Dynamics of stochastic epidemics on heterogeneous networks

Epidemic models currently play a central role in our attempts to understand and control infectious diseases. Here, we derive a model for the diffusion limit of stochastic susceptible-infectious-removed (SIR) epidemic dynamics on a heterogeneous network. Using this, we consider analytically the early asymptotic exponential growth phase of such epidemics, showing how the higher order moments of the network degree distribution enter into the stochastic behaviour of the epidemic. We find that the first three moments of the network degree distribution are needed to specify the variance in disease prevalence fully, meaning that the skewness of the degree distribution affects the variance of the prevalence of infection. We compare these asymptotic results to simulation and find a close agreement for city-sized populations.     

The size of epidemics in populations with heterogeneous susceptibility

We formulate and study a general epidemic model allowing for an arbitrary distribution of susceptibility in the population. We derive the final-size equation which determines the attack rate of the epidemic, somewhat generalizing previous work. Our main aim is to use this equation to investigate how properties of the susceptibility distribution affect the attack rate. Defining an ordering among susceptibility distributions in terms of their Laplace transforms, we show that a susceptibility distribution dominates another in this ordering if and only if the corresponding attack rates are ordered for every value of the reproductive number R0. This result is used to prove a sharp universal upper bound for the attack rate valid for any susceptibility distribution, in terms of R0 alone, and a sharp lower bound in terms of R0 and the coefficient of variation of the susceptibility distribution. We apply some of these results to study two issues of epidemiological interest in a population with heterogeneous susceptibility: (1) the effect of vaccination of a fraction of the population with a partially effective vaccine, (2) the effect of an epidemic of a pathogen inducing partial immunity on the possibility and size of a future epidemic. In the latter case, we prove a surprising '50% law': if infection by a pathogen induces a partial immunity reducing susceptibility by less than 50%, then, whatever the value of R0>1 before the first epidemic, a second epidemic will occur, while if susceptibility is reduced by more than 50%, then a second epidemic will only occur if R0 is larger than a certain critical value greater than 1.     

Modeling epidemics of multidrug-resistant M. tuberculosis of heterogeneous fitness

Mathematical models have recently been used to predict the future burden of multidrug-resistant tuberculosis (MDRTB). These models suggest the threat of multidrug resistance to TB control will depend on the relative 'fitness' of MDR strains and imply that if the average fitness of MDR strains is considerably less than that of drug-sensitive strains, the emergence of resistance will not jeopardize the success of tuberculosis control efforts. Multidrug resistance in M. tuberculosis is conferred by the sequential acquisition of a number of different single-locus mutations that have been shown to have heterogeneous phenotypic effects. Here we model the impact of initial fitness estimates on the emergence of MDRTB assuming that the relative fitness of MDR strains is heterogeneous. We find that even when the average relative fitness of MDR strains is low and a well-functioning control program is in place, a small subpopulation of a relatively fit MDR strain may eventually outcompete both the drug-sensitive strains and the less fit MDR strains. These results imply that current epidemiological measures and short-term trends in the burden of MDRTB do not provide evidence that MDRTB strains can be contained in the absence of specific efforts to limit transmission from those with MDR disease.     

On the spread of epidemics in a closed heterogeneous population

Heterogeneity is an important property of any population experiencing a disease. Here we apply general methods of the theory of heterogeneous populations to the simplest mathematical models in epidemiology. In particular, an SIR (susceptible-infective-removed) model is formulated and analyzed when susceptibility to or infectivity of a particular disease is distributed. It is shown that a heterogeneous model can be reduced to a homogeneous model with a nonlinear transmission function, which is given in explicit form. The widely used power transmission function is deduced from the model with distributed susceptibility and infectivity with the initial gamma-distribution of the disease parameters. Therefore, a mechanistic derivation of the phenomenological model, which is believed to mimic reality with high accuracy, is provided. The equation for the final size of an epidemic for an arbitrary initial distribution of susceptibility is found. The implications of population heterogeneity are discussed, in particular, it is pointed out that usual moment-closure methods can lead to erroneous conclusions if applied for the study of the long-term behavior of the models.     

Occurrence of conjugated R-plasmids in bacteria of the Aeromonas genus isolated from purified urban sewage water]

The aim of the study was to determine a participation of Aeromonas sp., having conjugation R plasmids, in a population of the bacteria present in purified urban sewage . In 1 ml of sewage 1.8 x 10(3) - 50 x 10(3) of Aeromonas sp., were found. The isolated strains belonged to the following genera: A. hydrophila, A. caviae, A. sorbia. All tested strains were sensitive to gentamicin, tetracycline, kanamycin and nalidixic acid. Among 186 strains of Aeromonas sp., two belonging to A. caviae genus transferred resistance to streptomycin to A. hydrophila and E. coli recipients on the other hand two strains of A. caviae transferred resistance to streptomycin exclusively to A. hydrophila recipient.     

