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.     

Genealogy with seasonality,the basic reproduction number,and the influenza pandemic

The basic reproduction number R ₀ has been used in population biology, especially in epidemiology, for several decades. But a suitable definition in the case of models with periodic coefficients was given only in recent years. The definition involves the spectral radius of an integral operator. As in the study of structured epidemic models in a constant environment, there is a need to emphasize the biological meaning of this spectral radius. In this paper we show that R ₀ for periodic models is still an asymptotic per generation growth rate. We also emphasize the difference between this theoretical R ₀ for periodic models and the “reproduction number” obtained by fitting an exponential to the beginning of an epidemic curve. This difference has been overlooked in recent studies of the H1N1 influenza pandemic.     

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.     

The epidemic threshold of vector-borne diseases with seasonality

Cutaneous leishmaniasis is a vector-borne disease transmitted to humans by sandflies. In this paper, we develop a mathematical model which takes into account the seasonality of the vector population and the distribution of the latent period from infection to symptoms in humans. Parameters are fitted to real data from the province of Chichaoua, Morocco. We also introduce a generalization of the definition of the basic reproduction number R ₀ which is adapted to periodic environments. This R ₀ is estimated numerically for the epidemic in Chichaoua; 1.94. The model suggests that the epidemic could be stopped if the vector population were reduced by a factor 3.76.     

Estimation of the reproduction number of dengue fever from spatial epidemic data

Dengue, a vector-borne disease, thrives in tropical and subtropical regions worldwide. A retrospective analysis of the 2002 dengue epidemic in Colima located on the Mexican central Pacific coast is carried out. We estimate the reproduction number from spatial epidemic data at the level of municipalities using two different methods: (1) Using a standard dengue epidemic model and assuming pure exponential initial epidemic growth and (2) Fitting a more realistic epidemic model to the initial phase of the dengue epidemic curve. Using Method I, we estimate an overall mean reproduction number of 3.09 (95% CI: 2.34,3.84) as well as local reproduction numbers whose values range from 1.24 (1.15,1.33) to 4.22 (2.90,5.54). Using Method II, the overall mean reproduction number is estimated to be 2.0 (1.75,2.23) and local reproduction numbers ranging from 0.49 (0.0,1.0) to 3.30 (1.63,4.97). Method I systematically overestimates the reproduction number relative to the refined Method II, and hence it would overestimate the intensity of interventions required for containment. Moreover, optimal intervention with defined resources demands different levels of locally tailored mitigation. Local epidemic peaks occur between the 24th and 35th week of the year, and correlate positively with the final local epidemic sizes (rho=0.92, P-value<0.001). Moreover, final local epidemic sizes are found to be linearly related to the local population size (P-value<0.001). This observation supports a roughly constant number of female mosquitoes per person across urban and rural regions.     

The effect of nutrition on the seasonality of reproduction in ewes

The beneficial effects of nutrition on reproduction in sheep have been described, particularly on ovulation rate. However, the relationships between nutrition and reproductive seasonality are not well known. This review will deal with the effects of body fat or food intake on sexual and hypothalamic/pituitary activity in sheep, mainly focused on Mediterranean genotypes. Although only severe malnutrition can significantly extend the length of the seasonal anestrous period, the level of fat reserves can play a significant role on reproductive seasonality delaying the onset of seasonal anoestrus, particularly on the Mediterranean environment. The effect of overfeeding on LH secretion has also been reported, specially at short term. Several experimental approaches have elucidated that both high body fat and food intake are able to modify the sensitivity of the hypothalamus to oestradiol negative feedback during seasonal anoestrus, with those effects being associated to a reduced amount of NPY mRNA and to an increase of plasma insulin, glucose and leptin concentrations, particularly in the late scenario. However, the highest receptivity to nutritional stimulation in terms of increasing LH occurs when ewes are subjected to a photoperiodic state of early anoestrus or late breeding season rather than under a photoperiod characteristic of the end of anoestrus or the beginning of the breeding season.     

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.     

