Showing papers on "Basic reproduction number published in 2006"
TL;DR: A large-scale stochastic simulation model is introduced and used to investigate the spread of a pandemic strain of influenza virus through the U.S. population and suggests that the rapid production and distribution of vaccines could significantly slow disease spread and limit the number ill to <10% of the population, particularly if children are preferentially vaccinated.
Abstract: Recent human deaths due to infection by highly pathogenic (H5N1) avian influenza A virus have raised the specter of a devastating pandemic like that of 1917-1918, should this avian virus evolve to become readily transmissible among humans. We introduce and use a large-scale stochastic simulation model to investigate the spread of a pandemic strain of influenza virus through the U.S. population of 281 million individuals for R0 (the basic reproductive number) from 1.6 to 2.4. We model the impact that a variety of levels and combinations of influenza antiviral agents, vaccines, and modified social mobility (including school closure and travel re- strictions) have on the timing and magnitude of this spread. Our simulations demonstrate that, in a highly mobile population, restricting travel after an outbreak is detected is likely to delay slightly the time course of the outbreak without impacting the eventual number ill. For R0 < 1.9, our model suggests that the rapid production and distribution of vaccines, even if poorly matched to circulating strains, could significantly slow disease spread and limit the number ill to <10% of the population, particularly if children are preferentially vaccinated. Alternatively, the aggressive deploy- ment of several million courses of influenza antiviral agents in a targeted prophylaxis strategy may contain a nascent outbreak with low R0, provided adequate contact tracing and distribution capacities exist. For higher R0, we predict that multiple strategies in combination (involving both social and medical interventions) will be required to achieve similar limits on illness rates. antiviral agents infectious diseases simulation modeling social network dynamics vaccines
1,085 citations
TL;DR: A mathematical model that describes HIV infection of CD4(+) T cells is analyzed and it is proved that, if the basic reproduction number R(0) < or = 1, the HIV infection is cleared from the T-cell population; if R( 0) > 1,The HIV infection persists.
Abstract: A mathematical model that describes HIV infection of CD4(+) T cells is analyzed. Global dynamics of the model is rigorously established. We prove that, if the basic reproduction number R(0) 1, the HIV infection persists. For an open set of parameter values, the chronic-infection equilibrium P* can be unstable and periodic solutions may exist. We establish parameter regions for which P* is globally stable.
279 citations
TL;DR: It is concluded 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 �”0, not by knowledge of any other epidemiological details.
269 citations
TL;DR: A simple ordinary differential equation model to study the epidemiological consequences of the drift mechanism for influenza A viruses is developed and a fourth class is introduced for the cross-immune individuals in the population, i.e., those that recovered after being infected by different strains of the same viral subtype in the past years.
Abstract: We develop a simple ordinary differential equation model to study the epidemiological consequences of the drift mechanism for influenza A viruses. Improving over the classical SIR approach, we introduce a fourth class (C) for the cross-immune individuals in the population, i.e., those that recovered after being infected by different strains of the same viral subtype in the past years. The SIRC model predicts that the prevalence of a virus is maximum for an intermediate value of R(0), the basic reproduction number. Via a bifurcation analysis of the model, we discuss the effect of seasonality on the epidemiological regimes. For realistic parameter values, the model exhibits a rich variety of behaviors, including chaos and multi-stable periodic outbreaks. Comparison with empirical evidence shows that the simulated regimes are qualitatively and quantitatively consistent with reality, both for tropical and temperate countries. We find that the basins of attraction of coexisting cycles can be fractal sets, thus predictability can in some cases become problematic even theoretically. In accordance with previous studies, we find that increasing cross-immunity tends to complicate the dynamics of the system.
162 citations
TL;DR: This review synthesizes the conflicting outbreak predictions generated by different biological assumptions in host-vector disease models and demonstrates how the choice of transmission term qualitatively and quantitatively alters R(0) and therefore alters predicted disease dynamics and control implications.
