A metapopulation model for the spread and persistence of contagious bovine pleuropneumonia (CBPP) in African sedentary mixed crop-livestock systems

Contagious bovine pleuropneumonia (CBPP) is endemic in several developing countries. Our objective is to evaluate the regional CBPP spread and persistence in a mixed crop-livestock system in Africa. A stochastic compartmental model in metapopulation is used, in which between-herd animal movements and the within-herd infection dynamics are explicitly represented. Hundred herds of varying size are modelled, each sending animals to n other herds (network degree). Animals are susceptible, latent, infectious, chronic carrier or resistant. The role of chronic carriers in CBPP spread being still debated, several chronic periods and infectiousness are tested. A sensitivity analysis is performed to evaluate the influence on model outputs of these parameters and of pathogen virulence, between-herd movement rate, network degree, and calves recruitment. Model outputs are the probability that individual- and group-level reproductive numbers R₀ and R_* are above one, the metapopulation infection duration, the probability of CBPP endemicity (when CBPP persists over 5 years), and the epidemic size in infected herds and infected animals. The most influential parameters are related to chronic carriers (infectiousness and chronic period), pathogen virulence, and recruitment rate. When assuming no CBPP re-introduction in the region, endemicity is only probable if chronic carriers are assumed infectious for at least 1 year and to shed the pathogen in not too low an amount. It becomes highly probable when assuming high pathogen virulence and high recruitment rate.     

The basic reproduction number and the probability of extinction for a dynamic epidemic model

We consider the spread of an epidemic through a population divided into n sub-populations, in which individuals move between populations according to a Markov transition matrix Σ and infectives can only make infectious contacts with members of their current population. Expressions for the basic reproduction number, R₀, and the probability of extinction of the epidemic are derived. It is shown that in contrast to contact distribution models, the distribution of the infectious period effects both the basic reproduction number and the probability of extinction of the epidemic in the limit as the total population size N → ∞. The interactions between the infectious period distribution and the transition matrix Σ mean that it is not possible to draw general conclusions about the effects on R₀ and the probability of extinction. However, it is shown that for n = 2, the basic reproduction number, R₀, is maximised by a constant length infectious period and is decreasing in ?, the speed of movement between the two populations.     

On the definition and utilization of heritable variation among hosts in reproduction ratio R0 for infectious diseases

Infectious diseases have a major role in evolution by natural selection and pose a worldwide concern in livestock. Understanding quantitative genetics of infectious diseases, therefore, is essential both for understanding the consequences of natural selection and for designing artificial selection schemes in agriculture. The basic reproduction ratio, R₀, is the key parameter determining risk and severity of infectious diseases. Genetic improvement for control of infectious diseases in host populations should therefore aim at reducing R₀. This requires definitions of breeding value and heritable variation for R₀, and understanding of mechanisms determining response to selection. This is challenging, as R₀ is an emergent trait arising from interactions among individuals in the population. Here we show how to define breeding value and heritable variation for R₀ for genetically heterogeneous host populations. Furthermore, we identify mechanisms determining utilization of heritable variation for R₀. Using indirect genetic effects, next-generation matrices and a SIR (Susceptible, Infected and Recovered) model, we show that an individual''s breeding value for R₀ is a function of its own allele frequencies for susceptibility and infectivity and of population average susceptibility and infectivity. When interacting individuals are unrelated, selection for individual disease status captures heritable variation in susceptibility only, yielding limited response in R₀. With related individuals, however, there is a secondary selection process, which also captures heritable variation in infectivity and additional variation in susceptibility, yielding substantially greater response. This shows that genetic variation in susceptibility represents an indirect genetic effect. As a consequence, response in R₀ increased substantially when interacting individuals were genetically related.     

