Compartmental Epidemic Model Research Articles

Uncertainty and sensitivity analysis of the basic reproduction number of a vaccinated epidemic model of influenza

The basic reproduction number and the point of endemic equilibrium are two very important factors in any deterministic compartmental epidemic model as the basic reproduction number and the point of endemic equilibrium represent the nature of disease transmission and disease prevalence respectively. In this article the sensitivity analysis based on mathematical as well as statistical techniques has been performed to determine the importance of the epidemic model parameters. It is observed that 6 out of the 11 input parameters play a prominent role in determining the magnitude of the basic reproduction number. It is shown that the basic reproduction number is the most sensitive to the transmission rate of disease. It is also shown that control of transmission rate and recovery rate of the clinically ill are crucial to stop the spreading of influenza epidemics.

Vaccination against 2009 pandemic H1N1 in a population dynamical model of Vancouver, Canada: timing is everything

BackgroundMuch remains unknown about the effect of timing and prioritization of vaccination against pandemic (pH1N1) 2009 virus on health outcomes. We adapted a city-level contact network model to study different campaigns on influenza morbidity and mortality.MethodsWe modeled different distribution strategies initiated between July and November 2009 using a compartmental epidemic model that includes age structure and transmission network dynamics. The model represents the Greater Vancouver Regional District, a major North American city and surrounding suburbs with a population of 2 million, and is parameterized using data from the British Columbia Ministry of Health, published studies, and expert opinion. Outcomes are expressed as the number of infections and deaths averted due to vaccination.ResultsThe model output was consistent with provincial surveillance data. Assuming a basic reproduction number = 1.4, an 8-week vaccination campaign initiated 2 weeks before the epidemic onset reduced morbidity and mortality by 79-91% and 80-87%, respectively, compared to no vaccination. Prioritizing children and parents for vaccination may have reduced transmission compared to actual practice, but the mortality benefit of this strategy appears highly sensitive to campaign timing. Modeling the actual late October start date resulted in modest reductions in morbidity and mortality (13-25% and 16-20%, respectively) with little variation by prioritization scheme.ConclusionDelays in vaccine production due to technological or logistical barriers may reduce potential benefits of vaccination for pandemic influenza, and these temporal effects can outweigh any additional theoretical benefits from population targeting. Careful modeling may provide decision makers with estimates of these effects before the epidemic peak to guide production goals and inform policy. Integration of real-time surveillance data with mathematical models holds the promise of enabling public health planners to optimize the community benefits from proposed interventions before the pandemic peak.

Threshold dynamics for compartmental epidemic models with impulses

The basic reproductive number and its calculation for general impulsive compartmental epidemic models, with pulses on both the infected and the uninfected compartments, are established. Theoretical results show that the basic reproductive number serves as a threshold parameter: the disease dies out if the basic reproductive number is smaller than unity, and breaks out if it is larger than unity. The global dynamics of a viral dynamical model with impulsive immune response is analyzed to study how the vaccination strength and the vaccination interval affect the basic reproductive number and virus progression.

The potential impact of an HIV vaccine with rapidly waning protection on the epidemic in Southern Africa: Examining the RV144 trial results

Background The prime-boost HIV vaccine regimen used in the recent RV144 trial resulted in modest efficacy of 31% over 3.5 years, but was substantially higher in the first year post-vaccination. We sought to explore the potential impact of a vaccine with rapidly waning efficacy in a South African population. Methods We explored two strategies using a dynamic compartmental epidemic model for heterosexual transmission of HIV: [1] vaccination of a single cohort (30%, 60% or 90% of the initial population), with exponentially waning efficacy, but booster vaccinations at 5- or 2-year intervals, and [2] continuous vaccination of the unvaccinated population at the same coverage levels (30%, 60% or 90%) but with a constant efficacy vaccine of short duration. We also examined potential changes in post-vaccination condom use. Results The single cohort vaccination strategies did not have a substantial impact on HIV prevalence, although without boosters they still prevented 2–6% of the expected infections at 20 years, depending on the population coverage. The 5-year and 2-year booster strategies prevented 8–24% and 17–45% of the expected infections, respectively. Continuous vaccination to maintain population coverage levels resulted in more substantial reductions in population HIV prevalence and greater numbers of infections prevented: HIV prevalence at 20 years was reduced from 23% to 8–14% and the number of expected infections was decreased by 34–59%, depending on the population coverage level. Moderate changes in post-vaccination condom use did not substantially affect these outcomes. Conclusions An HIV vaccine with partial efficacy and declining protection similar to the RV144 vaccine could prevent a substantial proportion of HIV infections if booster vaccinations were effective and available. Our estimates of the population impact of vaccination would be improved by further understanding of the duration of protection, the effectiveness of booster vaccination, and whether the vaccine efficacy varies between subpopulations at higher and lower risk of exposure.

