Stochastic amplification in an epidemic model with seasonal forcing

We study the stochastic susceptible-infected-recovered (SIR) model with time-dependent forcing using analytic techniques which allow us to disentangle the interaction of stochasticity and external forcing. The model is formulated as a continuous time Markov process, which is decomposed into a deterministic dynamics together with stochastic corrections, by using an expansion in inverse system size. The forcing induces a limit cycle in the deterministic dynamics, and a complete analysis of the fluctuations about this time-dependent solution is given. This analysis is applied when the limit cycle is annual, and after a period doubling when it is biennial. The comprehensive nature of our approach allows us to give a coherent picture of the dynamics which unifies past work, but which also provides a systematic method for predicting the periods of oscillations seen in whooping cough and measles epidemics.     

Growth retardation in weight and catch-up growth following infection by measles and whooping cough among babies of rural Zaire (Lake Tumba)

This study has shown that measles and whooping cough affect growth in weight among babies of rural Zaire (Lake Tumba). The average loss in weight due to measles, is more than twice the amount due to whooping cough. However, the time needed to regain the original growth channel after infection by measles, is about half the time needed to fully catch up after whooping cough. The amount of weight loss is proportional to the child's weight at the onset of the disease. There is no difference in the susceptibility to lose weight after measles or whooping cough, between boys and girls, nor between Oto and Twa. This study has also shown that the effect of measles and whooping cough on the growth in weight, is independent of the season of the hear, although the incidence of these diseases showed a significant seasonal pattern. Finally, this study gives evidence for the fact that the incidence of weight loss one month after the onset of measles or whooping cough is age-related.     

Transients and attractors in epidemics

Historical records of childhood disease incidence reveal complex dynamics. For measles, a simple model has indicated that epidemic patterns represent attractors of a nonlinear dynamic system and that transitions between different attractors are driven by slow changes in birth rates and vaccination levels. The same analysis can explain the main features of chickenpox dynamics, but fails for rubella and whooping cough. We show that an additional (perturbative) analysis of the model, together with knowledge of the population size in question, can account for all the observed incidence patterns by predicting how stochastically sustained transient dynamics should be manifested in these systems.     

Distinguishing error from chaos in ecological time series

Over the years, there has been much discussion about the relative importance of environmental and biological factors in regulating natural populations. Often it is thought that environmental factors are associated with stochastic fluctuations in population density, and biological ones with deterministic regulation. We revisit these ideas in the light of recent work on chaos and nonlinear systems. We show that completely deterministic regulatory factors can lead to apparently random fluctuations in population density, and we then develop a new method (that can be applied to limited data sets) to make practical distinctions between apparently noisy dynamics produced by low-dimensional chaos and population variation that in fact derives from random (high-dimensional) noise, such as environmental stochasticity or sampling error. To show its practical use, the method is first applied to models where the dynamics are known. We then apply the method to several sets of real data, including newly analysed data on the incidence of measles in the United Kingdom. Here the additional problems of secular trends and spatial effects are explored. In particular, we find that on a city-by-city scale measles exhibits low-dimensional chaos (as has previously been found for measles in New York City), whereas on a larger, country-wide scale the dynamics appear as a noisy two-year cycle. In addition to shedding light on the basic dynamics of some nonlinear biological systems, this work dramatizes how the scale on which data is collected and analysed can affect the conclusions drawn.     

Quantitative research on the cyclic fluctuations of the epidemic process in a number of infectious diseases in Bulgaria

The cyclic nature of the epidemic process in Bulgaria was studied by various methods (spectral analysis, etc.), forming a system. The morbidity dynamics in 10 infectious diseases (scarlet fever, rubella, measles, epidemic parotitis, whooping cough, diphtheria, typhoid fever, enterocolitis, bacterial dysentery, viral hepatitis) over the years of 1909-1983 were studied and cycles covering the periods of 3-4, 5-6, 10-11 and over 16 years were established. The data on the relative part of cyclic processes in the registered morbidity of infectious diseases, as well as information on the prognostication of the spread of infections in the absence of vaccinal prophylaxis, are presented.     

Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics.

We address the question of whether or not childhood epidemics such as measles and chickenpox are chaotic, and argue that the best explanation of the observed unpredictability is that it is a manifestation of what we call chaotic stochasticity. Such chaos is driven and made permanent by the fluctuations from the mean field encountered in epidemics, or by extrinsic stochastic noise, and is dependent upon the existence of chaotic repellors in the mean field dynamics. Its existence is also a consequence of the near extinctions in the epidemic. For such systems, chaotic stochasticity is likely to be far more ubiquitous than the presence of deterministic chaotic attractors. It is likely to be a common phenomenon in biological dynamics.     

Integrating life history and cross-immunity into the evolutionary dynamics of pathogens

Models for the diversity and evolution of pathogens have branched into two main directions: the adaptive dynamics of quantitative life-history traits (notably virulence) and the maintenance and invasion of multiple, antigenically diverse strains that interact with the host's immune memory. In a first attempt to reconcile these two approaches, we developed a simple modelling framework where two strains of pathogens, defined by a pair of life-history traits (infectious period and infectivity), interfere through a given level of cross-immunity. We used whooping cough as a potential example, but the framework proposed here could be applied to other acute infectious diseases. Specifically, we analysed the effects of these parameters on the invasion dynamics of one strain into a population, where the second strain is endemic. Whereas the deterministic version of the model converges towards stable coexistence of the two strains in most cases, stochastic simulations showed that transient epidemic dynamics can cause the extinction of either strain. Thus ecological dynamics, modulated by the immune parameters, eventually determine the adaptive value of different pathogen genotypes. We advocate an integrative view of pathogen dynamics at the crossroads of immunology, epidemiology and evolution, as a way towards efficient control of infectious diseases.     

Revisited measles and chickenpox dynamics through orthogonal transformation

The question addressed is whether or not childhood epidemics such as measles and chickenpox are characterized by low-dimensional chaos. We propose a new method for the detection and extraction of hidden periodic components embedded in an irregular cyclical series, and study the characterization of the epidemiological series in terms of the characteristic features or periodicity attributes of the extracted components. It is shown that the measles series possesses two periodic components each having a period of one year. Both the periodic components have time-varying pattern, and the process is nonlinear and deterministic; there is no evidence of strong chaoticity in the measles dynamics. The chickenpox series has one seasonal component with stable pattern, and the process is deterministic but linear, and hence non-chaotic. We also propose surrogate generators based on null hypotheses relating to the variability of the periodicity attributes to analyse the dynamics in the epidemic series. The process dynamics is also studied using seasonally forced SEIR epidemic model, and the characterization performance of the proposed schemes is assessed.     

Injury by hypertonic phosphate enema.

Over a 2-year period 67 strains of Bordetella pertussis were identified in 231 single specimens of nasopharyngeal secretions submitted from patients suspected to have whooping cough in the National Capital Region; 89.5% of the identifications were made by culture. Serotype 1,3 was predominant. At least 75% of the patients with bacteriologically confirmed whooping cough had not been fully immunized. There was no evidence that adenoviruses or other viruses played any important etiologic role in the 204 cases of whooping cough or whooping cough syndrome studied virologically.     

Study of intellectual performance of children in ordinary schools after certain serious complications of whooping cough. Swansea Research Unit of the Royal College of General Practitioners.

In a case-control study 27 index children from ordinary schools who had had convulsions or apnoea as a complication of whooping cough about eight years previously were compared with 27 children who had never had whooping cough and 15 who had had whooping cough without complications. Other factors likely to cause intellectual impairment after conception were considered. The index group had a significantly lower median intelligence quotient and poorer school attainment than either of the control groups. The results support the hypothesis that convulsions or apnoea as a complication of whooping cough may be associated with subsequent intellectual impairment.     