Epidemic modelling: Aspects where stochasticity matters

Epidemic models are always simplifications of real world epidemics. Which real world features to include, and which simplifications to make, depend both on the disease of interest and on the purpose of the modelling. In the present paper we discuss some such purposes for which a stochastic model is preferable to a deterministic counterpart. The two main examples illustrate the importance of allowing the infectious and latent periods to be random when focus lies on the probability of a large epidemic outbreak and/or on the initial speed, or growth rate, of the epidemic. A consequence of the latter is that estimation of the basic reproduction number R₀ is sensitive to assumptions about the distributions of the infectious and latent periods when using data from the early stages of an outbreak, which we illustrate with data from the H1N1 influenza A pandemic. Some further examples are also discussed as are some practical consequences related to these stochastic aspects.     

An SIS patch model with variable transmission coefficients

In this paper, an SIS patch model with non-constant transmission coefficients is formulated to investigate the effect of media coverage and human movement on the spread of infectious diseases among patches. The basic reproduction number R₀ is determined. It is shown that the disease-free equilibrium is globally asymptotically stable if R₀?1, and the disease is uniformly persistent and there exists at least one endemic equilibrium if R₀>1. In particular, when the disease is non-fatal and the travel rates of susceptible and infectious individuals in each patch are the same, the endemic equilibrium is unique and is globally asymptotically stable as R₀>1. Numerical calculations are performed to illustrate some results for the case with two patches.     

The value of decreasing the duration of the infectious period of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection

Finding medications or vaccines that may decrease the infectious period of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) could potentially reduce transmission in the broader population. We developed a computational model of the U.S. simulating the spread of SARS-CoV-2 and the potential clinical and economic impact of reducing the infectious period duration. Simulation experiments found that reducing the average infectious period duration could avert a median of 442,852 [treating 25% of symptomatic cases, reducing by 0.5 days, reproductive number (R₀) 3.5, and starting treatment when 15% of the population has been exposed] to 44.4 million SARS-CoV-2 cases (treating 75% of all infected cases, reducing by 3.5 days, R₀ 2.0). With R₀ 2.5, reducing the average infectious period duration by 0.5 days for 25% of symptomatic cases averted 1.4 million cases and 99,398 hospitalizations; increasing to 75% of symptomatic cases averted 2.8 million cases. At $500/person, treating 25% of symptomatic cases saved $209.5 billion (societal perspective). Further reducing the average infectious period duration by 3.5 days averted 7.4 million cases (treating 25% of symptomatic cases). Expanding treatment to 75% of all infected cases, including asymptomatic infections (R₀ 2.5), averted 35.9 million cases and 4 million hospitalizations, saving $48.8 billion (societal perspective and starting treatment after 5% of the population has been exposed). Our study quantifies the potential effects of reducing the SARS-CoV-2 infectious period duration.     

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.     

Sheep Movement Networks and the Transmission of Infectious Diseases

Background and Methodology

Various approaches have been used to investigate how properties of farm contact networks impact on the transmission of infectious diseases. The potential for transmission of an infection through a contact network can be evaluated in terms of the basic reproduction number, R ₀. The magnitude of R ₀ is related to the mean contact rate of a host, in this case a farm, and is further influenced by heterogeneities in contact rates of individual hosts. The latter can be evaluated as the second order moments of the contact matrix (variances in contact rates, and co-variance between contacts to and from individual hosts). Here we calculate these quantities for the farms in a country-wide livestock network: >15,000 Scottish sheep farms in each of 4 years from July 2003 to June 2007. The analysis is relevant to endemic and chronic infections with prolonged periods of infectivity of affected animals, and uses different weightings of contacts to address disease scenarios of low, intermediate and high animal-level prevalence.

Principal Findings and Conclusions

Analysis of networks of Scottish farms via sheep movements from July 2003 to June 2007 suggests that heterogeneities in movement patterns (variances and covariances of rates of movement on and off the farms) make a substantial contribution to the potential for the transmission of infectious diseases, quantified as R ₀, within the farm population. A small percentage of farms (<20%) contribute the bulk of the transmission potential (>80%) and these farms could be efficiently targeted by interventions aimed at reducing spread of diseases via animal movement.     