Optimal dispatching of multipriority jobs to two heterogeneous workstations

The general problem scenario of this paper is the following: Jobs of various priorities, stationed in a common storage area, are waiting to be dispatched to two non-identical workstations. Any of the waiting jobs can be accessed from the storage at any given time. Each job can be processed on either of the workstations, but once a job has been assigned it may not be preempted. By job priority it is meant that a higher priority job has disptach preference over a lower priority job. The processing time of a job on a given workstation is assumed to be random, the distribution being dependent on the job type and the configuration of the workstation. Specifically, the first problem studied considers only two classes of jobs: (1) “hot” jobs, whose processing is to be expedited and thus have the higher dispatch priority, and (2) “routine” jobs which may be assigned to an available workstation only if the workstation has been rejected by all “hot” jobs. The processing times are assumed to be exponentially distributed with means depending on the job class and workstation. We assume that, on the average, one workstation is faster than the other with regard to processing any job. The dispatching objective for each job class is to minimize its expected flowtime. It is shown that threshold dispatching policies are optimal for this problem. That is, the faster processor should be utilized whenever possible, and for each class there exists an explicit threshold such that when the number of jobs of that class in the buffer exceeds this threshold then a job of that class is dispatched to the slower processor, otherwise these jobs wait for the faster processor to become available. For the higher priority jobs, this threshold is shown to be a function only of the various processing rates of the two workstations. For the lower priority jobs, the threshold also depends on the number of higher priority jobs in the buffer. The results is extended to a system with n priority classes. Again, it is shown that when the processing times are exponentially distributed with different rates and the dispatching objective for each class is to minimize its expected flowtime, the optimal dispatching policies are of threshold type. Explicit thresholds are easily derived.     

Life not lived due to disequilibrium in heterogeneous age-structured populations

Three models of age-structured populations with demographically heterogeneous subpopulations are analyzed. In the first model, each subpopulation has its own age-specific vital rates which are fixed in time. In the second model, the vital rates of each subpopulation are uniformly inhibited by increasing total numbers of individuals. In the third, the vital rates of groups of subpopulations are inhibited by the total numbers of individuals in other groups of subpopulations with an intensity that depends on the interacting pair of groups. Three functions are defined to measure disequilibrium in the subpopulation frequencies, subpopulation age structures, and total population size. For the first model, we show that disequilibrium will shift the trajectory of the total numbers of individuals forward or backward in time by an asymptotic constant that is proportional to the sum of the disequilibrium measures. For the second model, we establish sufficient conditions for the existence of a globally stable equilibrium and we show that disequilibrium will result in a finite loss or gain in life which is proportional to the sum of the disequilibrium measures. For the last model, we show that the loss or gain in life for each group of subpopulations is a linear combination over all groups of the sums of the three disequilibrium measures. We illustrate these results with numerical examples and give possible biological interpretations of the models. We relate these new results to previous work on the cost of natural selection and measures of demographic disequilibrium.     

Vibrio parahaemolyticus in shellfish and clinical samples during two large epidemics of diarrhoea in southern Chile

Large epidemics of diarrhoea associated with seafood consumption and Vibrio parahaemolyticus occurred during the austral summers of 2004 and 2005 in the environs of Puerto Montt, Chile (41 degrees 29'S 72 degrees 24'W). There are no reports of V. parahaemolyticus infections before 2004 in this region, their absence being explained by the low ocean temperatures which seldom reach 16 degrees C. We analysed V. parahaemolyticus obtained from shellfish and clinical samples during epidemics. Isolates were examined using conventional protocols and an improved method for restriction enzyme analysis using total bacterial DNA which permits direct genome restriction enzyme analysis by conventional gel electrophoresis (DGREA) with a similar discrimination index as restriction fragment length polymorphism-pulsed field gel electrophoresis (RFLP-PFGE). Analysis of clinical samples showed that the epidemics were caused by the V. parahaemolyticus O3:K6 pandemic clonal group. On the other hand, analysis of shellfish samples during both epidemics showed that 53% contained V. parahaemolyticus (3-93 g(-1)). Detailed analysis of 50 positive shellfish samples showed that only three contained detectable levels of the pandemic clone. Most V. parahaemolyticus isolates obtained from shellfish corresponded to non-pandemic clones differentiated into 14 groups by DGREA. In summary, the causative agent during epidemics was only a minor component of a small but diverse population of V. parahaemolyticus in shellfish.     

Occurrence of R-plasmids in porcine faecal waste and comparison of their transfer rates in laboratory mating systems and anaerobic digesters

Summary Most of the coliform bacteria (77–100%) isolated from porcine faecal waste were resistant to 6 antibiotics. In up to 51% of cases, the genetic determinants for tetracycline, streptomycin and ampicillin resistances were found to be transmissible to a laboratory strain ofEscherichia coli in laboratory mating techniques. Transfer frequency varied according to the mating technique employed. Transfer of resistance determinants during treatment of the faecal waste in a laboratory-scale anaerobic digester was detected erratically in very few cases.     