Epidemiology and transmission dynamics of the 1918–19 pandemic influenza in Florence, Italy

To investigate the 1918/19 influenza pandemic daily number of new hospitalizations in the only hospital in Florence (Central Italy) were analyzed. In order to describe the transmission dynamics of the 1918/1919 pandemic influenza a compartmental epidemic model was used. Model simulations show a high level of agreement with the observed epidemic data. By assuming both latent and infectious period equal to 1.5 days, the estimated basic reproduction number was R 0 1 = 1.03 (95% CI: 1.00–1.08) during the summer wave and R 0 2 = 1.38 (95% CI: 1.32–1.48) during the fall wave. Varying the length of the generation time or the estimation method, R 0 2 ranges from 1.32 to 1.71. The hospitalization rate was found significantly different between summer and fall waves. Notably, the estimated basic reproductive numbers are lower compared to those observed in other countries, while the age distribution of deaths resulted to be consistent with the patterns generally observed during of the 1918–1919 pandemic. Our knowledge on past pandemics, as for the 1918–19 Spanish influenza, would help improving mathematical modeling accuracy and understanding the mechanisms underlying the dynamics of future pandemics.

On the backward bifurcation of a vaccination model with nonlinear incidence

A compartmental epidemic model, introduced by Gumel and Moghadas [1], is considered. The model incorporates a nonlinear incidence rate and an imperfect preventive vaccine given to susceptible individuals. A bifurcation analysis is performed by applying the bifurcation method introduced in [2], which is based on the use of the center manifold theory. Conditions ensuring the occurrence of backward bifurcation are derived. The obtained results are numerically validated and then discussed from both the mathematical and the epidemiological perspective.

Scaling up circumcision programs in Southern Africa: the potential impact of gender disparities and changes in condom use behaviors on heterosexual HIV transmission.

Circumcision significantly reduces female-to-male transmission of HIV infection, but changes in behavior may influence the overall impact on transmission. We sought to explore these effects, particularly for societies where women have less power to negotiate safe sex. We developed a compartmental epidemic model to simulate the population-level impact of various circumcision programs on heterosexual HIV transmission in Soweto. We incorporated gender-specific negotiation of condom use in sexual partnerships and explored post-circumcision changes in condom use. A 5-year prevention program in which only an additional 10% of uncircumcised males undergo circumcision each year, for example, would prevent 13% of the expected new HIV infections over 20 years. Outcomes were sensitive to potential changes in behavior and differed by gender. For Southern Africa, even modest programs offering circumcision would result in significant benefits. Because decreases in male condom use could diminish these benefits, particularly for women, circumcision programs should emphasize risk-reduction counseling.

General compartmental epidemic models

The age of infection approach introduced by Kermack and Mckendrick in 1927 gives a unified way of describing and analyzing a variety of epidemic models, including models with multiple stages, treatment, and heterogeneous mixing. The author gives a description of the main results for such models, emphasizing the use of the final size relation to estimate the size of the epidemic.

Effectiveness and cost-effectiveness of vaccination against pandemic influenza (H1N1) 2009.

Decisions on the timing and extent of vaccination against pandemic (H1N1) 2009 virus are complex. To estimate the effectiveness and cost-effectiveness of pandemic influenza (H1N1) vaccination under different scenarios in October or November 2009. Compartmental epidemic model in conjunction with a Markov model of disease progression. Literature and expert opinion. Residents of a major U.S. metropolitan city with a population of 8.3 million. Lifetime. Societal. Vaccination in mid-October or mid-November 2009. Infections and deaths averted, costs, quality-adjusted life-years (QALYs), and incremental cost-effectiveness. Assuming each primary infection causes 1.5 secondary infections, vaccinating 40% of the population in October or November would be cost-saving. Vaccination in October would avert 2051 deaths, gain 69 679 QALYs, and save $469 million compared with no vaccination; vaccination in November would avert 1468 deaths, gain 49 422 QALYs, and save $302 million. Vaccination is even more cost-saving if longer incubation periods, lower rates of infectiousness, or increased implementation of nonpharmaceutical interventions delay time to the peak of the pandemic. Vaccination saves fewer lives and is less cost-effective if the epidemic peaks earlier than mid-October. The model assumed homogenous mixing of case-patients and contacts; heterogeneous mixing would result in faster initial spread, followed by slower spread. Additional costs and savings not included in the model would make vaccination more cost-saving. Earlier vaccination against pandemic (H1N1) 2009 prevents more deaths and is more cost-saving. Complete population coverage is not necessary to reduce the viral reproductive rate sufficiently to help shorten the pandemic. Agency for Healthcare Research and Quality and National Institute on Drug Abuse.