On the Exact Measure of Disease Spread in Stochastic Epidemic Models

The basic reproduction number, R ₀, is probably the most important quantity in epidemiology. It is used to measure the transmission potential during the initial phase of an epidemic. In this paper, we are specifically concerned with the quantification of the spread of a disease modeled by a Markov chain. Due to the occurrence of repeated contacts taking place between a typical infective individual and other individuals already infected before, R ₀ overestimates the average number of secondary infections. We present two alternative measures, namely, the exact reproduction number, R _e0, and the population transmission number, R _p , that overcome this difficulty and provide valuable insight. The applicability of R _e0 and R _p to control of disease spread is also examined.     

Early real-time estimation of the basic reproduction number of emerging or reemerging infectious diseases in a community with heterogeneous contact pattern: Using data from Hong Kong 2009 H1N1 Pandemic Influenza as an illustrative example

Emerging and re-emerging infections such as SARS (2003) and pandemic H1N1 (2009) have caused concern for public health researchers and policy makers due to the increased burden of these diseases on health care systems. This concern has prompted the use of mathematical models to evaluate strategies to control disease spread, making these models invaluable tools to identify optimal intervention strategies. A particularly important quantity in infectious disease epidemiology is the basic reproduction number, R_0. Estimation of this quantity is crucial for effective control responses in the early phase of an epidemic. In our previous study, an approach for estimating the basic reproduction number in real time was developed. This approach uses case notification data and the structure of potential transmission contacts to accurately estimate R₀ from the limited amount of information available at the early stage of an outbreak. Based on this approach, we extend the existing methodology; the most recent method features intra- and inter-age groups contact heterogeneity. Given the number of newly reported cases at the early stage of the outbreak, with parsimony assumptions on removal distribution and infectivity profile of the diseases, experiments to estimate real time R₀ under different levels of intra- and inter-group contact heterogeneity using two age groups are presented. We show that the new method converges more quickly to the actual value of R₀ than the previous one, in particular when there is high-level intra-group and inter-group contact heterogeneity. With the age specific contact patterns, number of newly reported cases, removal distribution, and information about the natural history of the 2009 pandemic influenza in Hong Kong, we also use the extended model to estimate R₀ and age-specific R₀.     

Social contact patterns in Vietnam and implications for the control of infectious diseases

Background

The spread of infectious diseases from person to person is determined by the frequency and nature of contacts between infected and susceptible members of the population. Although there is a long history of using mathematical models to understand these transmission dynamics, there are still remarkably little empirical data on contact behaviors with which to parameterize these models. Even starker is the almost complete absence of data from developing countries. We sought to address this knowledge gap by conducting a household based social contact diary in rural Vietnam.

Methods and Findings

A diary based survey of social contact patterns was conducted in a household-structured community cohort in North Vietnam in 2007. We used generalized estimating equations to model the number of contacts while taking into account the household sampling design, and used weighting to balance the household size and age distribution towards the Vietnamese population. We recorded 6675 contacts from 865 participants in 264 different households and found that mixing patterns were assortative by age but were more homogenous than observed in a recent European study. We also observed that physical contacts were more concentrated in the home setting in Vietnam than in Europe but the overall level of physical contact was lower. A model of individual versus household vaccination strategies revealed no difference between strategies in the impact on R ₀.

Conclusions and Significance

This work is the first to estimate contact patterns relevant to the spread of infections transmitted from person to person by non-sexual routes in a developing country setting. The results show interesting similarities and differences from European data and demonstrate the importance of context specific data.     

Estimated epidemiologic parameters and morbidity associated with pandemic H1N1 influenza

Background

In the face of an influenza pandemic, accurate estimates of epidemiologic parameters are required to help guide decision-making. We sought to estimate epidemiologic parameters for pandemic H1N1 influenza using data from initial reports of laboratory-confirmed cases.

