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Results for “Basic Reproduction Number”

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PubMed2005

Mathematical analysis of an age-structured population model with space-limited recruitment.

In this paper, we investigate structured population model of marine invertebrate whose life stage is composed of sessile adults and pelagic larvae, such as barnacles contained in a local habitat. First we formulate the basic model as an Cauchy problem on a Banach space to discuss the existence and uniqueness of non-negative solution. Next we define the basic reproduction number R0 to formulate the invasion condition under which the larvae can successfully settle down in the completely vacant habitat. Subsequently we examine existence and stability of steady states. We show that the trivial steady state is globally asymptotically stable if R0 < or = 1, whereas it is unstable if R0 > 1. Furthermore, we show that a positive (non-trivial) steady state uniquely exists if R0 > 1 and it is locally asymptotically stable as far as absolute value of R0 - 1 is small enough.

Animals↗

PubMed2006

On the transmission of HIV with self-protective behavior and preferred mixing.

An epidemic model of HIV transmission with self-protective behavior and preferred mixing is presented. Individuals in the model are assumed to choose their levels of risk behavior by comparing the costs and benefits of self-protective actions. Unlike in models which treat individual risk behavior as exogenously given and fixed, the condition under which an endemic steady state equilibrium exists does not depend on the extent of assortative mixing in the population. Specifically, a unique endemic equilibrium exists when the basic reproductive number of the disease, which is given in the model by the expected number of secondary infections caused by an infected individual in the absence of any self-protection, is strictly greater than one. Otherwise, the disease-free equilibrium is the only steady state equilibrium. With respect to changes in contact patterns, it is shown that, if the degree of preferred mixing is increased, the disease prevalence can decrease in the high-risk subpopulation consisting of individuals who are more likely to engage in unsafe practices. The situation is reversed for the low-risk subpopulation, which is composed of individuals who are less willing to engage in risky practices, so that increasing the likelihood of mixing with members of one's own group may increase the prevalence level within the low-risk subpopulation.

Behavior Therapy↗

PubMed2006

Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells.

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) < or = 1, the HIV infection is cleared from the T-cell population; if 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.

Algorithms↗

PubMed2006

The SIRC model and influenza A.

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.

Antigenic Variation↗

PubMed2006

A semi-stochastic model for Salmonella infection in a multi-group herd.

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.

Animals↗

PubMed2006

Persistence and permanence in a metapopulation model with space-limited recruitment.

In this paper we analyze a metapopulation model with space-limited recruitment. The model describes the population dynamics of sessile adult and planktonic larvae in a common larval pool. We introduce the basic reproduction number of each species which is the expected number of future larvae reproduced by one larva. We consider the conditions for the persistence of the multi-species and multi-habitats model and the permanence of the single-species model. Subsequently, we consider the conditions for the existence of the non-trivial steady state of the single-species model and its global stability, and the permanence of the two species and two habitats model.

Animals↗

PubMed2006

Modeling relapse in infectious diseases.

An integro-differential equation is proposed to model a general relapse phenomenon in infectious diseases including herpes. The basic reproduction number R(0) for the model is identified and the threshold property of R(0) established. For the case of a constant relapse period (giving a delay differential equation), this is achieved by conducting a linear stability analysis of the model, and employing the Lyapunov-Razumikhin technique and monotone dynamical systems theory for global results. Numerical simulations, with parameters relevant for herpes, are presented to complement the theoretical results, and no evidence of sustained oscillatory solutions is found.

Computer Simulation↗

PubMed2005

Forecasting versus projection models in epidemiology: the case of the SARS epidemics.

In this work we propose a simple mathematical model for the analysis of the impact of control measures against an emerging infection, namely, the severe acute respiratory syndrome (SARS). The model provides a testable hypothesis by considering a dynamical equation for the contact parameter, which drops exponentially with time, simulating control measures. We discuss the role of modelling in public health and we analyse the distinction between forecasting and projection models as assessing tools for the estimation of the impact of intervention strategies. The model is applied to the communities of Hong Kong and Toronto (Canada) and it mimics those epidemics with fairly good accuracy. The estimated values for the basic reproduction number, R0, were 1.2 for Hong Kong and 1.32 for Toronto (Canada). The model projects that, in the absence of control, the final number of cases would be 320,000 in Hong Kong and 36,900 in Toronto (Canada). In contrast, with control measures, which reduce the contact rate to about 25% of its initial value, the expected final number of cases is reduced to 1778 in Hong Kong and 226 in Toronto (Canada). Although SARS can be a devastating infection, early recognition, prompt isolation, and appropriate precaution measures, can be very effective to limit its spread.

Canada↗

PubMed2005

A modelling framework to describe the spread of scrapie between sheep flocks in Great Britain.

