A periodic SEIRS epidemic model with a time-dependent latent period

Journal of Mathematical Biology - Many infectious diseases have seasonal trends and exhibit variable periods of peak seasonality. Understanding the population dynamics due to seasonal changes...     

An impulsive delayed SEIRS epidemic model with saturation incidence

A delayed SEIRS epidemic model with pulse vaccination and saturation incidence rate is investigated. Using Krasnoselskii's fixed-point theorem, we obtain the existence of infection-free periodic solution of the impulsive delayed epidemic system. We define some new threshold values R(1), R(2) and R(3). Further, using the comparison theorem, we obtain the explicit formulae of R(1) and R(2). Under the condition R(1) < 1, the infection-free periodic solution is globally attractive, and that R(2) > 1 implies that the disease is permanent. Theoretical results show that the disease will be extinct if the vaccination rate is larger than θ* and the disease is uniformly persistent if the vaccination rate is less than θ(*). Our results indicate that a long latent period of the disease or a large pulse vaccination rate will lead to eradication of the disease. Moreover, we prove that the disease will be permanent as R(3) > 1.     

Parameter identification of a dynamic model of CHO cell cultures: an experimental case study

In this paper, we address the problem of parameter identification in dynamic models of animal cultures, and we propose a step-by-step procedure, which gradually considers more detailed models. This procedure allows subsets of parameters to be estimated at each step, which can be used in the initialization of the next identification step. Finally, the full parameter set can be re-estimated starting from the results of the last step. The efficiency of the procedure is illustrated with a simulation case study and with the identification of a dynamic model from experimental data collected in CHO cell culture.     

Analysis of an SEIRS epidemic model with two delays

 A disease transmission model of SEIRS type with exponential demographic structure is formulated. All newborns are assumed susceptible, there is a natural death rate constant, and an excess death rate constant for infective individuals. Latent and immune periods are assumed to be constants, and the force of infection is assumed to be of the standard form, namely proportional to I(t)/N(t) where N(t) is the total (variable) population size and I(t) is the size of the infective population. The model consists of a set of integro-differential equations. Stability of the disease free proportion equilibrium, and existence, uniqueness, and stability of an endemic proportion equilibrium, are investigated. The stability results are stated in terms of a key threshold parameter. More detailed analyses are given for two cases, the SEIS model (with no immune period), and the SIRS model (with no latent period). Several threshold parameters quantify the two ways that the disease can be controlled, by forcing the number or the proportion of infectives to zero. Received 8 May 1995; received in revised form 7 November 1995     

Real-time forecasting of an epidemic using a discrete time stochastic model: a case study of pandemic influenza (H1N1-2009)

Background  

Real-time forecasting of epidemics, especially those based on a likelihood-based approach, is understudied. This study aimed to develop a simple method that can be used for the real-time epidemic forecasting.     

Parameter identification in epidemic models

Starting from a recent paper of Pollicott, Wang and Weiss we try to obtain improved representation formulas for the estimation of the time-dependent transmission rate of an epidemic in terms of either incidence or prevalence data. Although the formulas are (trivially) mathematically equivalent to previous formulas, the new representations need no additional estimates and they should be more stable numerically.We review the discrete time and the stochastic continuous time approach. We replace the assumption that recovery follows an exponential distribution and get estimates for the transmission rate for constant duration of the infectious phase.     

Avian-human influenza epidemic model

A mathematical model is proposed to interpret the spread of avian influenza from the bird world to the human world. Our mathematical model warns that two types of the outbreak of avian influenza may occur if the humans do not prevent the spread of avian influenza. Moreover, it suggests that we cannot feel relieved although the total infected humans are kept at low level. In order to prevent spread of avian influenza in the human world, we must take the measures not only for the birds infected with avian influenza to exterminate but also for the humans infected with mutant avian influenza to quarantine when mutant avian influenza has already occurred. In particular, the latter measure is shown to be important to stop the second pandemic of avian influenza.     

Estimating the initial relative infection rate for a stochastic epidemic model

Consider the problem of making inference about the initial relative infection rate of a stochastic epidemic model. A relatively complete analysis of infectious disease data is possible when it is assumed that the latent and infectious periods are non-random. Here two related martingale-based techniques are used to derive estimates and associated standard errors for the initial relative infection rate. The first technique requires complete information on the epidemic, the second only the total number of people who were infected and the population size. Explicit expressions for the estimates are obtained. The estimates of the parameter and its associated standard error are easily computed and compare well with results of other methods in an application to smallpox data. Asymptotic efficiency differences between the two martingale techniques are considered.     

Explaining rapid reinfections in multiple-wave influenza outbreaks: Tristan da Cunha 1971 epidemic as a case study

Influenza usually spreads through the human population in multiple-wave outbreaks. Successive reinfection of individuals over a short time interval has been explicitly reported during past pandemics. However, the causes of rapid reinfection and the role of reinfection in driving multiple-wave outbreaks remain poorly understood. To investigate these issues, we focus on a two-wave influenza A/H3N2 epidemic that occurred on the remote island of Tristan da Cunha in 1971. Over 59 days, 273 (96%) of 284 islanders experienced at least one attack and 92 (32%) experienced two attacks. We formulate six mathematical models invoking a variety of antigenic and immunological reinfection mechanisms. Using a maximum-likelihood analysis to confront model predictions with the reported incidence time series, we demonstrate that only two mechanisms can be retained: some hosts with either a delayed or deficient humoral immune response to the primary influenza infection were reinfected by the same strain, thus initiating the second epidemic wave. Both mechanisms are supported by previous empirical studies and may arise from a combination of genetic and ecological causes. We advocate that a better understanding and account of heterogeneity in the human immune response are essential to analysis of multiple-wave influenza outbreaks and pandemic planning.     

