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An age-dependent epidemic model with application to measles

DOI:10.1016/0025-5564(85)90081-1
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David Tudor at Germinal Knowledge. SARL
David Tudor
  • Germinal Knowledge. SARL
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Abstract

A combined epidemic-demographic model is developed which models the spread of an infectious disease throughout a population of constant size. The model allows for births, deaths, temporary or permanent immunity, and immunization. The relationship of this model to previously studied epidemic and demographic models is illustrated. An advantage of this model is that all epidemic and demographic parameters may be estimated. The stability of the equilibrium point corresponding to the elimination of the disease is studied and a threshold value is found which indicates whether the disease will die out or remain endemic in the population. The application of the model to measles indicates that immunization levels needed to reduce the incidence to near zero may not be as high as previously predicted.
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... This yields a nonlinear finite dimensional model for which some known theories can be applied. Inspired by Tudor's article [28], Model (1) is discretized in n classes of age, [0, a 1 ) , [a 1 , a 2 ) , ..., [a n−1 , L). The proportion of S-individuals in the k−th class of age represents the fraction of individuals in class age k that is susceptible at time t, which gives ...
... Remark that, since L is the maximum age, ρ n equals 0. The transfer rate ρ k is also used for I (t, a k ) and R (t, a k ). As mentioned in [28], integrating the equations of Model (1) with respect to the age variable, from a k−1 to a k , for k = 1, ..., n and using previous assumptions and initial conditions of Model (1), lead to the following system of nonlinear ODEs, ...
... Another condition for the feedback design is that the feedback law has to keep the positivity of the variables in the model (more precisely it has to keep them between 0 and 1) in order to have a physical meaning. Inspired by [28], we highlight the following positivity condition. ...
State feedback control law design for an age-dependent SIR model
Preprint
  • May 2022
An age-dependent SIR model is considered with the aim to develop a state-feedback vaccination law in order to eradicate a disease. A dynamical analysis of the system is performed using the principle of linearized stability and shows that, if the basic reproduction number is larger than 1, the disease free equilibrium is unstable. This result justifies the developement of a vaccination law. Two approaches are used. The first one is based on a dicretization of the partial integro-differential equations (PIDE) model according to the age. In this case a linearizing feedback law is found using Isidori's theory. Conditions guaranteeing stability and positivity are established. The second approach yields a linearizing feedback law developed for the PIDE model. This law is deduced from the one obtained for the ODE case. Using semigroup theory, stability conditions are also obtained. Finally, numerical simulations are presented to reinforce the theoretical arguments.
... Actually, the work of D. Bernoulli cited above dealt with measles epidemic based on the chronological age of the population, while the pioneer model of was based on the infection age of individuals. Starting in the 1970s, researchers have noticed that the chronological age of the host population plays a crucial role in the transmission process of infectious diseases and have proposed various age-structured models to study the transmission dynamics of childhood diseases, in particular measles, see for instance, Anderson and May [8], Corey and Noymer [9], Greenhalgh [10,11], Halloran et al. [12], Hethcote [13], Huang and Rohani [14], Kang et al. [15], Manfredi and Williams [16], McLean and Anderson [17,18], Schenzle [19], and Tudor [20]. We refer to the monographs of Iannelli [21], Inaba [22], Li et al. [23], and Webb [24] for fundamental theories on age-structured epidemic models. ...
... These facts indicate that it is reasonable to assume the mixing between susceptible and infectious individuals and use standard incidence rate to describe the transmission. Since one of the main measures in controlling measles is to find the optimal age to vaccinate children in order to have the maximum impact on the incidence of disease-related morbidity and mortality for a given rate of vaccination coverage, age-structured epidemic models have been extensively used to study the transmission dynamics and control of measles, see Anderson and May [8], Corey and Noymer [9], Greenhalgh [10,11], Halloran et al. [12], Hethcote [13], Huang and Rohani [14], Kang et al. [15], Manfredi and Williams [16], McLean and Anderson [17,18], Schenzle [19], and Tudor [20]. ...
Stability analysis of an age-structured epidemic model with vaccination and standard incidence rate
Article
Age structure of the host population is a crucial factor in the transmission and control of infectious diseases, since the risk from an infection increases along with age, different age groups interact heterogeneously, vaccination programs focus on specific age groups, and epidemiological data are reported according to ages. In this paper we consider an age-structured epidemic model of the susceptible–exposed–infectious–recovered (SEIR) type with vaccination and standard incidence rate. After establishing the well-posedness of the initial–boundary value problem, we study the existence and stability of the disease-free and endemic steady states based on the basic reproduction number R0. It is shown that the disease-free steady state is globally asymptotically stable if R0<1, the endemic steady state is unique if R0<1 and is locally asymptotically stable under some additional conditions. Some numerical simulations are presented to illustrate the theoretical results.