Domiciliary immunisation for preschool child defaulters.

A selective domiciliary immunisation service has been introduced for preschool children living in the Central Manchester Health District. The service is provided for the children of parents who have given written consent for immunisation but have subsequently failed to bring their children to the clinic or their general practitioner''s surgery to start or complete the basic course of immunisation against diphtheria, tetanus, whooping cough, poliomyelitis, and measles. This failure has occurred despite at least two written invitations for each injection and several reminders from the health visitor. Many additional children have been protected by this scheme.     

Natural course of 500 consecutive cases of whooping cough: a general practice population study.

OBJECTIVE--To describe the natural course of whooping cough. DESIGN--Observational study of a general practice population. SETTING--Discrete semirural East Midlands practice of 11,500 patients. SUBJECTS--500 consecutive cases of whooping cough diagnosed clinically during 1977-92. MAIN OUTCOME MEASURES--Incidence of vomiting, whooping, apnoea, admission to hospital, and complications; duration and frequency of paroxysms. Pattern of spread. RESULTS--The incidence in the practice population was 4347/100,000 population compared with a notification rate for England and Wales of 717/100,000. Most cases were relatively mild. 284 patients vomited after paroxysms, 242 whooped, and 57 had apnoea. Duration and frequency of paroxysms varied widely. Female and unimmunised patients suffered more severe disease. Bordetella was isolated from fewer immunised patients (24/96 v 63/122 unimmunised). Infection was usually spread through contacts with someone with clinical whooping cough. Five patients developed pneumonia, three of whom had been immunised. Three patients required hospital admission. CONCLUSIONS--Most cases of whooping cough are relatively mild. Such cases are difficult to diagnose without a high index of suspicion because doctors are unlikely to hear the characteristic cough, which may be the only symptom. Parents can be reassured that a serious outcome is unlikely. Adults also get whooping cough, especially from their children, and get the same symptoms as children. The difficulty of early diagnosis and probability of missed cases reinforces the need to keep the incidence low through immunisation in order to protect infants, who are the most vulnerable.     

Robust permanence and impermanence for stochastic replicator dynamics

Garay and Hofbauer (SIAM J. Math. Anal. 34 (2003)) proposed sufficient conditions for robust permanence and impermanence of the deterministic replicator dynamics. We reconsider these conditions in the context of the stochastic replicator dynamics, which is obtained from its deterministic analogue by introducing Brownian perturbations of payoffs. When the deterministic replicator dynamics is permanent and the noise level small, the stochastic dynamics admits a unique ergodic distribution whose mass is concentrated near the maximal interior attractor of the unperturbed system; thus, permanence is robust against small unbounded stochastic perturbations. When the deterministic dynamics is impermanent and the noise level small or large, the stochastic dynamics converges to the boundary of the state space at an exponential rate.     

Population dynamic interference among childhood diseases.

Epidemiologists usually study the interaction between a host population and one parasitic infection. However, different parasite species effectively compete, in an ecological sense, for the same finite group of susceptible hosts, so there may be an indirect effect on the population dynamics of one disease due to epidemics of another. In human populations, recovery from any serious infection is normally preceded by a period of convalescence, during which infected individuals stay at home and are effectively shielded from exposure to other infectious diseases. We present a model for the dynamics of two infectious diseases, incorporating a temporary removal of susceptibles. We use this model to explore population-level consequences of a temporary insusceptibility in childhood diseases, the dynamics of which are partly driven by differences in contact rates in and out of school terms. Significant population dynamic interference is predicted and cannot be dismissed in the limited case-study data available for measles and whooping cough in England before the vaccination era.     

Whooping cough in adults.

During the 1970s whooping cough returned in Sweden after an absence of more than 10 years and is now seen in all age groups, During a three-year period 174 adults with culture-verified whooping cough were identified in Gothenburg. Most of the patients had typical symptoms with whooping attacks and often vomiting. The disease was long lasting but complications were rare. Physicians should be aware that whooping cough may occur in adults, since adults may be an important source of infection for infants and erythromycin given in the catarrhal phase may modify the clinical course.     