Abstract: This review synthesizes the conflicting outbreak predictions generated by different biological assumptions in host–vector disease models. It is motivated by the North American outbreak of West Nile virus, an emerging infectious disease that has prompted at least five dynamical modelling studies. Mathematical models have long proven successful in investigating the dynamics and control of infectious disease systems. The underlying assumptions in these epidemiological models determine their mathematical structure, and therefore influence their predictions. A crucial assumption is the host–vector interaction encapsulated in the disease-transmission term, and a key prediction is the basic reproduction number, ℛ0. We connect these two model elements by demonstrating how the choice of transmission term qualitatively and quantitatively alters ℛ0 and therefore alters predicted disease dynamics and control implications. Whereas some transmission terms predict that reducing the host population will reduce disease outbreaks, others predict that this will exacerbate infection risk. These conflicting predictions are reconciled by understanding that different transmission terms apply biologically only at certain population densities, outside which they can generate erroneous predictions. For West Nile virus, ℛ0 estimates for six common North American bird species indicate that all would be effective outbreak hosts.
141 citations
TL;DR: It is proved that the global dynamics are completely determined by the basic reproduction number R0, which means that the disease-free equilibrium P(0) is globally asymptotically stable and the disease always dies out.
Abstract: We analyze a mathematical model for infectious diseases that
progress through distinct stages within infected hosts. An example
of such a disease is AIDS, which results from HIV infection. For a
general $n$-stage stage-progression (SP) model with bilinear
incidences, we prove that the global dynamics are completely
determined by the basic reproduction number $R_0.$ If $R_0\le 1,$
then the disease-free equilibrium $P_0$ is globally asymptotically
stable and the disease always dies out. If $R_0>1,$ $P_0$ is
unstable, and a unique endemic equilibrium $P^*$ is globally
asymptotically stable, and the disease persists at the endemic
equilibrium. The basic reproduction numbers for the SP model with
density dependent incidence forms are also discussed.
102 citations
TL;DR: More effort is still needed to convince those outside of epidemiology to more fully use epidemiological results and insights into the development and evaluation of disease controls.
Abstract: Epidemiology involves the study of the temporal, spatial, and spatio-temporal dynamics of disease in populations, and the utilization of results of experiments and surveys to describe, understand, compare, and predict epidemics. Such understanding and description of epidemics can lead directly to the development and evaluation of efficient control strategies and tactics. Mathematical and statistical models are key tools of the epidemiologist. Recent advances in statistics, including linear and nonlinear mixed models, are allowing a more appropriate matching of data type and experimental (or survey) design to the statistical model used for analysis, in order to meet the objectives of the investigator. Coupled ordinary and partial differential equations, as well as simpler growth-curve equations, are especially useful deterministic models for representing plant disease development in fields in time and space over single seasons or many years, and their use can lead to appraisal of control strategies through metrics such as the basic reproduction number, a summary parameter that may be calculated for many general epidemic scenarios. Recently, compelling arguments have been made for the use of Bayesian decision theory in developing and evaluating real-time disease prediction rules, based on measured disease or weather conditions and either empirical or mechanistic models for disease or control intervention. Through some simple calculations of predictor accuracy and (prior) probability of an epidemic (or the need for control), the success of any predictor can be quantified in terms of the estimated probability of random observations being epidemics when predicted to be epidemics or not epidemics. Overall, despite the many contributions in epidemiology over the past four decades, more effort is still needed to convince those outside of epidemiology to more fully use epidemiological results and insights into the development and evaluation of disease controls.
88 citations
TL;DR: It is shown that by controlling the rate of vertical transmission, the spread of the disease can be reduced significantly and consequently the equilibrium values of infective and AIDS population can be maintained at desired levels.
88 citations
01 Jan 2006
TL;DR: This chapter considers an alternative approach for developing theory in evolutionary epidemiology that uses the instantaneous rate of change of the number of infected hosts instead of using the total number of new infections generated by an infected individual as a measure of pathogen fitness.