Observations on commensal rats and their status to plague in Bombay

The distribution pattern of various types of commensal rodents in Bombay city reveals that Bandicota bengalensis constitutes the predominant commensal rodent species followed by R. rattus and Rattus norvegicus. Apart from these three types, Bandicota indica, M. musculus and an insectivore (Suncus murinus) are the three species of commensal small mammals that are frequently encountered in or near human habitations. These small mammals are prevalent throughout the year and their percentage distribution varies very little during different months of the year. None of the rodent species examined during the years 1976-85 revealed presence of Y. pestis infection by bacteriological or serological methods. From these findings, it could be concluded that in the city of Bombay a focus of zoonotic plague infection does not exist.     

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.     

Influence of exogenous melatonin on seasonality of reproduction in sheep

Two experiments were performed to examine the effects of exogenous melatonin on seasonality of reproduction in ewes. To simulate the secretory pattern observed on the shortest day of the year, 2.5 mg melatonin were administered i.m. at 1600 hr each day. This dosage has been shown to keep serum concentrations of melatonin elevated until the onset of darkness on the longest day of the year (i.e. ～ 6 hr). In experiment 1, ewes were administered melatonin (n=10) or vehicle (n=10) from June 20 until the onset of the breeding season. Jugular blood samples were collected three times weekly and the serum was stored for analysis of progesterone by radioimmunoassay. The interval from the first day of treatment until serum concentrations of progesterone remained above 1 ng/ml in two consecutive samples (indicative of the formation of a corpus luteum) was 39.3 ± 4.1 days in the melatonin-treated ewes and 61.4 ± 4.6 days in the ewes receiving vehicle (P < 0.05). In the second experiment, ewes were given injections of melatonin (n=6) or vehicle (n=6) from December 21 until September 1. Again, jugular blood samples were collected three times weekly for analysis of progesterone. The interval from initiation of treatment until serum concentrations of progesterone remained below 1 ng/ml for at least 5 consecutive samples was 80.2 ± 3.3 days in the melatonin-treated ewes and 37.8 ± 6.5 days in the control ewes (P < 0.05). However, when treatment was begun on January 1 the interval from the initiation of treatment until serum concentrations of progesterone were greater than 1 ng/ml in two consecutive samples after June 1 was not different between melatonin-treated and control ewes. These data indicate that adminis-tration of melatonin to ewes can hasten the onset of the breeding season in the fall or extend the length of the breeding season in the spring, but after prolonged administration the ewes may become refractory to the treatment.     

The state-reproduction number for a multistate class age structured epidemic system and its application to the asymptomatic transmission model

In this paper, we develop the theory of a state-reproduction number for a multistate class age structured epidemic system and apply it to examine the asymptomatic transmission model. We formulate a renewal integral equation system to describe the invasion of infectious diseases into a multistate class age structured host population. We define the state-reproduction number for a class age structured system, which is the net reproduction number of a specific host type and which plays an analogous role to the type-reproduction number [M.G. Roberts, J.A.P. Heesterbeek, A new method for estimating the effort required to control an infectious disease, Proc. R. Soc. Lond. B 270 (2003) 1359; J.A.P. Heesterbeek, M.G. Roberts, The type-reproduction number T in models for infectious disease control, Math. Biosci. 206 (2007) 3] in discussing the critical level of public health intervention. The renewal equation formulation permits computations not only of the state-reproduction number, but also of the generation time and the intrinsic growth rate of infectious diseases.Subsequently, the basic theory is applied to capture the dynamics of a directly transmitted disease within two types of infected populations, i.e., asymptomatic and symptomatic individuals, in which the symptomatic class is observable and hence a target host of the majority of interventions. The state-reproduction number of the symptomatic host is derived and expressed as a measurable quantity, leading to discussion on the critical level of case isolation. The serial interval and other epidemiologic indices are computed, clarifying the parameters on which these indices depend. As a practical example, we illustrate the eradication threshold for case isolation of smallpox. The generation time and serial interval are comparatively examined for pandemic influenza.     

