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.      

Functional outline of the epidemic process

The present study has shown that on the level of the parasitic system the epidemic process is a biological system, wherein the host population serves as the internal regulator, the mechanism of transmission serves as the external regulator and the parasite population, as the regulated object. The biological regulating mechanisms of the epidemic process have fundamental differences in the groups of infectious with various mechanisms of transmission, and the specific nature of the mechanism of transmission determines the peculiar features of the biological mechanism which governs the self-regulation of the epidemic process. In contrast, on a higher level of the organization of the epidemic process, i. e. on the level of the socio-ecological system, the epidemic process is a biosocial system, wherein the human society serves as the regulator, the parasitic system serves as the regulated object and the mechanism of transmission plays the role of the filter which determines the scope of social factors, most important in the regulation of the epidemic process in a given infection. The spontaneous regulation of the epidemic process is the freed forward channel from the regulator to the regulated object, and the controlled regulation is the feedback channel.     

The system of the epidemic process

An original social-ecological concept of the epidemic process has been constructed on the basis of using social ecology, systemic approach and the basic principles of cybernetics. According to this concept, the epidemic process is regarded as a biosocial, hierarchic, integral system providing for the reproduction of the species of human parasites. At a higher level of organization, the epidemic process is an epidemiological social-ecological system consisting of two interacting subsystems: the biological (epidemiological ecosystem) and the social (social and economic conditions of life of the society) subsystems where the biological subsystem plays the role of the governed object and the social acts as the internal regulator of these interactions. On the basis of this concept a rational structure of the system of epidemiological surveillance over infectious diseases has been proposed according to which each level of the structure of the epidemic process should be subject to adequate monitoring.     

The epidemic process as a system. The social regulators of the epidemic process

The article deals with the mechanisms of spontaneous (urbanization, the international migration of population, the anthropogenic transformation of nature) and goal-oriented regulatory action on the epidemic process. Social factors have been shown to transform into ecological ones with the subsequent transformation of information in the parasitic system and their reflection in the social subsystem as the indices of the risk of infection and its socio-economical importance. Using the processes of urbanization as an example, the present work demonstrates that the mechanism of transmission acts as a filter whose specific features are determined by a range of social factors playing the most important role in the regulation of the epidemic process on the socio-ecosystemic level of its organization.     

Estimation of the size of the vCJD epidemic

Estimates of the final size of the variant Creuzfeldt-Jakob Disease epidemic have been made by fitting theoretical curves of the incubation period distribution to the histogram of observed annual deaths to 2002, using various assumptions of the mean and standard deviation of this distribution, and also of the efficacy of the Specified Bovine Offals ban of 1989. Unless the mean incubation time is greater than 15 to 20 years the estimates lie in the low hundreds to about a thousand, and the most likely situation, of a mean between 11 and 15 years, gives estimates of about 150 to 500 deaths. Numbers above a few thousands would only occur if the mean incubation period is of the order of 25 to 30 years and reasons are adduced to indicate this is very unlikely. These numbers are not greatly increased if the ban was poorly observed. This method of analysis may be applicable to other situations where a cause that is limited in space and time is expected to have late effects.     

The epidemic process as a system. I. The structure of the epidemic process

A new epidemiological concept (socio-ecological) has been formulated on the basis of the principles of the theory of systems and the theory of information. In accordance with this concept, the epidemic process is organized on the same principle as living matter, and the stability of this process at all levels of its organization is ensured by the processes of self-regulation. The conditions of the life of human society have been shown to be organically incorporated into the structure of the epidemic process as a regulating subsystem on the socio-ecological level.     

Viral haemorrhagic fevers--evolution of the epidemic potential

In this review modern data on dangerous and particularly dangerous viral haemorrhagic fevers caused by a group of viruses belonging to the families of phylo-, arena-, flavi-, bunya- and togaviruses are presented. Morbidity rates and epidemics caused by Marburg virus, Ebola fever virus, Lassa fever virus, Argentinian and Bolivian haemorrhagic fever viruses, dengue haemorrhagic fever virus, Crimean haemorrhagic fever virus, Hantaviruses are analyzed. Mechanisms of the evolution of the epidemic manifestation of these infections are considered. The importance of the development of tools and methods of diagnosis, rapid prevention and treatment of exotic haemorrhagic fevers is emphasized.     

On the formulation of discrete-time epidemic models

Jacquez constructed a properly posed, more general model for the Reed-Frost epidemic process by assuming independent behaviors for the susceptibles and introducing the generating function for the number of contacts per person. An alternative approach is proposed here that relies on similar hypotheses for the infectives and allows the usual chain-binomial structure of the infection process to be extended. For this new model, the derivation of the final size and the threshold phenomenon becomes much simpler. A detailed analysis and its generalization to heterogeneous populations and continuous-time models will be the subject of a forthcoming paper.