Methods

We obtained data on laboratory-confirmed cases of pandemic H1N1 influenza reported in the province of Ontario, Canada, with dates of symptom onset between Apr. 13 and June 20, 2009. Incubation periods and duration of symptoms were estimated and fit to parametric distributions. We used competing-risk models to estimate risk of hospital admission and case-fatality rates. We used a Markov Chain Monte Carlo model to simulate disease transmission.

Results

The median incubation period was 4 days and the duration of symptoms was 7 days. Recovery was faster among patients less than 18 years old than among older patients (hazard ratio 1.23, 95% confidence interval 1.06–1.44). The risk of hospital admission was 4.5% (95% CI 3.8%–5.2%) and the case-fatality rate was 0.3% (95% CI 0.1%–0.5%). The risk of hospital admission was highest among patients less than 1 year old and those 65 years or older. Adults more than 50 years old comprised 7% of cases but accounted for 7 of 10 initial deaths (odds ratio 28.6, 95% confidence interval 7.3–111.2). From the simulation models, we estimated the following values (and 95% credible intervals): a mean basic reproductive number (R₀, the number of new cases created by a single primary case in a susceptible population) of 1.31 (1.25–1.38), a mean latent period of 2.62 (2.28–3.12) days and a mean duration of infectiousness of 3.38 (2.06–4.69) days. From these values we estimated a serial interval (the average time from onset of infectiousness in a case to the onset of infectiousness in a person infected by that case) of 4–5 days.

Interpretation

The low estimates for R₀ indicate that effective mitigation strategies may reduce the final epidemic impact of pandemic H1N1 influenza.The emergence and global spread of pandemic H1N1 influenza led the World Health Organization to declare a pandemic on June 11, 2009. As the pandemic spreads, countries will need to make decisions about strategies to mitigate and control disease in the face of uncertainty.For novel infectious diseases, accurate estimates of epidemiologic parameters can help guide decision-making. A key parameter for any new disease is the basic reproductive number (R₀), defined as the average number of new cases created by a single primary case in a susceptible population. R₀ affects the growth rate of an epidemic and the final number of infected people. It also informs the optimal choice of control strategies. Other key parameters that affect use of resources, disease burden and societal costs during a pandemic are duration of illness, rate of hospital admission and case-fatality rate. Early in an epidemic, the case-fatality rate may be underestimated because of the temporal lag between onset of infection and death; the delay between initial identification of a new case and death may lead to an apparent increase in deaths several weeks into an epidemic that is an artifact of the natural history of the disease.We used data from initial reports of laboratory-confirmed pandemic H1N1 influenza to estimate epidemiologic parameters for pandemic H1N1 influenza. The parameters included R₀, incubation period and duration of illness. We also estimated risk of hospital admission and case-fatality rates, which can be used to estimate the burden of illness likely to be associated with this disease.     

Impact of group mixing on disease dynamics

A general mathematical model is proposed to study the impact of group mixing in a heterogeneous host population on the spread of a disease that confers temporary immunity upon recovery. The model contains general distribution functions that account for the probabilities that individuals remain in the recovered class after recovery. For this model, the basic reproduction number R₀ is identified. It is shown that if R₀<1, then the disease dies out in the sense that the disease free equilibrium is globally asymptotically stable; whereas if R₀>1, this equilibrium becomes unstable. In this latter case, depending on the distribution functions and the group mixing strengths, the disease either persists at a constant endemic level or exhibits sustained oscillatory behavior.     

Estimation of the Basic Reproductive Number and Mean Serial Interval of a Novel Pathogen in a Small,Well-Observed Discrete Population

Background

Accurately assessing the transmissibility and serial interval of a novel human pathogen is public health priority so that the timing and required strength of interventions may be determined. Recent theoretical work has focused on making best use of data from the initial exponential phase of growth of incidence in large populations.

