4: Infectious Disease Modeling

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In the late \(1320^{\prime}\) s, an outbreak of the bubonic plague occurred in China. This disease is caused by the bacteria Yersinia pestis and is transmitted from rats to humans by fleas. The outbreak in China spread west, and the first major outbreak in Europe occurred in 1347 . During a five year period, 25 million people in Europe, approximately \(1 / 3\) of the population, died of the black death. Other more recent epidemics include the influenza pandemic known as the Spanish flu killing 50-100 million people worldwide during the years \(1918-1919\), and the present AIDS epidemic, originating in Africa and first recognized in the USA in 1981 , killing more than 25 million people. For comparison, the SARS epidemic for which Hong Kong was the global epicenter in the year 2003 resulted in 8096 known SARS cases and 774 deaths. Yet, we know well that this relatively small epidemic caused local social and economic turmoil.

Here, we introduce the most basic mathematical models of infectious disease epidemics and endemics. These models form the basis of the necessarily more detailed models currently used by world health organizations, both to predict the future spread of a disease and to develop strategies for containment and eradication.

4.1: The SI Model
4.2: The SIS Model
4.3: The SIR Epidemic Disease Model
4.4: Vaccination
4.5: The SIR Endemic Disease Model
4.6: Evolution of Virulence
Microorganisms continuously evolve due to selection pressures in their environments. Antibiotics are a common source of selection pressure on pathogenic bacteria, and the development of antibiotic-resistant strains presents a major health challenge to medical science. Bacteria and viruses also compete directly with each other for reproductive success resulting in the evolution of virulence. Here, using the SIR endemic disease model, we study how virulence may evolve.