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Results for “Basic Reproduction Number”

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PubMed2006

Theoretical assessment of public health impact of imperfect prophylactic HIV-1 vaccines with therapeutic benefits.

This paper presents a number of deterministic models for theoretically assessing the potential impact of an imperfect prophylactic HIV-1 vaccine that has five biological modes of action, namely "take," "degree," "duration," "infectiousness," and "progression," and can lead to increased risky behavior. The models, which are of the form of systems of nonlinear differential equations, are constructed via a progressive refinement of a basic model to incorporate more realistic features of HIV pathogenesis and epidemiology such as staged progression, differential infectivity, and HIV transmission by AIDS patients. The models are analyzed to gain insights into the qualitative features of the associated equilibria. This allows the determination of important epidemiological thresholds such as the basic reproduction numbers and a measure for vaccine impact or efficacy. The key findings of the study include the following (i) if the vaccinated reproduction number is greater than unity, each of the models considered has a locally unstable disease-free equilibrium and a unique endemic equilibrium; (ii) owing to the vaccine-induced backward bifurcation in these models, the classical epidemiological requirement of vaccinated reproduction number being less than unity does not guarantee disease elimination in these models; (iii) an imperfect vaccine will reduce HIV prevalence and mortality if the reproduction number for a wholly vaccinated population is less than the corresponding reproduction number in the absence of vaccination; (iv) the expressions for the vaccine characteristics of the refined models take the same general structure as those of the basic model.

AIDS Vaccines↗

PubMed2004

Transmissibility of 1918 pandemic influenza.

The 1918 influenza pandemic killed 20-40 million people worldwide, and is seen as a worst-case scenario for pandemic planning. Like other pandemic influenza strains, the 1918 A/H1N1 strain spread extremely rapidly. A measure of transmissibility and of the stringency of control measures required to stop an epidemic is the reproductive number, which is the number of secondary cases produced by each primary case. Here we obtained an estimate of the reproductive number for 1918 influenza by fitting a deterministic SEIR (susceptible-exposed-infectious-recovered) model to pneumonia and influenza death epidemic curves from 45 US cities: the median value is less than three. The estimated proportion of the population with A/H1N1 immunity before September 1918 implies a median basic reproductive number of less than four. These results strongly suggest that the reproductive number for 1918 pandemic influenza is not large relative to many other infectious diseases. In theory, a similar novel influenza subtype could be controlled. But because influenza is frequently transmitted before a specific diagnosis is possible and there is a dearth of global antiviral and vaccine stores, aggressive transmission reducing measures will probably be required.

Cities↗

PubMed2000

The pre-vaccination epidemiology of measles, mumps and rubella in Europe: implications for modelling studies.

Data on the pre-vaccination patterns of infection for measles, mumps and rubella are collated from a number of European countries in order to compare the epidemiology of the three viruses. Key epidemiological parameters, such as the age-specific force of infection and the basic reproduction number (R0) are estimated from case notification or serological data using standard techniques. A method is described to compare force of infection estimates derived from serological data. Analysis suggests that the pre-vaccination patterns of measles and mumps infection in the different countries were similar. In contrast, the epidemiology of rubella was highly variable between countries. This suggests that it may be acceptable to use parameter values estimated from other countries to model measles and mumps transmission, but that this approach to modelling rubella transmission requires more caution. Estimates of R0 depend on underlying mixing assumptions. Constraints were placed on R0 estimates by utilising knowledge of likely mixing patterns. The estimates for R0 were highest for measles, intermediate for mumps, and generally lowest for rubella. Analysis of within- and between-age-group transmission rates suggested that mumps transmission tends to be more concentrated within young children than the other two viruses. The implications for the design of immunization programmes are that mumps may be the easiest to control via infant immunization since it is predominantly transmitted between the very young and the variability in rubella epidemiology requires that careful consideration of the possible effects of vaccination options should be made using local data when planning rubella immunization programmes.

