model with generalized nonlinear incidence rate

We formulate an epidemic model with age of vaccination and generalized nonlinear incidence rate, where the total population consists of the susceptible, the vaccinated, the infected and the removed. We then reach a stochastic SVIR model when the fluctuation is introduced into the transmission rate. By using Itô’s formula and Lyapunov methods, we first show that the stochastic epidemic model admits a unique global positive solution with the positive initial value. We then obtain the sufficient conditions of the stochastic epidemic model. Moreover, the threshold tells the disease spreads or not is derived. If the intensity of the white noise is small enough and R˜0<1, then the disease eventually becomes extinct with negative exponential rate. If R˜0>1, then the disease is weakly permanent. The persistence in the mean of the infected is also obtained when the indicator Rˆ0>1, which means the disease will prevail in a long run. As a consequence, several illustrative examples are separately carried out with numerical simulations to support the main results of this paper.

Stability Analysis of an Age-Structured SIR Epidemic Model with a Reduction Method to ODEs

In this paper, we are concerned with the asymptotic stability of the nontrivial endemic equilibrium of an age-structured susceptible-infective-recovered (SIR) epidemic model. For a special form of the disease transmission function, we perform the reduction of the model into a four-dimensional system of ordinary differential equations (ODEs). We show that the unique endemic equilibrium of the reduced system exists if the basic reproduction number for the original system is greater than unity. Furthermore, we perform the stability analysis of the endemic equilibrium and obtain a fourth-order characteristic equation. By using the Routh–Hurwitz criterion, we numerically show that the endemic equilibrium is asymptotically stable in some epidemiologically relevant parameter settings.

An age-structured epidemic model for the demographic transition.

In this paper, we formulate an age-structured epidemic model for the demographic transition in which we assume that the cultural norms leading to lower fertility are transmitted amongst individuals in the same way as infectious diseases. First, we formulate the basic model as an abstract homogeneous Cauchy problem on a Banach space to prove the existence, uniqueness, and well-posedness of solutions. Next based on the normalization arguments, we investigate the existence of nontrivial exponential solutions and then study the linearized stability at the exponential solutions using the idea of asynchronous exponential growth. The relative stability defined in the normalized system and the absolute (orbital) stability in the original system are examined. For the boundary exponential solutions corresponding to infection-free or totally infected status, we formulate the stability condition using reproduction numbers. We show that bi-unstability of boundary exponential solutions is one of conditions which guarantee the existence of coexistent exponential solutions.

An age-structured epidemic model with waning immunity and general nonlinear incidence rate

In this paper, a susceptible-vaccinated-exposed-infectious-recovered epidemic model with waning immunity and continuous age structures in vaccinated, exposed and infectious classes has been formulated. By using the Fluctuation lemma and the approach of Lyapunov functionals, we establish a threshold dynamics completely determined by the basic reproduction number. When the basic reproduction number is less than one, the disease-free steady state is globally asymptotically stable, and otherwise the endemic steady state is globally asymptotically stable.

Global analysis of age-structured multi-stage epidemic models for infectious diseases

We formulate a multi-stage SEIR model for infectious diseases with continuous age structure for each successive infectious stage during a long infective period. The model can describe disease progression through multiple infectious stages as in the case of HIV, hepatitis B and hepatitis C. Mathematical analysis shows that the global dynamics are completely determined by the basic reproductive number R0. If R0≤1, the disease-free equilibrium is globally asymptotically stable and the disease dies out. If R0>1, a unique endemic equilibrium is globally asymptotically stable, and the disease persists at the endemic equilibrium. The proof of global stability of endemic equilibria utilizes a Lyapunov functional. Numerical simulations are illustrated and model generalization is also discussed.

Analysis and control of an age-structured HIV-1 epidemic model with different transmission mechanisms

In this paper, we propose a within-host HIV-1 epidemic model with cell-to-virus and cell-to-cell transmission. By mathematical analysis, we obtain the basic reproduction number mathcal {R}_{0}, which determines the viral persistence and the basic reproduction number mathcal {R}_{0}^{mathrm{cc}} with respect to cell-to-cell transmission which is not strong enough, i.e., it is less than 1. If the basic reproduction number is less than 1, then the viral-free steady state E_{0} is globally asymptotically stable, which is proved by fluctuation lemma and comparison method; if mathcal {R}_{0}>1 is greater than 1, the endemic steady state E^{*} is globally asymptotically stable, which is proved by constructing the Lyapunov functional. Antiretoviral therapy is implemented to suppress the viral replication. Protease inhibitors for cell-to-cell transmission play an important role in controlling cell-to-cell infection. Under some circumstances, the effects of the cell-to-cell infection process are more sensitive than those of cell-to-virus transmission.

