Final size and convergence rate for an epidemic in heterogeneous populations

2021 ◽  
pp. 1-31 ◽  
Author(s):  
Luis Almeida ◽  
Pierre-Alexandre Bliman ◽  
Grégoire Nadin ◽  
Benoît Perthame ◽  
Nicolas Vauchelet
Keyword(s):  
Reproduction Number ◽  
Heterogeneous Population ◽  
Dominant Eigenvalue ◽  
Initial Condition ◽  
Initial Value ◽  
Final Size ◽  
Diffusion Term ◽  
Contact Matrix ◽  
Metapopulation Models ◽  
The Basic Reproduction Number

We formulate a general SEIR epidemic model in a heterogeneous population characterized by some trait in a discrete or continuous subset of a space [Formula: see text]. The incubation and recovery rates governing the evolution of each homogeneous subpopulation depend upon this trait, and no restriction is assumed on the contact matrix that defines the probability for an individual of a given trait to be infected by an individual with another trait. Our goal is to derive and study the final size equation fulfilled by the limit distribution of the population. We show that this limit exists and satisfies the final size equation. The main contribution of this work is to prove the uniqueness of this solution among the distributions smaller than the initial condition. We also establish that the dominant eigenvalue of the next-generation operator (whose initial value is equal to the basic reproduction number) decreases along every trajectory until a limit smaller than 1. The results are shown to remain valid in the presence of a diffusion term. They generalize previous works corresponding to finite number of traits (including metapopulation models) or to rank 1 contact matrices (modeling e.g. susceptibility or infectivity presenting heterogeneity independently of one another).

scholarly journals Prediction of confinement effects on the number of Covid-19 outbreak in Algeria

2020 ◽  
Vol 15 ◽  
pp. 37 ◽  
Author(s):  
Ali Moussaoui ◽  
Pierre Auger
Keyword(s):  
Reproduction Number ◽  
Control Measures ◽  
Cumulative Number ◽  
Time Data ◽  
Final Size ◽  
First Case ◽  
Algerian Population ◽  
Fast Change ◽  
Real Time Data ◽  
The Basic Reproduction Number

The first case of coronavirus disease 2019 (COVID-19) in Algeria was reported on 25 February 2020. Since then, it has progressed rapidly and the number of cases grow exponentially each day. In this article, we utilize SEIR modelling to forecast COVID-19 outbreak in Algeria under two scenarios by using the real-time data from March 01 to April 10, 2020. In the first scenario: no control measures are put into place, we estimate that the basic reproduction number for the epidemic in Algeria is 2.1, the number of new cases in Algeria will peak from around late May to early June and up to 82% of the Algerian population will likely contract the coronavirus. In the second scenario, at a certain date T, drastic control measures are taken, people are being advised to self-isolate or to quarantine and will be able to leave their homes only if necessary. We use SEIR model with fast change between fully protected and risky states. We prove that the final size of the epidemic depends strongly on the cumulative number of cases at the date when we implement intervention and on the fraction of the population in confinement. Our analysis shows that the longer we wait, the worse the situation will be and this very quickly produces.

scholarly journals Extended SEIQR type model for COVID-19 epidemic and data analysis

2020 ◽  
Author(s):  
Swarnali Sharma ◽  
Vitaly Volpert ◽  
Malay Banerjee
Keyword(s):  
Social Interaction ◽  
Disease Progression ◽  
Reproduction Number ◽  
European Countries ◽  
Type Model ◽  
Slow Disease Progression ◽  
Final Size ◽  
Social Contacts ◽  
Detailed Model ◽  
The Basic Reproduction Number

An extended SEIQR type model is considered in order to model the COVID-19 epidemic. It contains the classes of susceptible individuals, exposed, infected symptomatic and asymptomatic, quarantined, hospitalized and recovered. The basic reproduction number and the final size of epidemic are determined. The model is used to fit available data for some European countries. A more detailed model with two different subclasses of susceptible individuals is introduced in order to study the influence of social interaction on the disease progression. The coefficient of social interaction $K$ characterizes the level of social contacts in comparison with complete lockdown (K=0) and the absence of lockdown (K=1). The fitting of data shows that the actual level of this coefficient in some European countries is about 0.1, characterizing a slow disease progression. A slight increase of this value in the autumn can lead to a strong epidemic burst.