Adolescent↗

PubMed2006

Transmission potential of primary pneumonic plague: time inhomogeneous evaluation based on historical documents of the transmission network.

BACKGROUND: The transmission potential of primary pneumonic plague, caused by Yersinia pestis, is one of the key epidemiological determinants of a potential biological weapon, and requires clarification and time dependent interpretation. METHOD: This study estimated the reproduction number and its time dependent change through investigations of outbreaks in Mukden, China (1946), and Madagascar (1957). Reconstruction of an epidemic tree, which shows who infected whom, from the observed dates of onset was performed using the serial interval. Furthermore, a likelihood based approach was used for the time inhomogeneous evaluation of the outbreaks for which there was scarcity of cases. RESULTS: According to the estimates, the basic reproduction number, R(0), was on the order of 2.8 to 3.5, which is higher than previous estimates. The lower 95% confidence intervals of R(0) exceeded unity. The effective reproduction number declined below unity after control measures were introduced in Mukden, and before the official implementation in Madagascar. CONCLUSION: While the time course of the latter outbreak could be explained by intrinsic factors and stochasticity in this remote and scarcely populated area, the former in Mukden suggests the possible continued chains of transmission in highly populated areas. Using the proposed methods, the who infected whom information permitted the evaluation of the time inhomogeneous transmission potential in relation to public health measures. The study also tackles the problem of statistical estimation of R(0) based on similar information, which was previously performed simply by counting the number of secondary transmissions regardless of time.

China↗

PubMed2006

Transmission dynamics of the great influenza pandemic of 1918 in Geneva, Switzerland: Assessing the effects of hypothetical interventions.

Recurrent outbreaks of the avian H5N1 influenza virus in Asia represent a constant global pandemic threat. We characterize and evaluate hypothetical public health measures during the 1918 influenza pandemic in the Canton of Geneva, Switzerland. The transmission rate, the recovery rate, the diagnostic rate, the relative infectiousness of asymptomatic cases, and the proportion of clinical cases are estimated through least-squares fitting of the model to epidemic curve data of the cumulative number of hospital notifications. The latent period and the case fatality proportion are taken from published literature. We determine the variance and identifiability of model parameters via a simulation study. Our epidemic model agrees well with the observed epidemic data. We estimate the basic reproductive number for the spring wave R1;=1.49 (95% CI: 1.45-1.53) and the reproductive number for the fall wave R2;=3.75 (95% CI: 3.57-3.93). In addition, we estimate the clinical reporting for these two waves to be 59.7% (95% CI: 55.7-63.7) and 83% (95% CI: 79-87). We surmise that the lower reporting in the first wave can be explained by a lack of initial awareness of the epidemic and the relative higher severity of the symptoms experienced during the fall wave. We found that effective isolation measures in hospital clinics at best would only ensure control with probability 0.87 while reducing the transmission rate by >76.5% guarantees stopping an epidemic.

Disease Notification↗

PubMed2000

Nematode parasites of sheep: a survey of epidemiological parameters and their application in a simple model.

We review the literature on parameter values relevant to the epidemiology of strongyle nematode infections of domestic sheep. Information is subdivided by parasite genus, country of origin and climate type. While field observations have been made in a large number of countries, the bulk of studies under controlled conditions have been conducted in Australia, New Zealand and the UK. For these countries, experiments and parameters are interpreted in terms of a previously published model of nematode dynamics, and are used to calculate the basic reproduction number. Average values range from less than 6 for Haemonchus contortus in New Zealand and a winter rainfall region of Australia, to more than 16 for Ostertagia circumcincta in New Zealand and the UK. Additional considerations of the effects of climate and the annual replacement of host stock show that for conditions favourable for parasite transmission this is a robust indicator of parasite epidemiology. When climate variation and annual replacement are added to the model, it is shown to reasonably describe the qualitative behaviour of an experimental data set, indicating it to be a useful tool for further investigation of some of the underlying assumptions of sheep-nematode dynamics.

Animals↗

PubMed2002

The dual role of CD4 T helper cells in the infection dynamics of HIV and their importance for vaccination.