Methods

We measured generational transmissibility by the basic reproductive number R₀ and the serial interval by its mean T_g. First, we constructed a simulation algorithm for case data arising from a small population of known size with R₀ and T_g also known. We then developed an inferential model for the likelihood of these case data as a function of R₀ and T_g. The model was designed to capture a) any signal of the serial interval distribution in the initial stochastic phase b) the growth rate of the exponential phase and c) the unique combination of R₀ and T_g that generates a specific shape of peak incidence when the susceptible portion of a small population is depleted.

Findings

Extensive repeat simulation and parameter estimation revealed no bias in univariate estimates of either R₀ and T_g. We were also able to simultaneously estimate both R₀ and T_g. However, accurate final estimates could be obtained only much later in the outbreak. In particular, estimates of T_g were considerably less accurate in the bivariate case until the peak of incidence had passed.

Conclusions

The basic reproductive number and mean serial interval can be estimated simultaneously in real time during an outbreak of an emerging pathogen. Repeated application of these methods to small scale outbreaks at the start of an epidemic would permit accurate estimates of key parameters.     

Understanding Spatio-Temporal Variability in the Reproduction Ratio of the Bluetongue (BTV-1) Epidemic in Southern Spain (Andalusia) in 2007 Using Epidemic Trees

Andalusia (Southern Spain) is considered one of the main routes of introduction of bluetongue virus (BTV) into Europe, evidenced by a devastating epidemic caused by BTV-1 in 2007. Understanding the pattern and the drivers of BTV-1 spread in Andalusia is critical for effective detection and control of future epidemics. A long-standing metric for quantifying the behaviour of infectious diseases is the case-reproduction ratio (R_t), defined as the average number of secondary cases arising from a single infected case at time t (for t>0). Here we apply a method using epidemic trees to estimate the between-herd case reproduction ratio directly from epidemic data allowing the spatial and temporal variability in transmission to be described. We then relate this variability to predictors describing the hosts, vectors and the environment to better understand why the epidemic spread more quickly in some regions or periods. The R_t value for the BTV-1 epidemic in Andalusia peaked in July at 4.6, at the start of the epidemic, then decreased to 2.2 by August, dropped below 1 by September (0.8), and by October it had decreased to 0.02. BTV spread was the consequence of both local transmission within established disease foci and BTV expansion to distant new areas (i.e. new foci), which resulted in a high variability in BTV transmission, not only among different areas, but particularly through time, which suggests that general control measures applied at broad spatial scales are unlikely to be effective. This high variability through time was probably due to the impact of temperature on BTV transmission, as evidenced by a reduction in the value of R_t by 0.0041 for every unit increase (day) in the extrinsic incubation period (EIP), which is itself directly dependent on temperature. Moreover, within the range of values at which BTV-1 transmission occurred in Andalusia (20.6°C to 29.5°C) there was a positive correlation between temperature and R_t values, although the relationship was not linear, probably as a result of the complex relationship between temperature and the different parameters affecting BTV transmission. R_t values for BTV-1 in Andalusia fell below the threshold of 1 when temperatures dropped below 21°C, a much higher threshold than that reported in other BTV outbreaks, such as the BTV-8 epidemic in Northern Europe. This divergence may be explained by differences in the adaptation to temperature of the main vectors of the BTV-1 epidemic in Andalusia (Culicoides imicola) compared those of the BTV-8 epidemic in Northern Europe (Culicoides obsoletus). Importantly, we found that BTV transmission (R_t value) increased significantly in areas with higher densities of sheep. Our analysis also established that control of BTV-1 in Andalusia was complicated by the simultaneous establishment of several distant foci at the start of the epidemic, which may have been caused by several independent introductions of infected vectors from the North of Africa. We discuss the implications of these findings for BTV surveillance and control in this region of Europe.     