scholarly journals On a Stochastic SEIS Model with Treatment Rate of Latent Population

2014 ◽  
Vol 2014 ◽  
pp. 1-16
Author(s):  
Shujing Gao ◽  
Yanfei Dai ◽  
Yan Zhang ◽  
Yujiang Liu
Keyword(s):  
Asymptotic Behavior ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Deterministic System ◽  
Treatment Rate ◽  
Initial Value ◽  
Asymptotic Dynamics ◽  
Disease Free ◽  
The Basic Reproduction Number

The asymptotic dynamics of a stochastic SEIS epidemic model with treatment rate of latent population is investigated. First, we show that the system provides a unique positive global solution starting from the positive initial value. Then, the long-term asymptotic behavior of the model is studied: ifR0, which is called the basic reproduction number of the corresponding deterministic model, is not more than unity, the solution of the model is oscillating around the disease-free equilibrium of the corresponding deterministic system, whereas ifR0is larger than unity, we show how the solution spirals around the endemic equilibrium of deterministic system under certain parametric restrictions. Finally, numerical simulations are carried out to support our theoretical findings.

scholarly journals Graphs with specified degree distributions, simple epidemics, and local vaccination strategies

2007 ◽  
Vol 39 (04) ◽  
pp. 922-948 ◽  
Author(s):  
Tom Britton ◽  
Svante Janson ◽  
Anders Martin-Löf
Keyword(s):  
Infectious Disease ◽  
Social Structure ◽  
Basic Reproduction Number ◽  
Degree Distribution ◽  
Reproduction Number ◽  
Final Size ◽  
Vaccination Strategies ◽  
The Social ◽  
Major Outbreak ◽  
The Basic Reproduction Number

Consider a random graph, having a prespecified degree distribution F, but other than that being uniformly distributed, describing the social structure (friendship) in a large community. Suppose that one individual in the community is externally infected by an infectious disease and that the disease has its course by assuming that infected individuals infect their not yet infected friends independently with probability p. For this situation, we determine the values of R 0, the basic reproduction number, and τ0, the asymptotic final size in the case of a major outbreak. Furthermore, we examine some different local vaccination strategies, where individuals are chosen randomly and vaccinated, or friends of the selected individuals are vaccinated, prior to the introduction of the disease. For the studied vaccination strategies, we determine R v , the reproduction number, and τ v , the asymptotic final proportion infected in the case of a major outbreak, after vaccinating a fraction v.

scholarly journals Computational and Mathematical Methods to Estimate the Basic Reproduction Number and Final Size for Single-Stage and Multistage Progression Disease Models for Zika with Preventative Measures

2017 ◽  
Vol 2017 ◽  
pp. 1-17 ◽  
Author(s):  
P. Padmanabhan ◽  
P. Seshaiyer
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Single Stage ◽  
Final Size ◽  
Multistage Model ◽  
Preventative Measures ◽  
Final Size Relation ◽  
The Impact ◽  
Size Relation ◽  
The Basic Reproduction Number

We present new mathematical models that include the impact of using selected preventative measures such as insecticide treated nets (ITN) in controlling or ameliorating the spread of the Zika virus. For these models, we derive the basic reproduction number and sharp estimates for the final size relation. We first present a single-stage model which is later extended to a new multistage model for Zika that incorporates more realistic incubation stages for both the humans and vectors. For each of these models, we derive a basic reproduction number and a final size relation estimate. We observe that the basic reproduction number for the multistage model converges to expected values for a standard Zika epidemic model with fixed incubation periods in both hosts and vectors. Finally, we also perform several computational experiments to validate the theoretical results obtained in this work and study the influence of various parameters on the models.

scholarly journals Graphs with specified degree distributions, simple epidemics, and local vaccination strategies

2007 ◽  
Vol 39 (4) ◽  
pp. 922-948 ◽  
Author(s):  
Tom Britton ◽  
Svante Janson ◽  
Anders Martin-Löf
Keyword(s):  
Infectious Disease ◽  
Social Structure ◽  
Basic Reproduction Number ◽  
Degree Distribution ◽  
Reproduction Number ◽  
Final Size ◽  
Vaccination Strategies ◽  
The Social ◽  
Major Outbreak ◽  
The Basic Reproduction Number