Global stability of an age-structure epidemic model with imperfect vaccination and relapse

A new age-structured epidemic model with imperfect vaccination and relapse is proposed. The total population of our model is partitioned into five subclasses: susceptible class S, vaccinated class V, exposed class E, infectious class I and removed class R. Age-structures are equipped with in exposed and recovered classes. Furthermore, imperfect vaccination is also introduced in our model. The basic reproduction number R0 is defined and proved as a threshold parameter of the model. Asymptotic smoothness of solutions and uniform persistence of the system are showed via reformulating the system as a system of Volterra integral equation. Furthermore, by constructing proper Volterra-type Lyapunov functional we get when R0<1, the disease-free equilibrium is globally asymptotically stable. When R0>1, the endemic equilibrium is globally stable. Our results show that to increase the efficiency of vaccination and reduce influence of relapse are vital essential for controlling epidemic.

Numerical approximation of the basic reproduction number for a class of age-structured epidemic models

We are concerned with the numerical approximation of the basic reproduction number R 0 , which is the well-known epidemiological threshold value defined by the spectral radius of the next generation operator. For a class of age-structured epidemic models in infinite-dimensional spaces, R 0 has the abstract form and cannot be explicitly calculated in general. We discretize the linearized equation for the infective population into a system of ordinary differential equations in a finite n -dimensional space and obtain a corresponding threshold value R 0 , n , which can be explicitly calculated as the positive dominant eigenvalue of the next generation matrix. Under the compactness of the next generation operator, we show that R 0 , n → R 0 as n → + ∞ in terms of the spectral approximation theory.

Threshold dynamics of an age-structured epidemic model with relapse and nonlinear incidence

Threshold dynamics of an age-structured epidemic model with relapse and nonlinear incidence

Global stability of an age-structured SVEIR epidemic model with waning immunity, latency and relapse

The global dynamics of an SVEIR epidemic model with age-dependent waning immunity, latency and relapse are studied. Sharp threshold properties for global asymptotic stability of both disease-free equilibrium and endemic equilibrium are given. The asymptotic smoothness, uniform persistence and the existence of interior global attractor of the semi-flow generated by a family of solutions of the system are also addressed. Furthermore, some related strategies for controlling the spread of diseases are discussed.

An age-structured SIR epidemic model with fixed incubation period of infection

This paper is devoted to the study of an age-structured SIR epidemic system on the basis of the model of polycyclic population dynamics of susceptible, infected and recovered individuals. This model was considered as a nonlinear competitive system of three initial–boundary value problems for the nonlinear transport equations with non-local integral boundary conditions and discrete time delay (fixed incubation period of infection). It was obtained the explicit recurrent formulae for computing the traveling wave solution of such system provided the model parameters (coefficients of equations and initial values) satisfy the restrictions that guarantee the existence and uniqueness of continuous solution. Using some insignificant simplifications the age-structured SIR model was reduced to the nonlinear autonomous system of delay ODE. It was studied the dimensionless indicators and conditions of existence of trivial disease-free equilibrium, two non-trivial endemic equilibriums for the incurable and curable infection-induced diseases, respectively. By carrying out an analysis of the local asymptotical stability of the solution of such system we obtain the restrictions for the time delay that guarantee the stability of solutions in the neighborhood of equilibriums. The numerous simulations of dynamics of SIR population illustrate the results of theoretical analysis.

Dynamic Behaviors of Equilibrium Solution to an Age-Structured Epidemic Model for Monitoring and Control of HIV/AIDS

In this paper, the dynamic behaviors, such as local asymptotic stability and global asymptotic stability of a continuous SUI epidemic model with chronological age and infection age for HIV/AIDS are discussed. The theoretical results of the paper indicate that HIV/AIDS diseases can be eradicated theoretically when the reasonable measures are taken to guarantee the conditions to be satisfied and the vertical transmission of HIV/AIDS diseases can be cut off. The results help us to interpret the importance of monitoring and control of HIV/AIDS. According to our theoretical results, everyone of us should be optimistic and be full of confidence in the monitoring and control of HIV/AIDS epidemic although no infected individuals can be cured and obtain the immunity to HIV/AIDS epidemic till now.