Optimization of control strategies for epidemics in heterogeneous populations with symmetric and asymmetric transmission

There is growing interest in incorporating economic factors into epidemiological models in order to identify optimal strategies for disease control when resources are limited. In this paper we consider how to optimize the control of a pathogen that is capable of infecting multiple hosts with different rates of transmission within and between species. Our objective is to find control strategies that maximize the discounted number of healthy individuals. We consider two classes of host-pathogen system, comprising two host species and a common pathogen, one with asymmetrical and the other with symmetrical transmission rates, applicable to a wide range of SI (susceptible-infected) epidemics of plant and animal pathogens. We motivate the analyses with an example of sudden oak death in California coastal forests, caused by Phytophthora ramorum, in communities dominated by bay laurel (Umbellularia californica) and tanoak (Lithocarpus densiflorus). We show for the asymmetric case that it is optimal to give priority in treating disease to the more infectious species, and to treat the other species only when there are resources left over. For the symmetric case, we show that although a switching strategy is an optimum, in which preference is first given to the species with the lower level of susceptibles and then to the species with the higher level of susceptibles, a simpler strategy that favors treatment of infected hosts for the more susceptible species is a robust alternative for practical application when the optimal switching time is unknown. Finally, since transmission rates are notoriously difficult to estimate, we analyze the robustness of the strategies when the true state with respect to symmetry or otherwise is unknown but one or other is assumed.     

Occurrence of Eurasian Skylark Alauda arvensis territories in relation to urban area and heterogeneous farmland

Within heterogeneous arable habitats close to urbanized Vienna, Austria, Eurasian Skylarks Alauda arvensis occurred at slightly lower densities than in other mixed farmland areas and at a much lower density than would be found in natural habitats. Territory density was negatively correlated with the occurrence of roads, tracks and high-intensity agricultural cultivation, possibly due to the resulting increased disturbance by humans.     

A general model for stochastic SIR epidemics with two levels of mixing

This paper is concerned with a general stochastic model for susceptible→infective→removed epidemics, among a closed finite population, in which during its infectious period a typical infective makes both local and global contacts. Each local contact of a given infective is with an individual chosen independently according to a contact distribution ‘centred’ on that infective, and each global contact is with an individual chosen independently and uniformly from the whole population. The asymptotic situation in which the local contact distribution remains fixed as the population becomes large is considered. The concepts of local infectious clump and local susceptibility set are used to develop a unified approach to the threshold behaviour of this class of epidemic models. In particular, a threshold parameter R_* governing whether or not global epidemics can occur, the probability that a global epidemic occurs and the mean proportion of initial susceptibles ultimately infected by a global epidemic are all determined. The theory is specialised to (i) the households model, in which the population is partitioned into households and local contacts are chosen uniformly within an infective’s household; (ii) the overlapping groups model, in which the population is partitioned in several ways, with local uniform mixing within the elements of the partitions; and (iii) the great circle model, in which individuals are equally spaced on a circle and local contacts are nearest-neighbour.     

Occurrence of two different pathways in the activation of porcine pepsinogen to pepsin

Activation of porcine pepsinogen at pH 2.0 was found to proceed simultaneously by two different pathways. One pathway is the direct conversion process of pepsinogen to pepsin, releasing the intact activation segment. The isolation of the released 44-residue segment was direct evidence of this one-step process. At pH 5.5 the segment bound tightly to pepsin to form a 1:1 pepsin-activation segment complex, which was chromatographically indistinguishable from pepsinogen. The other is a stepwise-activating or sequential pathway, in which pepsinogen is activated to pepsin through intermediate forms, releasing activation peptides stepwisely. These intermediate forms were isolated and characterized. The major intermediate form was shown to be generated by removal of the amino-terminal 16 residues from pepsinogen. The released peptide mixture was composed of two major peptides comprising residues 1-16 and 17-44, and hence the stepwise-activating process was deduced to be mainly a two-step process.     

Occurrence of Goitre during Lithium Treatment

Of 330 patients given lithium for recurrent manic-depressive disorder 12 developed goitre after treatment periods of five months to two years. All the patients remained clinically euthyroid. Pressure symptoms necessitated subtotal thyroidectomy in two patients. In 9 out of 10 patients with goitre, and in two out of seven without goitre study with radioactive iodine showed abnormal findings in iodine metabolism. Discontinuance of lithium led to disappearance of goitres, while thyroid metabolism returned to normal. Thyroxine or desiccated thyroid produced shrinkage of the gland in spite of continued lithium medication.