Given the role of the CD4 T helper cells in the development of memory CTL precursors, it seems beneficial to boost the CD4 T helper response in the context of vaccination against the human immunodeficiency virus (HIV). However, CD4 T cells are also the preferred targets of infection by HIV. Here, we address the question as to whether it is advantageous to stimulate the CD4 T helper cell response, as this will increase the pool of potential target cells of infection. To do so we formulated a mathematical model describing the interactions between virus-infected cells, susceptible cells, HIV-specific CD4 helper T cells, and CTL precursor (CTLp) and effector cells (CTLe). The effect of increased initial CD4 helper and CTLp numbers on the outcome of infection, as well as the effect on viral set point of increased CD4 T helper growth rate, CTL responsiveness and the rate at which CTLp and CTLe are produced were studied. We found that only when the virus has a low basic reproductive number does the number of CTLp and CD4 T helper cells at the moment of infection influence the outcome of infection. In this situation, high initial T helper and CTL numbers can switch the outcome from full-blown infection to virus control. However, this holds for virus with infectivity in a limited range, and current estimates of virus infectivity suggest that it is higher. In that case, only a vaccination protocol that increases CTL responsiveness, ideally in combination with the rate of production of CD4 T helper cells, may offer a solution as it can reduce the viral set point considerably. If brought under a certain level, the viral population might be unable to replicate any further. However, changing these parameters of the immune response is only beneficial when infection is controlled by CTL in the long term. When a CD4 lymphoproliferative response is mounted but the CTL response is not maintained, increasing the CD4 T helper growth rate is deleterious.

HIV Infections↗

PubMed2004

Ring vaccination and smallpox control.

We present a stochastic model for the spread of smallpox after a small number of index cases are introduced into a susceptible population. The model describes a branching process for the spread of the infection and the effects of intervention measures. We discuss scenarios in which ring vaccination of direct contacts of infected persons is sufficient to contain an epidemic. Ring vaccination can be successful if infectious cases are rapidly diagnosed. However, because of the inherent stochastic nature of epidemic outbreaks, both the size and duration of contained outbreaks are highly variable. Intervention requirements depend on the basic reproduction number (R0), for which different estimates exist. When faced with the decision of whether to rely on ring vaccination, the public health community should be aware that an epidemic might take time to subside even for an eventually successful intervention strategy.

Contact Tracing↗

PubMed2005

A multi-species epidemic model with spatial dynamics.

A model is formulated that describes the spatial propagation of a disease that can be transmitted between multiple species. The spatial component consists, for each species, of a certain number of patches that make up the vertices of a digraph, the arcs of which represent the movement of the various species between the patches. In each of the patches and for each species, a susceptible-exposed-infectious-recovered (SEIR) epidemic model describes the evolution of the disease status of individuals. Also in each patch, there is transmission of the disease from species to species. An analysis of the system is given, beginning with results on the mobility component. A formula is derived for the computation of the basic reproduction number R(0) for sspecies and npatches, which then determines the global stability properties of the disease free equilibrium. Simulations for the spread of a disease in one species and two patches are presented.

Animals↗

PubMed2006

Generality of the final size formula for an epidemic of a newly invading infectious disease.

The well-known formula for the final size of an epidemic was published by Kermack and McKendrick in 1927. Their analysis was based on a simple susceptible-infected-recovered (SIR) model that assumes exponentially distributed infectious periods. More recent analyses have established that the standard final size formula is valid regardless of the distribution of infectious periods, but that it fails to be correct in the presence of certain kinds of heterogeneous mixing (e.g., if there is a core group, as for sexually transmitted diseases). We review previous work and establish more general conditions under which Kermack and McKendrick's formula is valid. We show that the final size formula is unchanged if there is a latent stage, any number of distinct infectious stages and/or a stage during which infectives are isolated (the durations of each stage can be drawn from any integrable distribution). We also consider the possibility that the transmission rates of infectious individuals are arbitrarily distributed--allowing, in particular, for the existence of super-spreaders--and prove that this potential complexity has no impact on the final size formula. Finally, we show that the final size formula is unchanged even for a general class of spatial contact structures. We conclude that whenever a new respiratory pathogen emerges, an estimate of the expected magnitude of the epidemic can be made as soon the basic reproduction number R0 can be approximated, and this estimate is likely to be improved only by more accurate estimates of R0, not by knowledge of any other epidemiological details.