Three-stage stochastic epidemic model: an application to AIDS.

A three-stage stochastic epidemic model extending the so-called classical epidemic process to one that includes time-dependent transition probabilities is described, and a solution to the appropriate set of forward differential-difference equations is given. When an individual can move from being a susceptible to one infected with the HIV virus to one diagnosed as having AIDS, we can use this general model to describe an AIDS epidemic process. We obtain expressions for the mean and variance of the number of AIDS cases for some special cases. By comparing these with actual data, it is suggested that, for some categories of cases (in particular, children), this model might be a plausible model to describe the underlying mechanism of the AIDS epidemic.     

Some properties of a simple stochastic epidemic model of SIR type

We investigate the properties of a simple discrete time stochastic epidemic model. The model is Markovian of the SIR type in which the total population is constant and individuals meet a random number of other individuals at each time step. Individuals remain infectious for R time units, after which they become removed or immune. Individual transition probabilities from susceptible to diseased states are given in terms of the binomial distribution. An expression is given for the probability that any individuals beyond those initially infected become diseased. In the model with a finite recovery time R, simulations reveal large variability in both the total number of infected individuals and in the total duration of the epidemic, even when the variability in number of contacts per day is small. In the case of no recovery, R=infinity, a formal diffusion approximation is obtained for the number infected. The mean for the diffusion process can be approximated by a logistic which is more accurate for larger contact rates or faster developing epidemics. For finite R we then proceed mainly by simulation and investigate in the mean the effects of varying the parameters p (the probability of transmission), R, and the number of contacts per day per individual. A scale invariant property is noted for the size of an outbreak in relation to the total population size. Most notable are the existence of maxima in the duration of an epidemic as a function of R and the extremely large differences in the sizes of outbreaks which can occur for small changes in R. These findings have practical applications in controlling the size and duration of epidemics and hence reducing their human and economic costs.     

A stochastic mover/stayer model for an HIV epidemic.

A multicompartmental model is proposed to deal with the spread of the HIV epidemic. Stratification with respect to high- and low-risk groups is taken into account, introducing a "fuzzy" decomposition of the compartment of susceptibles. The infectivity period is staged using different levels of CD4 cell counts. Asymptotic behavior is studied, and some simulation results are reported in tabular and graphic forms.     

A stochastic model for the development of an AIDS epidemic in a heterosexual population.

A non-age-dependent model, describing the evolution of a bisexual population, is developed in this paper and applied to projecting an AIDS epidemic in a heterosexual population. Included in the formulation are frequency- and non-frequency-dependent rules of partnership formation as well as five states of HIV disease, affecting the probability of infection per sexual contact. Results from computer experiments, designed to study the development of an AIDS epidemic in a heterosexual population fed by single males with a 50% prevalence of HIV infection prior to becoming active in heterosexual partnerships, are reported. In these experiments, the only source of HIV infection for females was sexual contacts with infected males within partnerships. Data on the probability of infection per sexual contact with an infected partner and the number of sexual contacts per month were incorporated into the model. However, the numbers used for the initial population of singles, couples, and those becoming sexually active per month were hypothetical. Even though the prevalence of HIV infection among males entering heterosexual partnerships was high, after 30 years the projected prevalence of HIV infection among females ranged from about 10 to 15% depending in part on the expected duration of partnerships and on whether the frequency- or non-frequency-dependent model was used. In these experiments, solutions of the embedded, nonlinear, deterministic equations for the incidence of HIV infection and the cumulative number of deaths due to AIDS proved to be good measures of central tendency for the sample functions of the stochastic population process.     

A discrete method for the identification of parameters of a deterministic epidemic model

We study a problem of identification of the parameters for a deterministic epidemic model of the Kermack-McKendrick type. Particular emphasis is put on the analysis of the conditions of numerical stability of the method of integration used to calculate the solutions of the system of differential equations which describe the model. The numerical method can be regarded as a discrete model which reproduces the basic qualitative properties of the continuous model, which are positivity of the solutions, points of equilibrium, and the “threshold theorem.” This allows us to identify the parameters with good reliability, by means of an iterative procedure to minimize the functional which is the measure of discrepancy between the data observed and the data obtained from the discrete model. The initial estimate of the parameters is obtained by a direct method applied to the discretized system of equations.     

Stochastic model of an influenza epidemic with drug resistance

A continuous-time Markov chain (CTMC) model is formulated for an influenza epidemic with drug resistance. This stochastic model is based on an influenza epidemic model, expressed in terms of a system of ordinary differential equations (ODE), developed by Stilianakis, N.I., Perelson, A.S., Hayden, F.G., [1998. Emergence of drug resistance during an influenza epidemic: insights from a mathematical model. J. Inf. Dis. 177, 863-873]. Three different treatments-chemoprophylaxis, treatment after exposure but before symptoms, and treatment after symptoms appear, are considered. The basic reproduction number, R(0), is calculated for the deterministic-model under different treatment strategies. It is shown that chemoprophylaxis always reduces the basic reproduction number. In addition, numerical simulations illustrate that the basic reproduction number is generally reduced with realistic treatment rates. Comparisons are made among the different models and the different treatment strategies with respect to the number of infected individuals during an outbreak. The final size distribution is computed for the CTMC model and, in some cases, it is shown to have a bimodal distribution corresponding to two situations: when there is no outbreak and when an outbreak occurs. Given an outbreak occurs, the total number of cases for the CTMC model is in good agreement with the ODE model. The greatest number of drug resistant cases occurs if treatment is delayed or if only symptomatic individuals are treated.