Abstract: The basic reproduction number, denoted by R0, is one of the most important quantities in epidemiological theory ([11], [23]). It is defined as the expected number of new infections generated by an infected individual in an otherwise wholly susceptible population ([2], [12], [23]). Part of the reason why R0 plays such a central role in this body of theory undoubtedly stems from its relatively simple and intuitively sensible interpretation as a measure of pathogen reproduction. If R0 is less than unity then we expect the pathogen to die out since each infected individual fails to generate at least one other infection during the lifetime of the infection. Given that R0 is a measure of pathogen reproductive success, it is not surprising that this quantity has also come to form the basis of most evolutionary considerations of host-pathogen interactions ([1], [18]). For example, mathematical models for numerous epidemiological settings have been used to demonstrate that natural selection is often expected to favour the pathogen strain that results in the largest value of R0 ([6], [18]). In more complex epidemiological settings such optimization criteria typically cannot be derived and instead a game-theoretic approach is taken ([5]). In this context a measure of the fitness of a rare mutant pathogen strain is used to characterize the evolutionarily stable strain (i.e., the strain that, if present within the population in sufficient numbers, cannot be displaced by any mutant strain that arises). Typically R0 again plays a central role as the measure of mutant fitness in such invasion analyses ([10], [18], [30]). In this chapter we consider an alternative approach for developing theory in evolutionary epidemiology. Rather than using the total number of new infections generated by an infected individual (i.e., R0) as a measure of pathogen fitness we use the instantaneous rate of change of the number of infected hosts instead (see also [3], [18]). This shifts the focus from a consideration of pathogen reproductive success per generation to pathogen reproductive success per unit time. One very useful result of this change in focus is that we can then model the time dynamics of evolutionary change in the pathogen population simultaneously with the epidemiological dynamics, rather than simply characterizing the evolutionary equilibria that are expected. Even more importantly, however, this seemingly slight change
78 citations
TL;DR: This analysis shows that transport-related infection intensifies the disease spread if infectious diseases break out to cause an endemic situation in each region, in the sense that both the absolute and relative size of patients increase.
69 citations
TL;DR: In this paper, an SEIR epidemic model with the infectious force in the latent (exposed), infected and recovered period is studied and sufficient conditions for the global stability of the endemic equilibrium are obtained by the compound matrix theory.
Abstract: An SEIR epidemic model with the infectious force in the latent (exposed), infected and recovered period is studied. It is assumed that susceptible and exposed individuals have constant immigration rates. The model exhibits a unique endemic state if the fraction p of infectious immigrants is positive. If the basic reproduction number R0 is greater than 1, sufficient conditions for the global stability of the endemic equilibrium are obtained by the compound matrix theory. 2005 Elsevier Ltd. All rights reserved.
TL;DR: The present modelling exercise and results show that it is possible to study the population dynamics of disease vectors, and hence oscillatory behaviour as it is often observed in most indirectly transmitted infectious diseases of humans, without recourse to external seasonal forcing.
TL;DR: It is shown for the first time that for epidemiological models with backward bifurcation an expression for the minimum effort required to eradicate the infection if the authors concentrate on control measures affecting the transmission rate constant β is given.
Abstract: We study an epidemiological model which assumes that the susceptibility after a primary infection is r times the susceptibility before a primary infection. For r = 0 (r = 1) this is the SIR (SIS) model. For r > 1 + (μ/α) this model shows backward bifurcations, where μ is the death rate and α is the recovery rate. We show for the first time that for such models we can give an expression for the minimum effort required to eradicate the infection if we concentrate on control measures affecting the transmission rate constant β. This eradication effort is explicitly expressed in terms of α,r, and μ As in models without backward bifurcation it can be interpreted as a reproduction number, but not necessarily as the basic reproduction number. We define the relevant reproduction numbers for this purpose. The eradication effort can be estimated from the endemic steady state. The classical basic reproduction number R0 is smaller than the eradication effort for r > 1 + (μ/α) and equal to the effort for other values of r. The method we present is relevant to the whole class of compartmental models with backward bifurcation.
TL;DR: 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.
Abstract: 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.
TL;DR: An age-duration-structured population model for HIV infection in a homosexual community and there could exist multiple endemic steady states even if R(0) is less than one is considered, in contrast with classical epidemic models.
