Predicting the course of meningococcal disease outbreaks in closed subpopulations

Some properties of a simple stochastic epidemic model of SIR type

We investigate the properties of a simple discrete time stochastic epidemic model. The model is Markovian of the SIR type in which the total population is constant and individuals meet a random number of other individuals at each time step. Individuals remain infectious for R time units, after which they become removed or immune. Individual transition probabilities from susceptible to diseased states are given in terms of the binomial distribution. An expression is given for the probability that any individuals beyond those initially infected become diseased. In the model with a finite recovery time R, simulations reveal large variability in both the total number of infected individuals and in the total duration of the epidemic, even when the variability in number of contacts per day is small. In the case of no recovery, R = ∞, a formal diffusion approximation is obtained for the number infected. The mean for the diffusion process can be approximated by a logistic which is more accurate for larger contact rates or faster developing epidemics. For finite R we then proceed mainly by simulation and investigate in the mean the effects of varying the parameters p (the probability of transmission), R, and the number of contacts per day per individual. A scale invariant property is noted for the size of an outbreak in relation to the total population size. Most notable are the existence of maxima in the duration of an epidemic as a function of R and the extremely large differences in the sizes of outbreaks which can occur for small changes in R. These findings have practical applications in controlling the size and duration of epidemics and hence reducing their human and economic costs.

Diversity in pathogenicity can cause outbreaks of meningococcal disease

Neisseria meningitidis, the meningococcus, is a major cause of bacterial meningitis and septicemia worldwide. Infection in most cases leads to asymptomatic carriage and only rarely to disease. Meningococcal disease often occurs in outbreaks, which are both sporadic and highly unpredictable. The occurrence of disease outbreaks in a host population in which the etiological agent is widely carried is not well understood. A potential explanation lies in the fact that meningococci are diverse with respect to disease-causing potential. We formulated a stochastic mathematical model to investigate whether diversity of the bacterial population is related to outbreaks of meningococcal disease. In the model, strains that occasionally cause the disease appear repeatedly in a population dominated by a nonpathogenic strain. When the pathogenicity, i.e., the disease-causing potential, of the pathogenic lineage was low, the model shows distinct outbreaks, the size distribution of the outbreaks follows a power law, and the ratio of the variance to the mean number of cases is high. Analysis of notification data of meningococcal disease showed that the ratio of the variance to the mean was significantly higher for meningococcal diseases than for other bacterial invasive diseases. This result lends support to the hypothesis that outbreaks of meningococcal disease are caused by diversity in the pathogenicity of meningococcal strains.

Predicting meningococcal disease outbreaks in structured populations

Rational decision making on whether some form of intervention would be necessary to control the spread of a meningococcal epidemic is based on predictions concerning its potential natural progression. Unfortunately, reliable predictions are difficult to make during the early stages of an outbreak. A stochastic discrete time epidemic model was applied to adaptively predict the development of outbreaks of meningococcal disease in 'closed' populations such as military garrisons or boarding schools, which are further divided into subgroups called 'units'. The performance of the adaptive method was assessed by using 3 simulated epidemics representing substantially different realizations in a 'garrison' of 20 units, with 68 men in each. Predictions of the weekly number of disease cases, of the number of carriers, and of the number of new infections were computed. Simulations suggest that predictions based only on the observed numbers of disease cases are generally inaccurate. These predictions can be improved if temporal observations on asymptomatic carriers in different units are utilized together with observed time series of the disease. A sample of 15 per cent from all units can be sufficient for a major improvement if the alternative is to obtain a full sample of only some units. Exploiting fully such information requires computer intensive Markov chain Monte Carlo methods.

A probabilistic transmission model of Salmonella in the primary broiler production chain

Annual data from the Finnish National Salmonella Control Programme were used to build up a probabilistic transmission model of salmonella in the primary broiler production chain. The data set consisted of information on grandparent, parent, and broiler flock populations. A probabilistic model was developed to describe the unknown true prevalences, vertical and horizontal transmissions, as well as the dynamical model of infections. By combining these with the observed data, the posterior probability distributions of the unknown parameters and variables could be derived. Predictive distributions were derived for the true number of infected broiler flocks under the adopted intervention scheme and these were compared with the predictions under no intervention. With the model, the effect of the intervention used in the programme, i.e., eliminating salmonella positive breeding flocks, could be quantitatively assessed. The 95% probability interval of the posterior predictive distribution for (broiler) flock prevalence under current (1999) situation was [1.3%-17.4%] (no intervention), and [0.9%-5.8%] (with intervention). In the scenario of one infected grandparent flock, these were [2.8%-43.1%] and [1.0%-5.9%], respectively. Computations were performed using WinBUGS and Matlab softwares.