My aim was to develop a stochastic, spatial model describing the spread of scrapie between sheep flocks in Great Britain; I wanted a model, which could subsequently be used to assess the efficacy of different control strategies. The structure of the model reflects the demography of the British sheep flock, including a description of the contact structure between flocks. The dynamics of scrapie were incorporated through two probabilities associated with each flock: of acquiring infection and of experiencing a within-flock outbreak following exposure. The acquisition of infection depends on whether or not a flock buys-in sheep and, if it does, whether or not it trades with an affected flock. Once a flock is exposed, the probability of a within-flock outbreak occurring and its duration depend on the basic reproductive number, the prion-protein (PrP) genotype profile and the flock size. The model was validated using regional data from two postal surveys conducted in 1998 and 2002, which demonstrated that the model captures the spatial dynamics of scrapie (at least at a regional level). Moreover, the predicted distribution for the duration of a within-flock outbreak reflects the duration of outbreaks reported in the literature. Using the model to predict long-term trends in the proportion of affected flocks suggested that, even without control measures beyond the removal of animals with clinical signs, scrapie ultimately will disappear from the national flock, though it is likely to be decades before the disease is eliminated. However, there were scenarios consistent with the available data which suggested that scrapie could remain endemic within the British sheep flock. Consequently, it is essential to take this uncertainty in the long-term dynamics of scrapie into account when considering the efficacy of control strategies. Although control strategies were not explicitly examined, the model suggests two aspects important for control: larger flocks remain affected for longer and provide infection for other, smaller flocks and animal movements must be traceable.

Animals↗

PubMed2005

A model of lineage-1 and lineage-2 rinderpest virus transmission in pastoral areas of East Africa.

The development of a stochastic, state-transition model of rinderpest transmission dynamics is described using parameter estimates obtained from both laboratory and participatory research. Using serological data, the basic reproduction numbers for lineage-1 rinderpest virus in southern Sudan and for lineage-2 rinderpest virus in Somali livestock were estimated as 4.4 and between 1.2 and 1.9, respectively. The model predictions for the inter-epidemic period in Sudan and Somalia (1.2 and 4.2 years, respectively) were in agreement with analysis of livestock-owner reports (1-2 years and 5 years, respectively).

Animals↗

PubMed2005

A model of contagious bovine pleuropneumonia transmission dynamics in East Africa.

The dynamics of contagious bovine pleuropneumonia (CBPP) transmission vary widely between livestock production systems. This paper describes the development of a homogeneous, stochastic, compartmental model for CBPP transmission in pastoral herds of East Africa. The model was built using parameter estimates based on data published in the literature and on observations of livestock owners obtained through participatory research. The basic reproduction number for CBPP in southern Sudan was estimated to range from 3.2 to 4.6. The homogeneous model indicates that the critical community size for the persistence of CBPP falls within the typical herd sizes for pastoral communities in East Africa suggesting that individual isolated herds are capable of maintaining infection indefinitely. Vaccination alone with currently available vaccines was unlikely to eradicate the disease.

Africa, Eastern↗

PubMed2006

Interval estimates for epidemic thresholds in two-sex network models.

Epidemic thresholds in network models of heterogeneous populations characterized by highly right-skewed contact distributions can be very small. When the population is above the threshold, an epidemic is inevitable and conventional control measures to reduce the transmissibility of a pathogen will fail to eradicate it. We consider a two-sex network model for a sexually transmitted disease which assumes random mixing conditional on the degree distribution. We derive expressions for the basic reproductive number (R(0)) for one and heterogeneous two-population in terms of characteristics of the degree distributions and transmissibility. We calculate interval estimates for the epidemic thresholds for stochastic process models in three human populations based on representative surveys of sexual behavior (Uganda, Sweden, USA). For Uganda and Sweden, the epidemic threshold is greater than zero with high confidence. For the USA, the interval includes zero. We discuss the implications of these findings along with the limitations of epidemic models which assume random mixing.

Acquired Immunodeficiency Syndrome↗

PubMed2006

On the role of cross-immunity and vaccines on the survival of less fit flu-strains.

A pathogen's route to survival involves various mechanisms including its ability to invade (host's susceptibility) and its reproductive success within an invaded host ("infectiousness"). The immunological history of an individual often plays an important role in reducing host susceptibility or it helps the host mount a faster immunological response de facto reducing infectiousness. The cross-immunity generated by prior infections to influenza A strains from the same subtype provide a significant example. The results of this paper are based on the analytical study of a two-strain epidemic model that incorporates host isolation (during primary infection) and cross-immunity to study the role of invasion mediated cross-immunity in a population where a precursor related strain (within the same subtype, i.e. H3N2, H1N1) has already become established. An uncertainty and sensitivity analysis is carried out on the ability of the invading strain to survive for given cross-immunity levels. Our findings indicate that it is possible to support coexistence even in the case when invading strains are "unfit", that is, when the basic reproduction number of the invading strain is less than one. However, such scenarios are possible only in the presence of isolation. That is, appropriate increments in isolation rates and weak cross-immunity can facilitate the survival of less fit strains. The development of "flu" vaccines that minimally enhance herd cross-immunity levels may, by increasing genotype diversity, help facilitate the generation and survival of novel strains.