Abstract: In this paper we consider an age-duration-structured population model for HIV infection in a homosexual community. First we investigate the invasion problem to establish the basic reproduction ratio R0 for the HIV/AIDS epidemic by which we can state the threshold criteria: The disease can invade into the completely susceptible population if R0 > 1, whereas it cannot if R0 < 1. Subsequently, we examine existence and uniqueness of endemic steady states. We will show sufficient conditions for a backward or a forward bifurcation to occur when the basic reproduction ratio crosses unity. That is, in contrast with classical epidemic models, for our HIV model there could exist multiple endemic steady states even if R0 is less than one. Finally, we show sufficient conditions for the local stability of the endemic steady states.
TL;DR: An epidemic model is proposed to incorporate population dispersals between patches and to include a constant infection period and it is found that a disease may spread when the population migrates in two patches, even though it dies out in each isolated patch.
Abstract: An epidemic model is proposed to incorporate population dispersals between patches and to include a constant infection period. A basic reproduction number of the model is established by means of a next generation matrix. It is found that a disease may spread when the population migrates in two patches, even though it dies out in each isolated patch. It is also found that the disease admits multiple exchanges between persistence and extinction for some types of migrations of individuals.
TL;DR: An SIQS model is proposed to study the effect of transport-related infection and entry screening and it is shown that the disease is endemic in the sense of permanence if and only if the endemic equilibrium exists.
TL;DR: Numerical studies suggest that group structure makes major outbreaks less likely than would be the case in a homogeneous population with the same basic reproduction number.
Abstract: A multi-group semi-stochastic model is formulated to identify possible causes of why different strains of Salmonella develop so much variation in their infection dynamics in UK dairy herds. The model includes demography (managed populations) and various types of transmission: direct, pseudovertical and indirect (via free-living infectious units in the environment). The effects of herd size and epidemiological parameters on mean prevalence of infection and mean time until fade out are investigated. Numerical simulation shows that higher pathogen-induced mortality, shorter infectious period, more persistent immune response and more rapid removal of faeces result in a lower mean prevalence of infection, a shorter mean time until fade out, and a greater probability of fade out of infection within 600 days. Combining these results and those for the deterministic counterpart could explain differences in observed epidemiological patterns and help to identify the factors inducing the decline in reported cases of epidemic strains such as DT104 in cattle. We further investigate the effect of group structure on the probability of a major outbreak by using the stochastic threshold theory in homogeneous populations and that in heterogeneous populations. Numerical studies suggest that group structure makes major outbreaks less likely than would be the case in a homogeneous population with the same basic reproduction number. Moreover, some control strategies are suggested by investigating the effect of epidemiological parameters on the probability of an epidemic.
TL;DR: Given that simple threshold vaccination coverage might not be realistic within racing stables, the combination of movement restrictions, isolation, and other countermeasures alongside vaccination is recommended to contain outbreaks of equine influenza.
TL;DR: A two-sex network model for a sexually transmitted disease which assumes random mixing conditional on the degree distribution is considered and expressions for the basic reproductive number (R(0)) for one and heterogeneous two-population are derived in terms of characteristics of the degree distributions and transmissibility.
TL;DR: It is shown that targeting any one sector of the commercial sex alone for prevention will be difficult to have a decided effect on eradicating the epidemic, but if the aim of the targeted intervention policy is not eradication of the epidemic but decrease in HIV incidence of a particular high-risk group, then concentrated targeting strategy could be sufficient, if properly implemented.
Dissertation•
01 Aug 2006
TL;DR: This investigation formulate and analyze multi-host and multi-patch mathematical models for disease emergence and examines SIS and SIR epidemic models, which have applications to Hantavirus in wild rodents as well as other zoonotic diseases with multiple hosts.