Algorithms↗

PubMedDate not supplied

The seasonal pattern of dengue in endemic areas: mathematical models of mechanisms.

In dengue-endemic areas such as Thailand, there is clear seasonality in the number of reported cases of dengue virus disease. However, the roles of different entomological and biological variables in determining this pattern have not been ascertained. To investigate this, seasonally-varying parameters were introduced in a step-wise fashion into a mathematical model of the transmission dynamics of dengue viruses. The predicted prevalence of infection was then compared to observed seasonal patterns of disease. The strongest influences on the pattern of infection and its seasonal variation were duration of infectiousness of the host, vector mortality, and biting rate. However, seasonally-varying parameters such as the latent period of infection in the vector had to be incorporated into the model to generate the correct timing of peak infection prevalence. A few limiting variables usually control the prevalence of an infectious disease because small changes in their values can carry the infection beyond the threshold at which its basic reproductive number is one. It was changes in such parameters (vector biting and mortality rate) which caused seasonal prevalence, but the timing of peak prevalence was a result of time delays within the system.

Dengue↗

PubMedDate not supplied

Implication of Ariaal sexual mixing on gonorrhea.

Recent research on sexual mixing in populations of sub-Saharan Africa raises the question as to whether STDs can persist in these populations without the presence of a core group. A mathematical model is constructed for the spread of gonorrhea among the Ariaal population of Northern Kenya. A formula for the basic reproduction number R(0) (the expected number of secondary infections caused by a single new infective introduced into a susceptible population) is determined for this population in the absence of a core group. Survey data taken in 2003 on sexual behavior from the Ariaal population are used in the model which is formulated for their age-set system including four subpopulations: single and married, female and male. Parameters derived from the data, and other information from sub-Saharan Africa are used to estimate R(0). Results indicate that, even with the elevating effect of the age-set system, the disease should die out since R(0) < 1. Thus, the persistence of gonorrhea in the population must be due to factors not included in the model, for example, a core group of commercial sex workers or concurrent partnerships.

Female↗

PubMed1998

The dynamics of drug action on the within-host population growth of infectious agents: melding pharmacokinetics with pathogen population dynamics.

The use of simple mathematical models to study the kinetics of drug action and decay within vertebrate hosts has a long history with a major objective being to derive drug dosage regimens that optimize efficacy and minimize toxicity to the patient. Mathematical models of the relationship between dosage, route of delivery, drug concentration in defined sites and effect on a particular pathogen are widely used in the pharmacological literature. A more recent literature is that concerned with the population dynamics of pathogen replication within the host subjected to pressures exerted by the human immune system. In this paper we develop a theoretical framework to meld both approaches with the aim of identifying threshold criteria that dictate the optimum pattern of drug administration for pathogen clearance from the host. In particular we show how the percentage reduction in microparasite abundance is related to the pharmacokinetic parameter, AUC, recording the area under the drug concentration-time curve within the treated patient, in terms of the parameters that define the population dynamics of the pathogen and the properties of the drug. Two particular pathogens are examined to illustrate the principles underpinning the dynamics of the pharmacokinetic-population dynamic models, namely HIV and Plasmodium falciparum. Criteria for pathogen persistence or elimination are derived for these specific models based on the definition of a basic reproductive number, R0, which measures the average number of secondary infected target cells in a host generated by a single infected cell (CD4 lymphocyte for HIV, and erythrocyte for P. falciparum) within a population of susceptible cells. For the pathogen to invade the host and persist over time, R0</=1. Under chemotherapeutic regimens, expressions for R0 are derived allowing estimates to be made of the ideal treatment regime required to eliminate the pathogen, both for HIV and P. falciparum malaria.

