disease modsurvey

disease modsurvey - SIAM REVIEW Vol. 42, No. 4, pp. 599653...

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SIAM REVIEW c ° 2000 Society for Industrial and Applied Mathematics Vol. 42, No. 4, pp. 599–653 The Mathematics of Infectious Diseases Herbert W. Hethcote Abstract. Many models for the spread of infectious diseases in populations have been analyzed math- ematically and applied to speciFc diseases. Threshold theorems involving the basic repro- duction number R 0 , the contact number σ , and the replacement number R are reviewed for the classic SIR epidemic and endemic models. Similar results with new expressions for R 0 are obtained for MSEIR and SEIR endemic models with either continuous age or age groups. Values of R 0 and σ are estimated for various diseases including measles in Niger and pertussis in the United States. Previous models with age structure, heterogeneity, and spatial structure are surveyed. Key words. thresholds, basic reproduction number, contact number, epidemiology, infectious diseases AMS subject classiFcations. Primary, 92D30; Secondary, 34C23, 34C60, 35B32, 35±25 PII. S0036144500371907 1. Introduction. The efectiveness oF improved sanitation, antibiotics, and vac- cination programs created a con±dence in the 1960s that inFectious diseases would soon be eliminated. Consequently, chronic diseases such as cardiovascular disease and cancer received more attention in the United States and industrialized countries. But inFectious diseases have continued to be the major causes oF sufering and mortality in developing countries. Moreover, inFectious disease agents adapt and evolve, so that new inFectious diseases have emerged and some existing diseases have reemerged [142]. Newly identi±ed diseases include Lyme disease (1975), Legionnaire’s disease (1976), toxic-shock syndrome (1978), hepatitis C (1989), hepatitis E (1990), and hantavirus (1993). The human immunode±ciency virus (HIV), which is the etiological agent For acquired immunode±ciency syndrome (AIDS), emerged in 1981 and has become an important sexually transmitted disease throughout the world. Antibiotic-resistant strains oF tuberculosis, pneumonia, and gonorrhea have evolved. Malaria, dengue, and yellow Fever have reemerged and are spreading into new regions as climate changes occur. Diseases such as plague, cholera, and hemorrhagic Fevers (Bolivian, Ebola, Lassa, Marburg, etc.) continue to erupt occasionally. Surprisingly, new inFectious agents called prions have recently joined the previously known agents: viruses, bac- teria, protozoa, and helminths (worms). There is strong evidence that prions are the cause oF spongiForm encephalopathies, e.g., bovine spongiForm encephalopathy (BSE, “mad cow disease”), CreutzFeldt-Jakob disease (CJD), kuru, and scrapie in sheep [168]. Recent popular books have given us exciting accounts oF the emergence and de- tection oF new diseases [82, 168, 170, 183]. It is clear that human or animal invasions Received by the editors March 6, 2000; accepted for publication (in revised form) May 7, 2000; published electronically October 30, 2000.
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This note was uploaded on 02/10/2012 for the course MTH 487 taught by Professor Jhonopera during the Spring '11 term at Cleveland State.

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disease modsurvey - SIAM REVIEW Vol. 42, No. 4, pp. 599653...

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