The basic reproduction number for scrapie.

The basic reproduction number R0 provides a quantitative assessment of the ability of an infectious agent to invade a susceptible host population. A mathematical expression for R0 is derived based on a recently developed model for the spread of scrapie through a flock of sheep. The model incorporates sheep demography, a long and variable incubation period, genetic variation in susceptibility to scrapie, and horizontal and vertical routes of transmission. The sensitivity of R0 to a range of epidemiologically important parameters is assessed and the effects of genetic variation in susceptibility are examined. A reduction in the frequency of the susceptibility allele reduces R0 most effectively when the allele is recessive, whereas inbreeding may increase R0 when the allele is recessive, increasing the chance of an outbreak. Using this formulation, R0 is calculated for an outbreak of scrapie in a flock of Cheviot sheep.     

On the basic reproduction number in a random environment

The concept of basic reproduction number $R_0$ in population dynamics is studied in the case of random environments. For simplicity the dependence between successive environments is supposed to follow a Markov chain. $R_0$ is the spectral radius of a next-generation operator. Its position with respect to 1 always determines population growth or decay in simulations, unlike another parameter suggested in a recent article (Hernandez-Suarez et al., Theor Popul Biol, doi:10.1016/j.tpb.2012.05.004, 2012). The position of the latter with respect to 1 determines growth or decay of the population’s expectation. $R_0$ is easily computed in the case of scalar population models without any structure. The main emphasis is on discrete-time models but continuous-time models are also considered.     

The model of Kermack and McKendrick for the plague epidemic in Bombay and the type reproduction number with seasonality

The figure showing how the model of Kermack and McKendrick fits the data from the 1906 plague epidemic in Bombay is the most reproduced figure in books discussing mathematical epidemiology. In this paper we show that the assumption of constant parameters in the model leads to quite unrealistic numerical values for these parameters. Moreover the reports published at the time show that plague epidemics in Bombay occurred in fact with a remarkable seasonal pattern every year since 1897 and at least until 1911. So the 1906 epidemic is clearly not a good example of epidemic stopping because the number of susceptible humans has decreased under a threshold, as suggested by Kermack and McKendrick, but an example of epidemic driven by seasonality. We present a seasonal model for the plague in Bombay and compute the type reproduction numbers associated with rats and fleas, thereby extending to periodic models the notion introduced by Roberts and Heesterbeek.     

Edge removal in random contact networks and the basic reproduction number

Understanding the effect of edge removal on the basic reproduction number ${\mathcal{R}_0}$ for disease spread on contact networks is important for disease management. The formula for the basic reproduction number ${\mathcal{R}_0}$ in random network SIR models of configuration type suggests that for degree distributions with large variance, a reduction of the average degree may actually increase ${\mathcal{R}_0}$ . To understand this phenomenon, we develop a dynamical model for the evolution of the degree distribution under random edge removal, and show that truly random removal always reduces ${\mathcal{R}_0}$ . The discrepancy implies that any increase in ${\mathcal{R}_0}$ must result from edge removal changing the network type, invalidating the use of the basic reproduction number formula for a random contact network. We further develop an epidemic model incorporating a contact network consisting of two groups of nodes with random intra- and inter-group connections, and derive its basic reproduction number. We then prove that random edge removal within either group, and between groups, always decreases the appropriately defined ${\mathcal{R}_0}$ . Our models also allow an estimation of the number of edges that need to be removed in order to curtail an epidemic.     

Growth rate and basic reproduction number for population models with a simple periodic factor

For continuous-time population models with a periodic factor which is sinusoidal, both the growth rate and the basic reproduction number are shown to be the largest roots of simple equations involving continued fractions. As an example, we reconsider an SEIS model with a fixed latent period, an exponentially distributed infectious period and a sinusoidal contact rate studied in Williams and Dye [B.G. Williams, C. Dye, Infectious disease persistence when transmission varies seasonally, Math. Biosci. 145 (1997) 77]. We show that apart from a few exceptional parameter values, the epidemic threshold depends not only on the mean contact rate, but also on the amplitude of fluctuations.     

Early estimation of the reproduction number in the presence of imported cases: pandemic influenza H1N1-2009 in New Zealand

We analyse data from the early epidemic of H1N1-2009 in New Zealand, and estimate the reproduction number R. We employ a renewal process which accounts for imported cases, illustrate some technical pitfalls, and propose a novel estimation method to address these pitfalls. Explicitly accounting for the infection-age distribution of imported cases and for the delay in transmission dynamics due to international travel, R was estimated to be (95% confidence interval: 107,1.47). Hence we show that a previous study, which did not account for these factors, overestimated R. Our approach also permitted us to examine the infection-age at which secondary transmission occurs as a function of calendar time, demonstrating the downward bias during the beginning of the epidemic. These technical issues may compromise the usefulness of a well-known estimator of R--the inverse of the moment-generating function of the generation time given the intrinsic growth rate. Explicit modelling of the infection-age distribution among imported cases and the examination of the time dependency of the generation time play key roles in avoiding a biased estimate of R, especially when one only has data covering a short time interval during the early growth phase of the epidemic.     