Effectiveness and Cost-Effectiveness of Expanded Antiviral Prophylaxis and Adjuvanted Vaccination Strategies for an Influenza A (H5N1) Pandemic

Background: The pandemic potential of influenza A (H5N1) virus is a prominent public health concern of the 21st century. Objective: To estimate the effectiveness and cost-effectiveness of alternative pandemic (H5N1) mitigation and response strategies. Design: Compartmental epidemic model in conjunction with a Markov model of disease progression. Data Sources: Literature and expert opinion. Target Population: Residents of a U.S. metropolitan city with a population of 8.3 million. Time Horizon: Lifetime. Perspective: Societal. Interventions: 3 scenarios: 1) vaccination and antiviral pharmacotherapy in quantities similar to those currently available in the U.S. stockpile (stockpiled strategy), 2) stockpiled strategy but with expanded distribution of antiviral agents (expanded prophylaxis strategy), and 3) stockpiled strategy but with adjuvanted vaccine (expanded vaccination strategy). All scenarios assumed standard nonpharmaceutical interventions. Outcome Measures: Infections and deaths averted, costs, quality-adjusted life-years (QALYs), and incremental cost-effectiveness. Results of Base-Case Analysis: Expanded vaccination was the most effective and cost-effective of the 3 strategies, averting 68% of infections and deaths and gaining 404 030 QALYs at $10 844 per QALY gained relative to the stockpiled strategy. Results of Sensitivity Analysis: Expanded vaccination remained incrementally cost-effective over a wide range of assumptions. Limitations: The model assumed homogenous mixing of cases and contacts; heterogeneous mixing would result in faster initial spread, followed by slower spread. We did not model interventions for children or older adults; the model is not designed to target interventions to specific groups. Conclusion: Expanded adjuvanted vaccination is an effective and cost-effective mitigation strategy for an influenza A (H5N1) pandemic. Expanded antiviral prophylaxis can help delay the pandemic while additional strategies are implemented. Primary Funding Source: National Institutes of Health and Agency for Healthcare Research and Quality.

The construction of next-generation matrices for compartmental epidemic models

The basic reproduction number (0) is arguably the most important quantity in infectious disease epidemiology. The next-generation matrix (NGM) is the natural basis for the definition and calculation of (0) where finitely many different categories of individuals are recognized. We clear up confusion that has been around in the literature concerning the construction of this matrix, specifically for the most frequently used so-called compartmental models. We present a detailed easy recipe for the construction of the NGM from basic ingredients derived directly from the specifications of the model. We show that two related matrices exist which we define to be the NGM with large domain and the NGM with small domain. The three matrices together reflect the range of possibilities encountered in the literature for the characterization of (0). We show how they are connected and how their construction follows from the basic model ingredients, and establish that they have the same non-zero eigenvalues, the largest of which is the basic reproduction number (0). Although we present formal recipes based on linear algebra, we encourage the construction of the NGM by way of direct epidemiological reasoning, using the clear interpretation of the elements of the NGM and of the model ingredients. We present a selection of examples as a practical guide to our methods. In the appendix we present an elementary but complete proof that (0) defined as the dominant eigenvalue of the NGM for compartmental systems and the Malthusian parameter r, the real-time exponential growth rate in the early phase of an outbreak, are connected by the properties that (0) > 1 if and only if r > 0, and (0) = 1 if and only if r = 0.

Tuberculosis elimination in the Canadian First Nations population: assessment by a state-transfer, compartmental epidemic model

Tuberculosis (TB) remains an important public health problem in Canadian Aboriginal (First Nations and Inuit) communities. The objectives of this study were to predict future disease burden and set feasible targets for the elimination of TB in the First Nations population, using retrospective data and an epidemic model. Reported TB incidence data (1974-2002), previously published TB meningitis data from the pre-chemotherapy era, and previous estimates of disease risk following infection were used to estimate a trend in the annual risk of infection from 1929 to 2002, and the age-specific prevalence of infection in 2002. A state-transfer, compartmental model was then developed to predict future disease burden. Two scenarios were simulated, with different disease risk parameters. The estimated prevalence of infection in 2002 was 20.9% in scenario 1 and 25.5% in scenario 2. Predicted incidence rates in 2015 were 16.8 per 100,000 and 11.7 per 100,000 for the two scenarios, respectively. The incidence of disease was not lower than 1 per 100,000 for either scenario in 2034, the arbitrarily chosen last year of the model. The goal of eliminating TB among Aboriginal peoples in Canada is a feasible one, but will only be achieved with continued investment in programs designed to control and prevent transmission. Reactivation disease cases may occur for a number of years to come, making rapid elimination a difficult goal.