... We utilized the social network model, instead of the population models with differential equations, such as the SIR models, because our aim is to clarify the consequences of the actions of people that either generate, prune, or perturb the local contact within neighbors, and accordingly, to derive the knowledge required for determining appropriate measures that need to be imposed for preventing the spread of infection. This advantage of the network-based model has been highlighted by Karaivanov in [1], wherein, its differences from population-based models such as SIR [2] and its extensions [3][4][5][6][7] were clarified. Here, the effects of preventive actions such as lockdown, its release, and of infections via edges bridging far-apart nodes, on infection spread have been evaluated. ...
Stay with your community: Bridges between clusters trigger expansion of COVID-19
Article
Full-text available
In this study, the spread of virus infection was simulated using artificial human networks. Here, real-space urban life was modeled as a modified scale-free network with constraints. To date, the scale-free network has been adopted for modeling online communities in several studies. However, in the present study, it has been modified to represent the social behaviors of people where the generated communities are restricted and reflect spatiotemporal constraints in real life. Furthermore, the networks have been extended by introducing multiple cliques in the initial step of network construction and enabling people to contact hidden (zero-degree) as well as popular (large-degree) people. Consequently, four findings and a policy proposal were obtained. First, “second waves” were observed in some cases of the simulations even without external influence or constraints on people’s social contacts or the releasing of the constraints. These waves tend to be lower than the first wave and occur in “fresh” clusters, that is, via the infection of people who are connected in the network but have not been infected previously. This implies that the bridge between infected and fresh clusters may trigger a new spread of the virus. Second, if the network changes its structure on the way of infection spread or after its suppression, a second wave larger than the first can occur. Third, the peak height in the time series of the number of infected cases depends on the difference between the upper bound of the number of people each member actually meets and the number of people they choose to meet during the period of infection spread. This tendency is observed for the two kinds of artificial networks introduced here and implies the impact of bridges between communities on the virus spreading. Fourth, the release of a previously imposed constraint may trigger a second wave higher than the peak of the time series without introducing any constraint so far previously, if the release is introduced at a time close to the peak. Thus, overall, both the government and individuals should be careful in returning to society where people enjoy free inter-community contact.
... This model is then used to describe measles by substituting in the model an immunity loss rate of zero. The application of the model to measles indicates that the immunization levels needed to reduce the incidence to near zero may not be as high as previously predicted by homogeneous mixing [27]. ...
Short review of mathematical model of measles
Conference Paper
  • Sep 2020
Measles is a very contagious and serious disease caused by a virus in the paramyxovirus family. This disease is a major cause of death in several countries, especially in children. According to WHO, the main strategy to eradicate measles involves vaccines and medical treatment. Various models have been introduced by many authors involving various factors in the measles dynamics model, such as the measles vaccine strategy, the phenomenon of measles co-infection, age structures, seasonal parame- ters, and human mobility model using metapopulation approach to describe the spread of measles among the human population. Most authors examine the qualitative behavior of their model, namely equilibrium points, basic reproduction numbers, local and global stability, and bifurcation analysis. In this article, we give a comprehensive overview of how these forms of literature are related to each other and to see another chance in future modelling of measles dynamics.
... The reason why we use the social network model instead of the population models with differential equations such as the SIR models is that we aim to clarify the effects of actions of people that generate, prune, or perturb local connections with neighbors so that we acquire knowledge about measures for preventing infection spread. This benefit of the network-based model has been pointed out in [1], where the difference from population-based models (such as SIR [2] and its extensions [3][4][5][6][7]), the effects of preventive actions such as a lock-down and its release and of infections via edges bridging far-apart nodes on infection spread has been evaluated. ...
Stay with Your Community: Bridges between Clusters Trigger Expansion of COVID-19
Preprint
Full-text available
  • Jun 2020
The spreading of virus infection is here simulated over artificial human networks. The real-space urban life of people is modeled as a modified scale-free network with constraints. A scale-free network has been adopted in several studies for modeling on-line communities so far but is modified here for the aim to represent peoples' social behaviors where the generated communities are restricted reflecting the spatiotemporal constraints in the real life. Furthermore, the networks have been extended by introducing multiple cliques in the initial step of network construction and enabling people to zero-degree people as well as popular (large degree) people. As a result, four findings and a policy proposal have been obtained. First, the "second waves" occur without external influence or constraints on contacts or the releasing of the constraints. These second waves, mostly lower than the first wave, implies the bridges between infected and fresh clusters may trigger new expansions of spreading. Second, if the network changes the structure on the way of infection spreading or after its suppression, the peak of the second wave can be larger than the first. Third, the peak height in the time series depends on the difference between the upper bound of the number of people each member accepts to meet and the number of people one chooses to meet. This tendency is observed for two kinds of artificial networks and implies the impact of the bridges between communities on the virus spreading. Fourth, the release of once given constraint may trigger a second wave higher than the peak of the time series without introducing any constraint from the beginning, if the release is introduced at a time close to the peak. Thus, both governments and individuals should be careful in returning to human society with inter-community contacts.