Consider a random graph, having a prespecified degree distribution F, but other than that being uniformly distributed, describing the social structure (friendship) in a large community. Suppose that one individual in the community is externally infected by an infectious disease and that the disease has its course by assuming that infected individuals infect their not yet infected friends independently with probability p. For this situation, we determine the values of R0, the basic reproduction number, and τ0, the asymptotic final size in the case of a major outbreak. Furthermore, we examine some different local vaccination strategies, where individuals are chosen randomly and vaccinated, or friends of the selected individuals are vaccinated, prior to the introduction of the disease. For the studied vaccination strategies, we determine Rv, the reproduction number, and τv, the asymptotic final proportion infected in the case of a major outbreak, after vaccinating a fraction v.

scholarly journals Estimation of measles vaccine efficacy and critical vaccination coverage in a highly vaccinated population

2010 ◽  
Vol 7 (52) ◽  
pp. 1537-1544 ◽  
Author(s):  
Michiel van Boven ◽  
Mirjam Kretzschmar ◽  
Jacco Wallinga ◽  
Philip D O'Neill ◽  
Ole Wichmann ◽  
...  
Keyword(s):  
Public School ◽  
Vaccination Coverage ◽  
Basic Reproduction Number ◽  
Vaccine Efficacy ◽  
Reproduction Number ◽  
Herd Immunity ◽  
Credible Interval ◽  
Final Size ◽  
Measles Elimination ◽  
The Basic Reproduction Number

Measles is a highly infectious disease that has been targeted for elimination from four WHO regions. Whether and under which conditions this goal is feasible is, however, uncertain since outbreaks have been documented in populations with high vaccination coverage (more than 90%). Here, we use the example of a large outbreak in a German public school to show how estimates of key epidemiological parameters such as the basic reproduction number ( R 0 ), vaccine efficacy (VE S ) and critical vaccination coverage ( p c ) can be obtained from partially observed outbreaks in highly vaccinated populations. Our analyses rely on Bayesian methods of inference based on the final size distribution of outbreak size, and use data which are easily collected. For the German public school the analyses indicate that the basic reproduction number of measles is higher than previously thought ( , 95% credible interval: 23.6–40.4), that the vaccine is highly effective in preventing infection ( , 95% credible interval: 0.993–0.999), and that a vaccination coverage in excess of 95 per cent may be necessary to achieve herd immunity ( , 95% credible interval: 0.961–0.978). We discuss the implications for measles elimination from highly vaccinated populations.

scholarly journals Epidemic Progression and Vaccination in a Heterogeneous Population. Application to the Covid-19 epidemic

2020 ◽  
Author(s):  
Vitaly Volpert ◽  
Malay Banerjee ◽  
Swarnali Sharma
Keyword(s):  
Disease Transmission ◽  
Reproduction Number ◽  
Heterogeneous Population ◽  
Time Behaviour ◽  
Final Size ◽  
Long Time Behaviour ◽  
Long Time ◽  
Epidemic Development ◽  
In The Beginning

AbstractThe paper is devoted to a compartmental epidemiological model of infection progression in a heterogeneous population which consists of two groups with high disease transmission (HT) and low disease transmission (LT) potentials. Final size and duration of epidemic, the total and current maximal number of infected individuals are estimated depending on the structure of the population. It is shown that with the same basic reproduction number R0 in the beginning of epidemic, its further progression depends on the ratio between the two groups. Therefore, fitting the data in the beginning of epidemic and the determination of R0 are not sufficient to predict its long time behaviour. Available data on the Covid-19 epidemic allows the estimation of the proportion of the HT and LT groups. Estimated structure of the population is used for the investigation of the influence of vaccination on further epidemic development. The result of vaccination strongly depends on the proportion of vaccinated individuals between the two groups. Vaccination of the HT group acts to stop the epidemic and essentially decreases the total number of infected individuals at the end of epidemic and the current maximal number of infected individuals while vaccination of the LT group only acts to protect vaccinated individuals from further infection.

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