An age-structured multi-strain epidemic model for antigenically diverse infectious diseases: A multi-locus framework

The establishment of cross-protective responses and development of immunity within a host exert pressure on pathogens through cross-immunity mediated competition between antigenic forms. In this paper, we incorporate age-specificity in the multi-locus epidemic model used to study the pathogen-specific dynamic behaviours for infectious diseases with diverse co-circulating antigenic types. We establish results on the existence of a unique mild solution, and on the necessary conditions for local stability of the steady-states. In particular, we find that, when the reproductive number R0<1, all strains go to extinction. When R0>1, we show that there exist additional conditions which determine the stability of different types of endemic equilibria, namely weak and strong endemicity, where the weak endemic equilibria correspond to the existence of principle of competitive exclusions of pathogen-specific clusters, while strong endemicity represents the co-existence of all strains. Using numerical simulations, we also show that weak endemic equilibria yield dynamic features in which only one of the clusters containing discrete strain structures (e.g., of minimally, or non-overlapping antigenic types) persists while others go to extinction. For unique strong endemicity, we observe no strain structure, where antigenic types co-exist or exhibit cyclical strain structure with diverse dynamical behaviours (e.g., (quasi-)periodicity, intermittency, chaos). This demonstrates that pathogenic-specific dynamic features are ubiquitous and shows how cross-immunity between antigenic variants shape the maintenance and evolution of strain structures.

Impact on Epidemic Measles of Vaccination Campaigns Triggered by Disease Outbreaks or Serosurveys: A Modeling Study.

BackgroundRoutine vaccination supplemented by planned campaigns occurring at 2–5 y intervals is the core of current measles control and elimination efforts. Yet, large, unexpected outbreaks still occur, even when control measures appear effective. Supplementing these activities with mass vaccination campaigns triggered when low levels of measles immunity are observed in a sample of the population (i.e., serosurveys) or incident measles cases occur may provide a way to limit the size of outbreaks.Methods and FindingsMeasles incidence was simulated using stochastic age-structured epidemic models in settings conducive to high or low measles incidence, roughly reflecting demographic contexts and measles vaccination coverage of four heterogeneous countries: Nepal, Niger, Yemen, and Zambia. Uncertainty in underlying vaccination rates was modeled. Scenarios with case- or serosurvey-triggered campaigns reaching 20% of the susceptible population were compared to scenarios without triggered campaigns. The best performing of the tested case-triggered campaigns prevent an average of 28,613 (95% CI 25,722–31,505) cases over 15 y in our highest incidence setting and 599 (95% CI 464–735) cases in the lowest incidence setting. Serosurvey-triggered campaigns can prevent 89,173 (95% CI, 86,768–91,577) and 744 (612–876) cases, respectively, but are triggered yearly in high-incidence settings. Triggered campaigns reduce the highest cumulative incidence seen in simulations by up to 80%. While the scenarios considered in this strategic modeling exercise are reflective of real populations, the exact quantitative interpretation of the results is limited by the simplifications in country structure, vaccination policy, and surveillance system performance. Careful investigation into the cost-effectiveness in different contexts would be essential before moving forward with implementation.ConclusionsSerologically triggered campaigns could help prevent severe epidemics in the face of epidemiological and vaccination uncertainty. Hence, small-scale serology may serve as the basis for effective adaptive public health strategies, although, in high-incidence settings, case-triggered approaches are likely more efficient.

Efficient Vaccine Distribution Based on a Hybrid Compartmental Model.

To effectively and efficiently reduce the morbidity and mortality that may be caused by outbreaks of emerging infectious diseases, it is very important for public health agencies to make informed decisions for controlling the spread of the disease. Such decisions must incorporate various kinds of intervention strategies, such as vaccinations, school closures and border restrictions. Recently, researchers have paid increased attention to searching for effective vaccine distribution strategies for reducing the effects of pandemic outbreaks when resources are limited. Most of the existing research work has been focused on how to design an effective age-structured epidemic model and to select a suitable vaccine distribution strategy to prevent the propagation of an infectious virus. Models that evaluate age structure effects are common, but models that additionally evaluate geographical effects are less common. In this paper, we propose a new SEIR (susceptible—exposed—infectious šC recovered) model, named the hybrid SEIR-V model (HSEIR-V), which considers not only the dynamics of infection prevalence in several age-specific host populations, but also seeks to characterize the dynamics by which a virus spreads in various geographic districts. Several vaccination strategies such as different kinds of vaccine coverage, different vaccine releasing times and different vaccine deployment methods are incorporated into the HSEIR-V compartmental model. We also design four hybrid vaccination distribution strategies (based on population size, contact pattern matrix, infection rate and infectious risk) for controlling the spread of viral infections. Based on data from the 2009–2010 H1N1 influenza epidemic, we evaluate the effectiveness of our proposed HSEIR-V model and study the effects of different types of human behaviour in responding to epidemics.