... Prior studies have developed accurate prediction and control models in the setting of other emerging outbreaks including West Nile Virus, Avian influenza and SARS, among others [8][9][10]. Some previous studies have focused on developing analytical or algorithmic understanding of policies for epidemic control or eradication [11][12][13][14][15][16], while others have examined techniques for constructing realistic epidemic simulation environments [17][18][19]. ...
Structure of epidemic models: toward further applications in economics
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We know that the exact solutions for most of stochastic age-structured capital systems are difficult to find. The numerical approximation method becomes an important tool to study properties for stochastic age-structured capital models. In the paper, we study the numerical solution for stochastic age-dependent capital system based on explicit-implicit algorithm and discuss the convergence of numerical solution. For the practical significance of capital, we need to consider the positivity of the numerical solution. Therefore, we introduce a penalty factor in the stochastic age-dependent capital system to maintain the positivity, and analyze the convergence of the positive numerical solution. Finally, an example is given to verify our theoretical results.
The asymptotic stability of numerical analysis for stochastic age-dependent cooperative Lotka-Volterra system
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In this study, we explore a stochastic age-dependent cooperative Lotka-Volterra (LV) system with an environmental noise. By applying the theory of M-matrix, we prove the existence and uniqueness of the global solution for the system. Since the stochastic age-dependent cooperative LV system cannot be solved explicitly, we then construct an Euler-Maruyama (EM) numerical solution to approach the exact solution of the system. The convergence rate and the pth-moment boundedness of the scheme have also been obtained. Additionally, numerical experiments have been conducted to verify our theoretical results.
The Opportunity and Obligation to Eliminate Measles From the United States
Article
  • Sep 1979
ON OCT 4, 1978, the Secretary of the Department of Health, Education, and Welfare, Joseph A. Califano, Jr, announced, "We are launching an effort that seeks to free the United States from measles by October 1, 1982" (statement made in Washington, DC, Oct 4, 1978). In announcing this initiative, the secretary noted the "remarkable progress made in controlling measles in the past 15 years" that made possible such an attempt to "eliminate indigenous cases of measles in this country." The purpose of this article is to describe the epidemiologic and operational history behind this initiative, to present its scientific rationale, and to summarize the major tactics viewed as essential to its successful execution.Background Measles has had severe impact in the United States. Outbreaks were reported as early as 1635; a major epidemic occurred in the colonies in the period 1713 to 1715. In 1772, it was reported that 800
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  • Jan 1985
The purpose of this note is to demonstrate the need of using models with explicit age-structure when dealing with questions of control of virus transmission.
Mathematical theories of populations: Demographics, genetics, and epidemics
Article
  • Jan 1975
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Hethcote, H. Qualitative analyses of communicable disease models. Mathematical Biosciences
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Deterministic communicable disease models which are initial value problems for a system of ordinary differential equations are considered, where births and deaths occur at equal rates with all newborns being susceptible. Asymptotic stability regions are determined for the equilibrium points for models involving temporary immunity, disease-related fatalities, carriers, migration, dissimilar interacting groups, and transmission by vectors. Epidemiological interpretations of all results are given.
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The Incidence of Infectious Diseases Under the Influence of Seasonal Fluctuations.
Article
  • Jan 1976
The present paper is concerned with the effect of seasonal variations of the contact rate on the incidence of infectious diseases. The regular oscillations of the number of cases around the average endemic level has attracted the attention of epidemiologists and mathematicians alike, In particular, the two-year period of measles in some large communities has been the object of many attempts of explanation in terms of deterministic and stochastic models, Soper’s [1] deterministic approach produced damped oscillations in contrast to the observations, Bartlett [2] suggested that a stochastic version of Soper’s model was more realistic, (See also Bailey [3], Chap, 7.) London and Yorke [4] however were able to obtain undamped oscillations with periods of one and two years using a deterministic model which includes a latent period between the time of infection and the beginning of the infectious period, From their simulations they conclude that the length of the latent period has to be within a small range for the occurrence of biennial outbreaks, Recently, Stirzaker [5] treated this problem from the point of view of the theory of nonlinear oscillations according to which the biennial cycles of measles epidemics could be understood as subharmonic parametric resonance.