Abstract: Most zoonotic pathogens are capable of infecting multiple hosts. These animal hosts can often move through a large territory and many different landscapes. In this investigation, we formulate and analyze multi-host and multi-patch mathematical models for disease emergence. In particular, we examine SIS and SIR epidemic models. We determine under what conditions the disease can emerge. In the multi-host models, we examine a single pathogen that can infect n different hosts. In the multi-patch models, a single host moves through n different patches. For the multi-host models, it is shown that the basic reproduction number increases with the number of hosts. Therefore, the possibility for disease emergence increases with the number of infected hosts. In the multi-patch models, the basic reproduction number for the system lies between the basic reproduction numbers for each disconnected patch. Therefore, connection of patches may or may not lead to disease emergence. It is shown that as the mobility of hosts between patches increases, the basic reproduction number approaches a limiting value. We also examine a three-patch model which supports two hosts. The basic reproduction number is found for this model. For each type of model, we also formulate a set of stochastic differential equations. These stochastic models introduce variability through the demographic changes. Numerical examples illustrate the dynamics of each of the models. The models have applications to Hantavirus in wild rodents as well as other zoonotic diseases with multiple hosts.
01 Jan 2006
TL;DR: In this article, the authors considered a compartmental model for the transmission dynamics of the HTLV-I infection and established that the global dynamics of T-cell infection are completely determined by a basic reproduction number R0.
Abstract: Human T-cell lymphotropic virus I (HTLVI) infection is linked to the development of adult T-cell leukemia/lymphoma (ATL). HTLV-I infection of healthy CD4 T cells is known to take place through cell-to-cell contact with infected T cells. We consider a compartmental model for the transmission dynamics of the HTLV-I infection. The force of infection is assumed to be of a general form, and the resulting incidence term contains, as special cases, the bilinear and the standard incidences. Our mathematical analysis establishes that the global dynamics of T-cell infection are completely determined by a basic reproduction number R0. If R0 ≤ 1, infected T cells always die out. If R0 > 1, HTLV-I infection becomes chronic, and a unique endemic equilibrium is globally stable in the interior of the feasible region.
TL;DR: By means of limit theory and Fonda's theorem, an SEIS epidemic model with constant recruitment and the disease-related rate is considered in this paper, where the incidence term is of the nonlinear form, and the basic reproduction number is found.
Abstract: By means of limit theory and Fonda’s theorem, an SEIS epidemic model with constant recruitment and the disease-related rate is considered. The incidence term is of the nonlinear form, and the basic reproduction number is found. If the basic reproduction number is less than one, there exists only the disease-free equilibrium, which is globally asymptotically stable, and the disease dies out eventually. If the basic reproduction number is greater than one, besides the unstable disease-free equilibrium, there exists also a unique endemic equilibrium, which is locally asymptotically stable, and the disease is uniformly persistent.
13 Jun 2006
TL;DR: In this article, an age-structured partial differential equation compartmental model was used to predict minimal vaccination strategies to eliminate hepatitis A in Bulgaria using single and double stage vaccination campaigns.
Abstract: In this paper we examine an age-structured partial differential equation compartmental model to predict minimal vaccination strategies to eliminate hepatitis A in Bulgaria. We describe the mathematical model and briefly summarise previous theoretical results. The basic reproduction number is a key parameter of the model. We consider proportional, assortative and symmetric mixing. Using pre-hepatitis A vaccination Bulgarian age-serological data we derive estimates for the basic reproduction number and minimum proportions of susceptibles to be vaccinated to eliminate hepatitis A in Bulgaria using single and double stage vaccination campaigns. 95 percentile confidence intervals are also given.
TL;DR: In this paper, a pulse vaccination SIR model with periodic infection rate β (t) has been proposed and studied, and the dynamical behaviors of the model are analyzed with the help of persistence, bifurcation and global stability.
Abstract: A pulse vaccination SIR model with periodic infection rate $\beta (t)$ have been proposed and studied. The basic reproductive number $R_0$ is defined. The dynamical behaviors of the model are analyzed with the help of persistence, bifurcation and global stability. It has been shown that the infection-free periodic solution is globally stable provided $R_0 1$. Standard bifurcation theory have been used to show the existence of the positive periodic solution for the case of $R_0 \to1^+$. Finally, the numerical simulations have been performed to show the uniqueness and the global stability of the positive periodic solution of the system.