Journal of Theoretical Biology 237 (2005) 302–315 Modeling polio as a disease of development Svetlana Bunimovich-Mendrazitsky, Lewi Stone à Biomathematics Unit, Department of Zoology, Tel-Aviv University, Ramat Aviv 69978, Israel Received 14 July 2004; received in revised form 21 April 2005; accepted 22 April 2005 Available online 21 June 2005 Abstract Poliomyelitisisadiseasewhichbegantoappearinepidemicproportionsinthelate19thcentury,paradoxically,justatthetime when living conditions and developments in health were transforming enormously for the better. We present a simple age-class model that explains this ‘‘disease of development’’ as a threshold phenomenon. Epidemics arise when improved conditions in hygiene are able to reduce disease transmission of polio amongst children below a critical threshold level. This generates a large susceptibleadultpopulationinwhich,underappropriateconditions,epidemicscanpropagate.Thepoliomodelisanalysedinterms of its bifurcation properties and in terms of its non-equilibrium outbreak dynamics. r 2005 Elsevier Ltd. All rights reserved. Keywords: Disease of development; Polio; Age-structured model; Threshold; Contact rate; Environmental factor; Epidemic 1. Introduction ‘‘Poliomyelitis’’isanepidemiologicaldiseasethathas accompanied humankind throughout history. The ear- liestidentifiablereferencetoparalyticpoliomyelitisisan Egyptian stone engraving that dates back to more than 3500 years ago, and depicts a crippled young man, apparentlyapriest,withallthecharacteristicfeaturesof polio (Paul, 1971). The name itself is derived from the Greek words ‘‘polios’’, or grey (referring to the grey matterofthenervoussystem)and‘‘myelos’’formarrow (referring to the myelin sheath membrane that sur- rounds the spinal cord) (Thomas and Robbins, 1997). Polio is also a prominent example of what is now referredtoasadiseaseofdevelopment(MillerandGay, 1997; Krause, 1998; Sutter et al., 1999). This is because inthelate19thandearly20thcenturies,duringaperiod of intense industrial development, social revolution and increased hygiene, there was a large increase of poliomyelitis with epidemics of a scale never seen previously (Figs.1and2). Poliomyelitis is caused by poliovirus, which invades local lymphoid tissue and enters the blood stream. Poliovirus enters through the mouth, attaches to receptors on the epithelium of the throat and intestine, and replicates inside these cells. Polioviruses are spread directly or indirectly from person to person by droplets oraerosolsandbyfecalcontaminationofhands,eating utensils, milk, food and water (Dowdle and Birming- ham, 1997). Exposure to poliomyelitis results in one of the following consequences: inapparent infection with- out symptoms (72% of people), minor illness (24%), non-paralytical poliomyelitis (4%) or paralytic polio- myelitis (o1%) (Sutter et al., 1999). Paralytic polio- myelitis is a severe form of the disease which occurs when a systemic infection moves to the central nervous system(CNS)anddestroysneuronalcells.Althoughthe paralytic form is an infrequent manifestation of polio, obviouslyalarge-scaleoutbreakofthediseasewithtens of thousands of polio cases, can give rise to a large number of paralytic cases. There is no simple all-encompassing theory that is capableofexplainingthehistoryanddynamicsofpolio epidemics. The best known theory is based on the somewhat controversial observation that the ratio of ARTICLE IN PRESS www.elsevier.com/locate/yjtbi 0022-5193/$-see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2005.04.017 à Correspondingauthor.Tel.:+97236409806;fax:+97236409403. E-mail address: lew521@yahoo.com (L. Stone).