Intensity of the epidemic process of whooping cough under conditions of mass immunization over a period of many years

On the basis of analysis of whooping cough incidence in 1959 to 1975 in Moscow the authors present characteristics of the epidemic process under conditions of planned many-year immunization of children against this infection. Whooping cough morbidity proved to undergo significant changes-from sharp falls the first 8 years of immunization to a relative stabilization with a gradual reduction the last 5 years. Marked seasonal and periodic elevations persist against the background of reduction of morbidity. Preschool- and schoolchildren are equally involved in the epidemic process; the most frequently involved are nonvaccinated children aged 1 year and 4 to 10 years, which lost postvaccinal immunity. Further reduction of the severity of whooping cough and increase in the number of unimanifested forms of the infection was noted. This was apparently connected not only with immunization, but also with the reduction of the virulence of H. pertusis detected in studying the cultures isolated in Moscow from 1967 to 1974. The intensity indices of the epidemic process in whooping cough pointed to the necessity of using a more effective vaccine for the protection of children from whooping cough, despite the noted reduction of morbidity, and diminished severity of the course of the disease, and of the causative agent virulence.     

Development and application of synthetic peptides as vaccines

Vaccination against bacterial and viral diseases has been one of the major achievements in medicine and immunology since the beginning of this century. Extensive vaccination programs have been able to control or, in the case of smallpox, virtually wipe out some of the most dangerous infectious diseases e.g. poliomyelitis, measles, whooping cough, diphtheria and tetanus. However, as this success has been limited mainly to the developed, affluent countries, infectious diseases still remain the worlds largest health problem. Furthermore, vaccines against human parasites are non-existent. Recent advances in immunology and molecular biology including recombinant DNA technology have provided the basis for new approaches to vaccine development.     

Vaccines and exposed occupations

The use of safe and efficacious vaccines in occupational settings to protect workers from diseases to which they may be exposed is obvious and has been included in the employment law. Healthcare workers are particular exposed. Immunization has two purposes : protect the worker from contracting a disease, but also prevent him from disseminating the disease to weakened patients. It is important not only to take into account existing recommendations for immunization, but also to envisage their extension to teachers and staff of nurseries and primary schools. Routine vaccination against whooping cough, varicella, measles and hepatitis A is particularly warranted in these categories. Recommendations should also extend to medical students who are too often poorly protected and insufficiently warned against potential occupational exposure to pathogens and dissemination to their patients.     

Whooping-cough vaccine. Statement by Joint Committee on Vaccination and Immunization of the Central Health Services Council and the Scottish Health Service Planning Council.

In 1974 it was recommended that pertussis vaccine should continue to be offered in a triple vaccine together with diphtheria and tetanus vaccines. Further data on the prevalence of whooping cough and the incidence of adverse reactions have shown no reason to change this policy; the hazard of whooping cough remains greater than that of immunization.     

Etiology, clinical manifestations, and epidemiology in community-acquired respiratory infection in children in the Khabarovsk region

Clinico-epidemiological analysis and etiological verification of the outbreak of respiratory infection among school children in a rural district of the Khabarovsk territory, registered in spring 1997, were made. According to clinical signs, one-third of the patients had whooping cough, while the rest of the children exhibited the signs of respiratory infection with the symptoms of longering bronchitis. A half of the children had not been vaccinated against whooping cough, as they had been given injections of adsorbed DT vaccine with reduced antigen content. Etiologically, the diagnosis of whooping cough was confirmed in 57% of the patients with 47.4% of them having Bordetella pertussis monoinfection and 52.6% having mixed infection, mainly in combination with chlamydiosis. Whooping cough took an abnormal course under these circumstances. Treatment with erythromycin produced a good effect.