Threshold Dynamics for Compartmental Epidemic Models in Periodic Environments

The basic reproduction ratio and its computation formulae are established for a large class of compartmental epidemic models in periodic environments. It is proved that a disease cannot invade the disease-free state if the ratio is less than unity and can invade if it is greater than unity. It is also shown that the basic reproduction number of the time-averaged autonomous system is applicable in the case where both the matrix of new infection rate and the matrix of transition and dissipation within infectious compartments are diagonal, but it may underestimate and overestimate infection risks in other cases. The global dynamics of a periodic epidemic model with patch structure is analyzed in order to study the impact of periodic contacts or periodic migrations on the disease transmission.

Lyapunov Functions and Global Stability for SIR and SIRS Epidemiological Models with Non-Linear Transmission

Lyapunov functions for two-dimension SIR and SIRS compartmental epidemic models with non-linear transmission rate of a very general form f(S, I) constrained by a few biologically feasible conditions are constructed. Global properties of these models including these with vertical and horizontal transmission, are thereby established. It is proved that, under the constant population size assumption, the concavity of the function f(S, I) with respect to the number of the infective hosts I ensures the uniqueness and the global stability of the positive endemic equilibrium state.

Lyapunov functions and global properties for SEIR and SEIS epidemic models.

Lyapunov functions for classic SEIR and SEIS compartmental epidemic models, including those with vertical and horizontal transmission, are introduced. Global properties of the models is thereby established.

Lyapunov functions and global properties for SEIR and SEIS epidemic models

Lyapunov functions for classic SEIR and SEIS compartmental epidemic models, including those with vertical and horizontal transmission, are introduced. Global properties of the models is thereby established.

A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence

Explicit Lyapunov functions for SIR and SEIR compartmental epidemic models with nonlinear incidence of the form betaI(p)S(q) for the case p </= 1 are constructed. Global stability of the models is thereby established.

The contribution of genetic diversity to the spread of infectious diseases in livestock populations.

This article uses stochastic simulations with a compartmental epidemic model to quantify the impact of genetic diversity within animal populations on the transmission of infectious disease. Genetic diversity is defined by the number of distinct genotypes in the population conferring resistance to microparasitic (e.g., viral or bacterial) infections. Scenarios include homogeneous populations and populations composed of few (finite-locus model) or many (infinitesimal model) genotypes. Genetic heterogeneity has no impact upon the expected value of the basic reproductive ratio (the primary description of the transmission of infection) but affects the variability of this parameter. Consequently, increasing genetic heterogeneity is associated with an increased probability of minor epidemics and decreased probabilities of both major (catastrophic) epidemics and no epidemics. Additionally, heterogeneity per se is associated with a breakdown in the expected relationship between the basic reproductive ratio and epidemic severity, which has been developed for homogeneous populations, with increasing heterogeneity generally resulting in fewer infected animals than expected. Furthermore, increased heterogeneity is associated with decreased disease-dependent mortality in major epidemics and a complex trend toward decreased duration of these epidemics. In summary, more heterogeneous populations are not expected to suffer fewer epidemics on average, but are less likely to suffer catastrophic epidemics.

Behavioral plasticity and virus propagation: the FIV-cat population example

A new theoretical approach is proposed to investigate the effect of intra-individual variability in behavior on the spread of directly transmitted diseases within host populations. The classical hawk–dove game is used to describe interactions between individuals on a fast time scale (the day). Individuals may exhibit both tactics according to their own experience, to environment conditions, and to the opponent. They are not able to recognize the epidemiological state of their opponents. This fast-time part of the model is coupled to a classical compartmental epidemic model describing the demography of the population and the transmission of the disease from an infected individual to a healthy one on a slow time scale (the year). The model is applied to the case of feline immunodeficiency virus (FIV)-domestic cat population system. Our model gives rise to different predictions according to values of cost and gain due to fights: extinction of the epidemic, FIV endemicity at low, intermediate and high prevalence. These predictions are in good agreement with results from domestic cat populations living in different environmental conditions.

Estimated population mixing by country and risk cohort for the HIV/AIDS epidemic in Western Europe

This paper applies a compartmental epidemic model to estimating the mixing relations that support the transfer of HIV infection between risk populations within the countries of Western Europe. To this end, a space-time epidemic model with compartments representing countries with populations specified to be at high (gay men and intravenous drug injectors ever with AIDS) and low (the remainder who are sexually active) risk is described. This model also allows for contacts between susceptible and infectious individuals by both local and international travel. This system is calibrated to recorded AIDS incidence and the best-fit solution provides estimates of variations in the rates of mixing between the compartments together with a reconstruction of the transmission pathway. This solution indicates that, for all the countries, AIDS incidence among those at low risk is expected to remain extremely small relative to their total number. A sensitivity analysis of the low risk partner acquisition rate, however, suggests this endemic state might be fragile within Europe during this century. The discussion examines the relevance of these mixing relationships for the maintenance of disease control.