A test of two hypotheses explaining the seasonality of reproduction in temperate mammals

1. Two proposed hypotheses about energy allocation were tested to explain the patterns of seasonal reproduction found in temperate mammals. The two hypotheses predict either that total demand for energy is greater during reproduction than during winter (when thermoregulatory costs are high) (Increased Demand Hypothesis) or that total costs during winter are greater than or equal to total costs during reproduction (Reallocation Hypothesis).
2. Data were compiled from the literature on summer (non-reproducing) and winter metabolic rates of temperate mammals, and were used on litter sizes and a published equation to predict metabolic rates during lactation.
3. All three measures of metabolic rate scaled to body mass with slopes significantly less than one. Metabolic rates during winter averaged ≈ 2 times greater than those of non-reproducing mammals during summer. On average, predicted metabolic rates during lactation were not significantly greater than during winter, but for some individual species they clearly were.
4. It is suggested that neither the Reallocation nor the Increased Demand Hypothesis can fully explain seasonal reproductive patterns in temperate mammals.     

Extending the type reproduction number to infectious disease control targeting contacts between types

A new quantity called the target reproduction number is defined to measure control strategies for infectious diseases with multiple host types such as waterborne, vector-borne and zoonotic diseases. The target reproduction number includes as a special case and extends the type reproduction number to allow disease control targeting contacts between types. Relationships among the basic, type and target reproduction numbers are established. Examples of infectious disease models from the literature are given to illustrate the use of the target reproduction number.     

Age-structured homogeneous epidemic systems with application to the MSEIR epidemic model

In this paper, we develop a new approach to deal with asymptotic behavior of the age-structured homogeneous epidemic systems and discuss its application to the MSEIR epidemic model. For the homogeneous system, there is no attracting nontrivial equilibrium, instead we have to examine existence and stability of persistent solutions. Assuming that the host population dynamics can be described by the stable population model, we rewrite the basic system into the system of ratio age distribution, which is the age profile divided by the stable age profile. If the host population has the stable age profile, the ratio age distribution system is reduced to the normalized system. Then we prove the stability principle that the local stability or instability of steady states of the normalized system implies that of the corresponding persistent solutions of the original homogeneous system. In the latter half of this paper, we prove the threshold and stability results for the normalized system of the age-structured MSEIR epidemic model.      

Effect of plane of nutrition on seasonality of reproduction in Spanish Payoya goats

The aim of this study was to determine if there is a seasonal pattern of sexual activity in female Payoya goats and if this seasonality could be modulated by nutrition. During the experimental period of 20 months, 43 non-pregnant adults goats were penned under natural photoperiod at latitude 37 degrees 15'N. At the onset of the experiment, the animals were allocated to three experimental groups differing in the level of nutrition and whether the animals were entire or ovariectomized does. The high nutrition group (H, n = 16 entire does) receiving 1.5 times maintenance requirements. The low nutrition group (L, n = 16 entire does) and an ovariectomized and oestradiol treated group (OVX, n = 11 ovariectomized does) received a diet supporting their maintenance requirements. The groups were balanced for live weight (LW) and body condition score (BCS) at the beginning of the study. In entire goats, oestrus was tested daily using aproned males, ovulation rate was assessed by laparoscopy 7 days after identification of oestrus and plasma samples were obtained twice per week for progesterone assay. OVX goats were isolated from the other groups and bucks, plasma samples were assayed twice per week for LH and there were four intensive sampling periods during the year to determine LH pulsatility. LW and BCS were recorded for all animals once a week. A clear circannual cycle in live weight change was observed in all experimental groups, being relatively stable or slightly decreasing in summer and autumn and increasing during winter and spring. The effect of exposure to high (H) rather than low (L) nutrition was to cause earlier onset of ovarian activity (5 versus 17 August; P < 0.05), and expression of oestrous (16 August versus 2 September; P < 0.01) and later cessation of reproductive activity (ovulation 11 February versus 17 January; P < 0.01). Consequently, seasonal anoestrus was 32 days shorter in does on the higher plane of nutrition (P < 0.01). The seasonality of reproductive activity was confirmed in the OVX does, with reduced LH concentrations during spring and summer, and increased LH concentrations in autumn and winter. There was no effect of nutrition on ovulation rate. These results demonstrate that the female Payoya goat exhibits marked reproductive seasonality which is modulated by nutrition but possibly not ovulation rate.