Final Size Equation Research Articles

Final epidemic size and critical times for susceptible-infectious-recovered models with a generalized contact rate.

During the spread of an infectious disease, the contact rate or the incidence rate may affect disease characteristics. For simplicity, most disease models assume standard incidence or mass action rates to calculate the basic reproduction number, final epidemic size, and peak time of an epidemic. For standard incidence, the contact rate remains constant resulting in the incidence rate is inversely proportional to the population size, while for the mass action rate, this contact rate is proportional to the total population size resulting in the incidence rate is independent of the population size. In this paper, we consider susceptible-infectious-recovered epidemic models with a generalized contact rate C(N) and a nonlinear incidence rate in view of the behavioral change from susceptible or infectious individuals when an infectious disease appears. The basic reproduction number and the final size equation are derived. The impact of different types of contact rates on them is studied. Moreover, two critical times (peak time and epidemic duration) of an epidemic are considered. Explicit formulas for the peak time and epidemic duration are obtained. These formulas are helpful not only for taking early effective epidemic precautions but also for understanding how the epidemic duration can be changed by acting on the model parameters, especially when the epidemic model is used to make public health policy.

Age-dependent final size equation to anticipate mortality impact of COVID-19 in China.

Before reopening society in December 2022, China had not achieved sufficiently high vaccination coverage among people aged 80 years and older, who are vulnerable to severe infection and death owing to COVID-19. Suddenly ending the zero-COVID policy was anticipated to lead to substantial mortality. To investigate the mortality impact of COVID-19, we devised an age-dependent transmission model to derive a final size equation, permitting calculation of the expected cumulative incidence. Using an age-specific contact matrix and published estimates of vaccine effectiveness, final size was computed as a function of the basic reproduction number, R0. We also examined hypothetical scenarios in which third-dose vaccination coverage was increased in advance of the epidemic, and also in which mRNA vaccine was used instead of inactivated vaccines. Without additional vaccination, the final size model indicated that a total of 1.4 million deaths (half of which were among people aged 80 years and older) were anticipated with an assumed R0 of 3.4. A 10% increase in third-dose coverage would prevent 30,948, 24,106, and 16,367 deaths, with an assumed second-dose effectiveness of 0%, 10%, and 20%, respectively. With mRNA vaccine, the mortality impact would have been reduced to 1.1 million deaths. The experience of reopening in China indicates the critical importance of balancing pharmaceutical and non-pharmaceutical interventions. Ensuring sufficiently high vaccination coverage is vital in advance of policy changes.

Final size and convergence rate for an epidemic in heterogeneous populations

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).

A method for estimating the transmissibility of influenza using serial cross-sectional seroepidemiological data

BackgroundSeroepidemiological surveillance data has been demonstrated to be useful for estimating the cumulative incidence of influenza, and measures the difference between pre- and post-epidemic seropositive fractions. Despite this, such studies relied on a chosen cut-off value for seropositivity. The aim of the present study is to develop a method to analyze distributions of serial cross-sectional seroepidemiological surveillance datasets using an epidemiological model so that the transmission potential can be estimated without imposing a cut-off value. MethodsA mathematical model of influenza transmission with a discrete antibody titer level was constructed. The final size equation for pre- and post-epidemic titer levels was derived. Subsequently, using the estimated distribution of the dilution increase caused by infection and the measurement error distribution, the model parameters were optimized using the maximum likelihood method. A bootstrap-based confidence interval calculation and sensitivity analysis were also performed. ResultsWithout imposing a cut-off value, the cumulative incidence was quantified, thereby yielding an estimate of the basic reproduction number. For the purpose of exposition, the proposed method was applied to influenza A/Victoria/3/75(H3N2) data, and serological data between 1975 and 1976 were compared. The estimated reproduction number was greater than that using the cut-off value of the hemagglutination inhibition level with titer level 20 (dilution 1:20) or above to define positives. ConclusionThe proposed method without a cut-off value offers an unbiased approach to estimating the cumulative incidence along with the reproduction number. If a cut-off value is required, the results imply that titer level 20 or above may better represent a reasonable cut-off value for calculating the incidence, but it could underestimate the basic reproduction number.

Calculation of final size for vector-transmitted epidemic model.

Calculation of final size of an epidemic model offers a useful estimation for the impact of an epidemic. Despite its usefulness, the majority of practical applications focuses on the classical Kermack McKendrick model for final size calculation. Estimation of final size for different types of epidemics such as vector-transmitted infection is a forthcoming target. In this paper, we derive an explicit form of a final size equation for a vector-transmitted epidemic model. Numerical calculation of a final size equation revealed the existence of a threshold curve which separates a region into two distinct bistable sub-regions if infection induced death is present. In other words, an epidemic outcome can be qualitatively different depending on the initial state of an epidemic.

Trade-off between disease resistance and crop yield: a landscape-scale mathematical modelling perspective.

The deployment of crop varieties that are partially resistant to plant pathogens is an important method of disease control. However, a trade-off may occur between the benefits of planting the resistant variety and a yield penalty, whereby the standard susceptible variety outyields the resistant one in the absence of disease. This presents a dilemma: deploying the resistant variety is advisable only if the disease occurs and is sufficient for the resistant variety to outyield the infected standard variety. Additionally, planting the resistant variety carries with it a further advantage in that the resistant variety reduces the probability of disease invading. Therefore, viewed from the perspective of a grower community, there is likely to be an optimal trade-off and thus an optimal cropping density for the resistant variety. We introduce a simple stochastic, epidemiological model to investigate the trade-off and the consequences for crop yield. Focusing on susceptible-infected-removed epidemic dynamics, we use the final size equation to calculate the surviving host population in order to analyse the yield, an approach suitable for rapid epidemics in agricultural crops. We identify a single compound parameter, which we call the efficacy of resistance and which incorporates the changes in susceptibility, infectivity and durability of the resistant variety. We use the compound parameter to inform policy plots that can be used to identify the optimal strategy for given parameter values when an outbreak is certain. When the outbreak is uncertain, we show that for some parameter values planting the resistant variety is optimal even when it would not be during the outbreak. This is because the resistant variety reduces the probability of an outbreak occurring.

