There is a long tradition of using mathematical models to generate insights into the transmission dynamics of infectious diseases and assess the potential impact of different intervention strategies. The increasing use of mathematical models for epidemic forecasting has highlighted the importance of designing reliable models that capture the baseline transmission characteristics of specific pathogens and social contexts. More refined models are needed however, in particular to account for variation in the early growth dynamics of real epidemics and to gain a better understanding of the mechanisms at play. Here, we review recent progress on modeling and characterizing early epidemic growth patterns from infectious disease outbreak data, and survey the types of mathematical formulations that are most useful for capturing a diversity of early epidemic growth profiles, ranging from sub-exponential to exponential growth dynamics. Specifically, we review mathematical models that incorporate spatial details or realistic population mixing structures, including meta-population models, individual-based network models, and simple SIR-type models that incorporate the effects of reactive behavior changes or inhomogeneous mixing. In this process, we also analyze simulation data stemming from detailed large-scale agent-based models previously designed and calibrated to study how realistic social networks and disease transmission characteristics shape early epidemic growth patterns, general transmission dynamics, and control of international disease emergencies such as the 2009 A/H1N1 influenza pandemic and the 2014-15 Ebola epidemic in West Africa.
A compartmental model is presented for the spread of HIV in a homosexual population divided into subgroups by degree of sexual activity. The model includes constant recruitment rates for the susceptibles in the subgroups. It incorporates the long infectious period of HIV-infected individuals and allows one to vary infectiousness over the infectious period. A new pattern of mixing, termed preferred mixing, is defined, in which a fraction of a group's contacts can be reserved for within-group contacts, the remainder being subject to proportional mixing. The fraction reserved may differ among groups. In addition, the classic definition of reproductive number is generalized to show that for heterogeneous populations in general the endemic threshold is BDc,, where cr is the mean number of contacts per infective. The most important finding is that the pattern of contacts between the different groups has a major effect on the spread of HIV, an effect inadequately recognized or studied heretofore.
A common procedure in paleodemography is to use mean skeletal age to estimate the expectation of life at birth in a population. This paper shows that skeletal age and expectation of life at birth are not equivalent unless the population is stationary. This assumption is not justified for most real populations. We show that mean skeletal age is approximately equivalent to the reciprocal of the birth rate and is not correlated with the death rate. Thus, the practice of inferring changes in life span and death rates from changes in mean age at death is not reliable and most conclusions of paleodemographic studies should be revised. On the other hand, skeletal age may provide high-quality information about fertility in archaeological populations. Several published paleodemographic studies are reinterpreted in light of the model presented.
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