Animals↗

PubMed2000

Transmission and dynamics of tuberculosis on generalized households.

Tuberculosis (TB) transmission is enhanced by systematic exposure to an infectious individual. This enhancement usually takes place at either the home, workplace, and/or school (generalized household). Typical epidemiological models do not incorporate the impact of generalized households on the study of disease dynamics. Models that incorporate cluster (generalized household) effects and focus on their impact on TB's transmission dynamics are developed. Detailed models that consider the effect of casual infections, that is, those generated outside a cluster, are also presented. We find expressions for the Basic Reproductive Number as a function of cluster size. The formula for R0 separates the contributions of cluster and casual infections in the generation of secondary TB infections. Relationships between cluster and classical epidemic models are discussed as well as the concept of critical cluster size.

Causality↗

PubMed2000

Carrying capacity and demographic stochasticity: scaling behavior of the stochastic logistic model.

The stochastic logistic model is the simplest model that combines individual-level demography with density dependence. It explicitly or implicitly underlies many models of biodiversity of competing species, as well as non-spatial or metapopulation models of persistence of individual species. The model has also been used to study persistence in simple disease models. The stochastic logistic model has direct relevance for questions of limiting similarity in ecological systems. This paper uses a biased random walk heuristic to derive a scaling relationship for the persistence of a population under this model, and discusses its implications for models of biodiversity and persistence. Time to extinction of a species under the stochastic logistic model is approximated by the exponential of the scaling quantity U=(R-1)(2) N/R(R+1), where N is the habitat size and R is the basic reproductive number.

Animals↗

PubMed2000

A model for tuberculosis with exogenous reinfection.

Following primary tuberculosis (TB) infection, only approximately 10% of individuals develop active T.B. Most people are assumed to mount an effective immune response to the initial infection that limits proliferation of the bacilli and leads to long-lasting partial immunity both to further infection and to reactivation of latent bacilli remaining from the original infection. Infected individuals may develop active TB as a consequence of exogenous reinfection, i.e., acquiring a new infection from another infectious individual. Our results in this paper suggest that exogenous reinfection has a drastic effect on the qualitative dynamics of TB. The incorporation of exogenous reinfection into our TB model allows the possibility of a subcritical bifurcation at the critical value of the basic reproductive number R(0)=1, and hence the existence of multiple endemic equilibria for R(0)<1 and the exogenous reinfection rate larger than a threshold. Our results suggest that reducing R(0) to be smaller than one may not be sufficient to eradicate the disease. An additional reduction in reinfection rate may be required. These results may also partially explain the recently observed resurgence of TB.

Disease Susceptibility↗

PubMed1998

Transmission and distribution of African horse sickness virus serotypes in South African zebra.

The prevalences of African horse sickness (AHS) virus serotypes in zebra foals from the Kruger National Park, South Africa were examined for possible associations between serotypes. Serotypes known to cross-react were combined for analysis. The distributions of serotypes between zebra were not always independent; in 7-8 month old zebra positive pairwise associations were observed between 3 serotypes. This could be generated by biological interactions between serotypes or heterogeneity in host-vector transmission. The data were also used to estimate the basic reproduction number, R0. For AHS virus overall, estimates of R0 ranged from 31-68. This underlines the need for a better understanding of serotype transmission and interactions in AHS.

African Horse Sickness↗

PubMed1996

Estimation from current-status data in continuous time.

The nonparametric maximum likelihood estimator for current-status data has been known for at least 40 years, but only recently have the mathematical-statistical properties been clarified. This note provides a case study in the important and often studied context of estimating age-specific immunization intensities from a seroprevalence survey. Fully parametric and spline-based alternatives (also based on continuous-time models) are given. The basic reproduction number R0 exemplifies estimation of a functional. The limitations implied by the necessarily rather restrictive epidemiological assumptions are briefly discussed.

Adolescent↗