An epidemic model with noisy parameters

We analyse an SIR model where the epidemiological parameters are subject to small amplitude random fluctuations. We derive a final size equation and extend the result to an SEIR model. We use a small amplitude perturbation to estimate the expected final size of the SIR model and its variance, and compare the result with numerical simulations. We show that although individual realisations may exhibit considerable variation around solutions of the deterministic model, the mean of the final size distribution is in good agreement with the deterministic final size, and its standard deviation is small compared to the mean.

On a Pandemic Threshold Theorem of the Early Kermack–McKendrick Model with Individual Heterogeneity

A pandemic threshold theorem of the Kermack–McKendrick epidemic system with individual heterogeneity is proved from the definition of R 0 by Diekmann, Heesterbeek, and Metz. The early Kermack–McKendrick epidemic model is extended to recognize individual heterogeneity, where the state variable indicates an epidemiological state or genetic, physiological, or behavioral characteristics such as risk of infection. With the basic reproduction number R 0 for the heterogeneous population, the final size equation of the limit epidemic starting from a completely susceptible steady state at t = −∞ has a unique positive solution if and only if R 0 > 1. The main result is that the positive solution of the final size equation gives the lower bound of the intensity of any epidemic starting from a host population composed of susceptible and a few infected individuals who spread on a noncompact domain of the trait variable.

Case-Based Surveillance of Pandemic (H1N1) 2009 in Maebashi City, Japan

After national case-based surveillance for pandemic influenza A (H1N1) ceased on July 23, 2009, a daily case-based surveillance system was implemented in Maebashi City, Japan. All medical facilities in the city reported all patients who had positive rapid antigen tests for influenza A. When the epidemic exploded in late October, case-based surveillance for influenza-like illness (ILI) was implemented from November 3, 2009 until the end of the epidemic. A total of 7,781 influenza cases were reported between July 25 and November 2, 2009, with a cumulative incidence rate of 22.5 per 1,000 population. Nearly 70% of the patients were under 15 years old. Between November 3, 2009 and the end of March 2010, a total of 16,394 ILI cases were reported, with a cumulative incidence rate of 47.4 per 1,000 population. Of the ILI cases reported, 63% were in patients younger than 15 years old. Only one death with laboratory confirmation of the H1N1 2009 virus was reported during the epidemic. The age-specific reproduction number among children under 15 years of age was almost 1.40, whereas between children and adults (15 years of age and above) it was considerably less than 1.0. The reproduction number derived from the next-generation matrix using data from September 30 to October 14 was estimated to be 1.48 (95% confidence interval, 1.41-1.56). Among individuals under 15 years of age, the infection rate calculated using the final size equation under the assumption of no mitigation measures was nearly twice the rate reported during the epidemic. These findings indicate that the majority of the transmission of influenza A (H1N1) 2009 in the city occurred among children.

Global dynamics of cholera models with differential infectivity

A general compartmental model for cholera is formulated that incorporates two pathways of transmission, namely direct and indirect via contaminated water. Non-linear incidence, multiple stages of infection and multiple states of the pathogen are included, thus the model includes and extends cholera models in the literature. The model is analyzed by determining a basic reproduction number R 0 and proving, by using Lyapunov functions and a graph-theoretic result based on Kirchhoff’s Matrix Tree Theorem, that it determines a sharp threshold. If R 0 ⩽ 1 , then cholera dies out; whereas if R 0 > 1 , then the disease tends to a unique endemic equilibrium. When input and death are neglected, the model is used to determine a final size equation or inequality, and simulations illustrate how assumptions on cholera transmission affect the final size of an epidemic.

The final size of an epidemic and its relation to the basic reproduction number.

We study the final size equation for an epidemic in a subdivided population with general mixing patterns among subgroups. The equation is determined by a matrix with the same spectrum as the next generation matrix and it exhibits a threshold controlled by the common dominant eigenvalue, the basic reproduction number {mathcal{R}_{0}}: There is a unique positive solution giving the size of the epidemic if and only if {mathcal{R}_{0}} exceeds unity. When mixing heterogeneities arise only from variation in contact rates and proportionate mixing, the final size of the epidemic in a heterogeneously mixing population is always smaller than that in a homogeneously mixing population with the same basic reproduction number {mathcal{R}_{0}}. For other mixing patterns, the relation may be reversed.

Age-of-infection and the final size relation

We establish the final size equation for a general age-of-infection epidemic model in a new simpler form if there are no disease deaths (total population size remains constant). If there are disease deaths, the final size relation is an inequality but we obtain an estimate for the final epidemic size.

A deterministic epidemic model taking account of repeated contacts between the same individuals

We introduce a certain population contact structure and derive, in three different ways, the final size equation for a quite general superimposed epidemic process. The contact structure is characterized by the following two properties: (i) each individual contacts exactly k other individuals; (ii) these k acquaintances are a random sample of the (infinite) population.

A deterministic epidemic model taking account of repeated contacts between the same individuals

We introduce a certain population contact structure and derive, in three different ways, the final size equation for a quite general superimposed epidemic process. The contact structure is characterized by the following two properties: (i) each individual contacts exactly k other individuals; (ii) these k acquaintances are a random sample of the (infinite) population.