An R1ρ expression for a spin in chemical exchange between two sites with unequal transverse relaxation rates

An analytical expression is derived for the rotating frame relaxation rate, R _1ρ , of a spin exchanging between two sites with different transverse relaxation times. A number of limiting cases are examined, with the equation reducing to formulae derived previously under the assumption of equivalent relaxation rates at each site. The measurement of a pair off-resonance R _1ρ values, with the carrier displaced equally on either side of the observed correlation, forms the basis of one of the approaches for obtaining signs of chemical shift differences, Δω, of exchanging nuclei. The results presented here establish that this method is relatively insensitive to differential transverse relaxation rates between the exchaning states, greatly simplifying the calculation of optimal parameters in R _1ρ based experiments that are used for measurement of signs of Δω.     

Practical considerations for measuring the effective reproductive number,Rt

Estimation of the effective reproductive number R_t is important for detecting changes in disease transmission over time. During the Coronavirus Disease 2019 (COVID-19) pandemic, policy makers and public health officials are using R_t to assess the effectiveness of interventions and to inform policy. However, estimation of R_t from available data presents several challenges, with critical implications for the interpretation of the course of the pandemic. The purpose of this document is to summarize these challenges, illustrate them with examples from synthetic data, and, where possible, make recommendations. For near real-time estimation of R_t, we recommend the approach of Cori and colleagues, which uses data from before time t and empirical estimates of the distribution of time between infections. Methods that require data from after time t, such as Wallinga and Teunis, are conceptually and methodologically less suited for near real-time estimation, but may be appropriate for retrospective analyses of how individuals infected at different time points contributed to the spread. We advise caution when using methods derived from the approach of Bettencourt and Ribeiro, as the resulting R_t estimates may be biased if the underlying structural assumptions are not met. Two key challenges common to all approaches are accurate specification of the generation interval and reconstruction of the time series of new infections from observations occurring long after the moment of transmission. Naive approaches for dealing with observation delays, such as subtracting delays sampled from a distribution, can introduce bias. We provide suggestions for how to mitigate this and other technical challenges and highlight open problems in R_t estimation.     

Variability order of the latent and the infectious periods in a deterministic SEIR epidemic model and evaluation of control effectiveness

We use distribution theory and ordering of non-negative random variables to study the Susceptible-Exposed-Infectious-Removed (SEIR) model with two control measures, quarantine and isolation, to reduce the spread of an infectious disease. We identify that the probability distributions of the latent period and the infectious period are primary features of the SEIR model to formulate the epidemic threshold and to evaluate the effectiveness of the intervention measures. If the primary features are changed, the conclusions will be altered in an importantly different way. For the latent and infectious periods with known mean values, it is the dilation, a generalization of variance, of their distributions that ranks the effectiveness of these control measures. We further propose ways to set quarantine and isolation targets to reduce the controlled reproduction number below the threshold using observed initial growth rate from outbreak data. If both quarantine and isolation are 100% effective, one can directly use the observed growth rate for setting control targets. If they are not 100% effective, some further knowledge of the distributions is required.     

Semi-stochastic models for Salmonella infection within finishing pig units in the UK

A multi-group semi-stochastic model is formulated to describe Salmonella   dynamics on a pig herd within the UK and assess whether farm structure has any effect on the dynamics. The models include both direct transmission and indirect (via free-living infectious units in the environment and airborne infection). The basic reproduction number _R0$R_0$ is also investigated. The models estimate approximately 24.6% and 25.4% of pigs at slaughter weight will be infected with Salmonella within a slatted-floored and solid-floored unit respectively, which corresponds to values found in previous abattoir and farm studies, suggesting that the model has reasonable validity. Analysis of the models identified the shedding rate to be of particular importance in the control of Salmonella   spread, a finding also evident in an increase in the _R0$R_0$ value.     

Empirical Estimation of R 0 for Unknown Transmission Functions: The Case of Chronic Wasting Disease in Alberta

We consider the problem of estimating the basic reproduction number R ₀ from data on prevalence dynamics at the beginning of a disease outbreak. We derive discrete and continuous time models, some coefficients of which are to be fitted from data. We show that prevalence of the disease is sufficient to determine R ₀. We apply this method to chronic wasting disease spread in Alberta determining a range of possible R ₀ and their sensitivity to the probability of deer annual survival.