On a new perspective of the basic reproduction number in heterogeneous environments

Although its usefulness and possibility of the well-known definition of the basic reproduction number R0 for structured populations by Diekmann, Heesterbeek and Metz (J Math Biol 28:365-382, 1990) (the DHM definition) have been widely recognized mainly in the context of epidemic models, originally it deals with population dynamics in a constant environment, so it cannot be applied to formulate the threshold principle for population growth in time-heterogeneous environments. Since the mid-1990s, several authors proposed some ideas to extend the definition of R0 to the case of a periodic environment. In particular, the definition of R0 in a periodic environment by Baca?r and Guernaoui (J Math Biol 53:421-436, 2006) (the BG definition) is most important, because their definition of periodic R0 can be interpreted as the asymptotic per generation growth rate, which is an essential feature of the DHM definition. In this paper, we introduce a new definition of R0 based on the generation evolution operator (GEO), which has intuitively clear biological meaning and can be applied to structured populations in any heterogeneous environment. Using the generation evolution operator, we show that the DHM definition and the BG definition completely allow the generational interpretation and, in those two cases, the spectral radius of GEO equals the spectral radius of the next generation operator, so it gives the basic reproduction number. Hence the new definition is an extension of the DHM definition and the BG definition. Finally we prove a weak sign relation that if the average Malthusian parameter exists, it is nonnegative when R0>1 and it is nonpositive when R0<1.     

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.     

A graph-theoretic method for the basic reproduction number in continuous time epidemiological models

In epidemiological models of infectious diseases the basic reproduction number is used as a threshold parameter to determine the threshold between disease extinction and outbreak. A graph-theoretic form of Gaussian elimination using digraph reduction is derived and an algorithm given for calculating the basic reproduction number in continuous time epidemiological models. Examples illustrate how this method can be applied to compartmental models of infectious diseases modelled by a system of ordinary differential equations. We also show with these examples how lower bounds for can be obtained from the digraphs in the reduction process.     

Model-consistent estimation of the basic reproduction number from the incidence of an emerging infection

We investigate the merit of deriving an estimate of the basic reproduction number early in an outbreak of an (emerging) infection from estimates of the incidence and generation interval only. We compare such estimates of with estimates incorporating additional model assumptions, and determine the circumstances under which the different estimates are consistent. We show that one has to be careful when using observed exponential growth rates to derive an estimate of , and we quantify the discrepancies that arise.      

Vector-borne diseases and the basic reproduction number: a case study of African horse sickness

Abstract. The basic reproduction number, R ₀, can be used to determine factors important in the ability of a disease to invade or persist. We show how this number can be derived or estimated for vector-borne diseases with different complicating factors. African horse sickness is a viral disease transmitted mainly by the midge Culicoides imicola. We use this as an example of such a vector-transmitted disease where latent periods, seasonality in vector populations, and multiple host types may be important. The effect of vector population dynamics which are dependent on either host or vector density are also addressed. If density-dependent constraints on vector population density are less severe, R_o is more sensitive to vector mortality and the virus development rate. Host-dependent vector dynamics change the relationship between R₀ and host population size. Seasonality can either increase or decrease the estimate of R₀ , depending on the lag between the peak of the midge population and the infective host population. The relative abundance of two host types is a factor in the ability of a disease to invade, but the strength of this factor depends on the differences between the hosts in recovery from infection, mortality and transmission. Removal of a reservoir host may increase R_Q.     

Epidemic models with uncertainty in the reproduction number

One of the first quantities to be estimated at the start of an epidemic is the basic reproduction number, ${\mathcal{R}_0}$ . The progress of an epidemic is sensitive to the value of ${\mathcal{R}_0}$ , hence we need methods for exploring the consequences of uncertainty in the estimate. We begin with an analysis of the SIR model, with ${\mathcal{R}_0}$ specified by a probability distribution instead of a single value. We derive probability distributions for the prevalence and incidence of infection during the initial exponential phase, the peaks in prevalence and incidence and their timing, and the final size of the epidemic. Then, by expanding the state variables in orthogonal polynomials in uncertainty space, we construct a set of deterministic equations for the distribution of the solution throughout the time-course of the epidemic. The resulting dynamical system need only be solved once to produce a deterministic stochastic solution. The method is illustrated with ${\mathcal{R}_0}$ specified by uniform, beta and normal distributions. We then apply the method to data from the New Zealand epidemic of H1N1 influenza in 2009. We apply the polynomial expansion method to a Kermack–McKendrick model, to simulate a forecasting system that could be used in real time. The results demonstrate the level of uncertainty when making parameter estimates and projections based on a limited amount of data, as would be the case during the initial stages of an epidemic. In solving both problems we demonstrate how the dynamical system is derived automatically via recurrence relationships, then solved numerically.     

Estimation of the basic reproduction number of BSE: the intensity of transmission in British cattle

The basic reproduction number, R0, of an infectious agent is a key factor determining the rate of spread and the proportion of the host population affected. We formulate a general mathematical framework to describe the transmission dynamics of long incubation period diseases with complex pathogenesis. This is used to derive expressions for R0 of bovine spongiform encephalopathy (BSE) in British cattle, and back-calculation methods are used to estimate R0 throughout the time-course of the BSE epidemic. We show that the 1988 meat and bonemeal ban was effective in rapidly reducing R0 below 1, and demonstrate that this indicates that BSE will be unable to become endemic in the UK cattle population even when case clustering is taken into account. The analysis provides some insight into absolute infectiousness for bovine-to-bovine transmission, indicating maximally infectious animals may have infected up to 400 animals each. The relationship between R0 and the early stages of the BSE epidemic and the requirements for additional research are also discussed.     

Some properties of an estimator for the basic reproduction number of the general epidemic model.

In this paper some properties of a convenient estimator, derived from a martingale estimating function, for the basic reproduction number of the general epidemic model are given for both finite and large samples. These properties give some guidelines for using this convenient estimator. It is shown that it underestimates the parameter and that the bias tends to zero when the population size and the initial number of infectives are increased simultaneously. The bias cannot be removed for a fixed number of introductory infectives. However, the estimator is asymptotically unbiased, conditional on a major outbreak. A simulation study shows that the central limit theorem applies for moderate population sizes.