Animals↗

PubMed2000

Subcritical endemic steady states in mathematical models for animal infections with incomplete immunity.

Many classical mathematical models for animal infections assume that all infected animals transmit the infection at the same rate, all are equally susceptible, and the course of the infection is the same in all animals. However for some infections there is evidence that seropositives may still transmit the infection, albeit at a lower rate. Animals can also experience more than one episode of the infection although those who have already experienced it have a partial immune resistance. Animals who experience a second or subsequent period of infection may not necessarily exhibit clinical symptoms. The main example discussed is bovine respiratory syncytial virus (BRSV) amongst cattle. We consider simple models with vaccination and homogeneous and proportional mixing between seropositives and seronegatives. We derive an expression for the basic reproduction number, R(o), and perform an equilibrium and stability analysis. We find that it may be possible for there to be two endemic equilibria (one stable and one unstable) for R(o)<1 and in this case at R(o)=1 there is a backwards bifurcation of an unstable endemic equilibrium from the infection-free equilibrium. Then the implications for control strategies are considered. Finally applications to Aujesky's disease (pseudorabies virus) in pigs are discussed.

Animals↗

PubMed2000

Analysis and simulation of a stochastic, discrete-individual model of STD transmission with partnership concurrency.

Deterministic differential equation models indicate that partnership concurrency and non-homogeneous mixing patterns play an important role in the spread of sexually transmitted infections. Stochastic discrete-individual simulation studies arrive at similar conclusions, but from a very different modeling perspective. This paper presents a stochastic discrete-individual infection model that helps to unify these two approaches to infection modeling. The model allows for both partnership concurrency, as well as the infection, recovery, and reinfection of an individual from repeated contact with a partner, as occurs with many mucosal infections. The simplest form of the model is a network-valued Markov chain, where the network's nodes are individuals and arcs represent partnerships. Connections between the differential equation and discrete-individual approaches are constructed with large-population limits that approximate endemic levels and equilibrium probability distributions that describe partnership concurrency. A more general form of the discrete-individual model that allows for semi-Markovian dynamics and heterogeneous contact patterns is implemented in simulation software. Analytical and simulation results indicate that the basic reproduction number R(0) increases when reinfection is possible, and the epidemic rate of rise and endemic levels are not related by 1-1/R(0), when partnerships are not point-time processes.

Computer Simulation↗

PubMed2000

The transmission dynamics of the aetiological agent of scrapie in a sheep flock.

We formulate and investigate the properties of a model framework to mimic the transmission dynamics of the aetiological agent of scrapie in a sheep flock. We derive expressions for summary parameters that characterize transmission scenarios, notably the basic reproduction number R(0) and the mean generation time T(g). The timescale of epidemic outbreaks is expressed in terms of R(0) and cumulants of the generation time distribution. We discuss the relative contributions to the overall rate of transmission of horizontal and vertical routes during invasion and in endemicity. Simplified models are used to obtain analytical insight into the characteristics of the endemic state.

Animals↗

PubMed2001

Global stability of an SEIS epidemic model with recruitment and a varying total population size.

This paper considers an SEIS epidemic model that incorporates constant recruitment, disease-caused death and disease latency. The incidence term is of the bilinear mass-action form. It is shown that the global dynamics is completely determined by the basic reproduction number R(0). If R(0) 1, a unique endemic equilibrium is globally stable in the interior of the feasible region and the disease persists at the endemic equilibrium.

Communicable Diseases↗

PubMed2001

Modeling epidemics caused by respiratory syncytial virus (RSV).

Respiratory syncytial virus (RSV) is the most common cause of acute lower respiratory tract infection in children. In this paper we use models of RSV transmission to interpret the pattern of seasonal epidemics of RSV disease observed in different countries, and to estimate epidemic and eradication thresholds for RSV infection. We compare the standard SIRS model with a more realistic model of RSV transmission in which individuals acquire immunity gradually after repeated exposure to infection. The models are fitted to series of monthly hospital case reports of RSV disease from developed and developing countries. The models can explain many of the observed patterns: regular yearly outbreaks in some countries, and in other countries cycles of alternating larger and smaller annual epidemics, with shifted maxima in alternate years. Previously these patterns have been attributed to the transmission of different strains of RSV. In some countries the timing of epidemics is not consistent with increased social contact among school children during term time being the major driving mechanism. Climatic factors appear to be more important. Qualitatively different models gave equally good fits to the data series, but estimates of the transmission parameter were different by a factor of 4. Estimates of the basic reproduction number (R(0)) ranged from 1.2 to 2.1 with the SIRS model, and from 5.4 to 7.1 with the model with gradual acquisition of partial immunity.

Child↗