{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,27]],"date-time":"2026-09-27T04:46:55Z","timestamp":1790484415533,"version":"4.1.0"},"reference-count":25,"publisher":"The Royal Society","issue":"1318","license":[{"start":{"date-parts":[[1994,12,29]],"date-time":"1994-12-29T00:00:00Z","timestamp":788659200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/royalsociety.org\/journals\/ethics-policies\/data-sharing-mining\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[1994,12,29]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Time series analysis has revealed two different patterns of smallpox epidemics in Britain in the seventeenth and eighteenth centuries: in large conurbations (exemplified by London) the disease was endemic whereas medium-sized rural towns (exemplified by Penrith, Cumbria) suffered from 5 year epidemics with no cases of smallpox in the inter-epidemic years. The oscillations (epidemics) persisted for over 150 years and it is suggested that both systems were pumped up by regular fluctuations in susceptibility (6j3). Modelling suggests that: (i) the natural frequency of oscillations in large cities is two years and the system is pumped up by a 1 year, seasonal input; (ii) it takes five years to build up a pool of susceptibles in medium-sized towns by new births and epidemics are then triggered by a 5 year input. The equations represent a system that has two components, a basic linear element with the remainder of the system being nonlinear; modelling a progressive increase in 6j8 in London illustrates theoretically how a predominantly linear response changes to a nonlinear response and ultimately to chaos. A variation in susceptibility is a theoretical condition for inducing chaos; the undriven system cannot become chaotic. Modelling populations of progressively increasing size\/density and applying a 1 year or 5 year sinusoidal oscillation in 5j8 illustrates the fundamental distinction in the response of medium-sized rural towns and large cities.<\/jats:p>","DOI":"10.1098\/rstb.1994.0158","type":"journal-article","created":{"date-parts":[[2006,12,18]],"date-time":"2006-12-18T19:03:56Z","timestamp":1166468636000},"page":"407-419","source":"Crossref","is-referenced-by-count":25,"title":["Modelling the different smallpox epidemics in England"],"prefix":"10.1098","volume":"346","author":[{"given":"S. R.","family":"Duncan","sequence":"first","affiliation":[{"name":"Control Systems Centre, Department of Electrical Engineering and Electronics, UMIST, P. O. Box 88, Manchester M60 1QD, U. K."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"S.","family":"Scott","sequence":"additional","affiliation":[{"name":"Department of Economic and Social History, University of Liverpool, Liverpool L69 3BX, U. K."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"C. J.","family":"Duncan","sequence":"additional","affiliation":[{"name":"Department of Environmental and Evolutionary Biology, University of Liverpool, Liverpool L69 3BX, U. K."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"175","reference":[{"key":"p_1","doi-asserted-by":"crossref","unstructured":"Anderson R .M . & M ay R .M . 1991 Infectious diseases of humans. Oxford University Press.","DOI":"10.1093\/oso\/9780198545996.001.0001"},{"key":"p_2","unstructured":"Appleby A.B. 1981 Epidem ics and fam ine in the L ittle Ice Age. In Climate and history (ed. R .I. R otberg & T .K . R ab b ) pp. 6 5 -8 3 . Princeton New Jersey: Princeton U niversity Press."},{"key":"p_3","first-page":"665","article-title":"Seasonality and p erioddoubling bifurcations in an epidem ic model. J. theor","volume":"110","author":"Schw I.B.","year":"1984","journal-title":"Biol."},{"key":"p_4","first-page":"321","article-title":"C hance or chaos (with discussion)? Jl R. statist","volume":"153","year":"1990","journal-title":"Soc. A"},{"key":"p_5","doi-asserted-by":"publisher","DOI":"10.1098\/rspb.1993.0011"},{"key":"p_6","unstructured":"Corfield P.J. 1987 In tro d u ctio n . In The speckled monster smallpox in England 1670-1970. H unstanton Norfolk: W itley Press."},{"key":"p_7","doi-asserted-by":"crossref","unstructured":"Dietz K. 1976 T h e incidence of infectious diseases u nder the influence of seasonal fluctuations. In Lecture notes in biomathematics (ed. J. Berger W. Buhler R. Repges & P. T a n tu ) pp. 1-15. H eidelberg Berlin: Springer-V erlag.","DOI":"10.1007\/978-3-642-93048-5_1"},{"key":"p_8","first-page":"293","article-title":"T im e series analysis of oscillations in a model population: the effects of plague, pestilence and famine. J. theor","volume":"158","author":"Scott S.","year":"1992","journal-title":"Biol."},{"key":"p_9","first-page":"231","article-title":"An hypothesis for the periodicity of smallpox epidemics as revealed by time series analysis. J. theor","volume":"160","author":"Scott S.","year":"1993","journal-title":"Biol."},{"key":"p_10","doi-asserted-by":"publisher","DOI":"10.2307\/2061648"},{"key":"p_11","first-page":"255","article-title":"Smallpox epidemics in cities in Britain. J. interdiscip","volume":"25","author":"Scott S.","year":"1994","journal-title":"Hist."},{"key":"p_12","unstructured":"H) infectious (F) and recovered Forsythe G .E. M alcolm M .A. & M oler C.B. 1977"},{"key":"p_13","unstructured":"and hence immune (Z). Thus T +Z . It is Computer methodsfor mathematical computations. Prentice-Hall."},{"key":"p_14","unstructured":"assumed that the net rate at which infections occur is"},{"key":"p_15","unstructured":"proportional to the number of encounters between"},{"key":"p_16","unstructured":"susceptibles and infectious {SXY (where \/3 is a"},{"key":"p_17","unstructured":"transmission coefficient). Individuals move from"},{"key":"p_18","unstructured":"latent to infectious at a per capita rate a and"},{"key":"p_19","unstructured":"recover so becoming immune at rate The"},{"key":"p_20","unstructured":"dynamics of the infection are then described by the"},{"key":"p_21","unstructured":"following equations: dX\/ dt =fxN -ixX-\/3A(1) dH\/dt = \u00a3XY{\/ x+ a)H-A(2) dY\/ dt = crH - (lx + y)Y-1 A(3) dZ\/dt - y Y -ixZ. A(4)"},{"key":"p_22","unstructured":"The criterion for triggering an epidemic is the"},{"key":"p_23","unstructured":"requirement (Anderson & May 1991) that the"},{"key":"p_24","unstructured":"susceptibles (A) exceed a threshold density X > p"},{"key":"p_25","unstructured":"where"}],"container-title":["Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/royalsocietypublishing.org\/doi\/pdf\/10.1098\/rstb.1994.0158","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/royalsocietypublishing.org\/rstb\/article-pdf\/346\/1318\/407\/83427\/rstb.1994.0158.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/royalsocietypublishing.org\/rstb\/article-pdf\/346\/1318\/407\/83427\/rstb.1994.0158.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,2]],"date-time":"2026-01-02T12:48:41Z","timestamp":1767358121000},"score":1,"resource":{"primary":{"URL":"https:\/\/royalsocietypublishing.org\/rstb\/article\/346\/1318\/407\/18676\/Modelling-the-different-smallpox-epidemics-in"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,12,29]]},"references-count":25,"journal-issue":{"issue":"1318","published-print":{"date-parts":[[1994,12,29]]}},"URL":"https:\/\/doi.org\/10.1098\/rstb.1994.0158","relation":{},"ISSN":["0962-8436","1471-2970"],"issn-type":[{"value":"0962-8436","type":"print"},{"value":"1471-2970","type":"electronic"}],"subject":[],"published":{"date-parts":[[1994,12,29]]}}}