72:, so that an infection rate at a given time was proportional to the number of susceptible and infectious ones at that time. It is effective for moderately large populations, but it does not take into account multiple infections that come into contact with the same individual. Therefore, in small populations, the model greatly overestimates the number of susceptibles that become infected.
53:
During the 1920s, mathematician Lowell Reed and physician Wade
Hampton Frost developed a binomial chain model for disease propagation, used in their biostatistics and epidemiology classes at Johns Hopkins University. Despite not having published their results, several other academics have done them
141:
With this information, a simple formula allows the calculation of how many individuals will be infected, and how many immune, in the next time interval. This is repeated until the entire population is immune, or no infective individuals remain. The model can then be run repeatedly, adjusting the
75:
Reed and Frost modified the Soper model to account for the fact that only one new case would be produced if a particular susceptible includes contact with two or more cases. The Reed-Frost model has been widely used and served as the basis for the development of more detailed disease propagation
88:
The Reed–Frost model is one of the simplest stochastic epidemic models. It was formulated by Lowell Reed and Wade Frost in 1928 (in unpublished work) and describes the evolution of an infection in generations. Each infected individual in generation t (t = 1,2,...) independently infects each
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susceptible individual in the population with some probability p. The individuals that become infected by the individuals in generation t then constitute generation t + 1 and the individuals in generation t are removed from the epidemic process.
45:. While originally presented in a talk by Frost in 1928 and used in courses at Hopkins for two decades, the mathematical formulation was not published until the 1950s, when it was also made into a TV episode.
452:
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Each individual has a fixed probability of coming into adequate contact with any other specified individual in the group within one time interval, and this probability is the same for every member of the
68:
in 1929 for measles. Soper's model was deterministic, in which all members of the population were equally susceptible to disease and had the ability to transmit disease. The model is also based on the
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individual in a given period, will develop the infection and will be infectious to others only within the following time period; in subsequent time periods, he is wholly and permanently immune.
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In the program, Lowell Reed, after explaining the formal definition of the model, demonstrates its application through experimentation with marbles of different colors.
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The infection is spread directly from infected individuals to others by a certain type of contact (termed "adequate contact") and in no other way.
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is the probability that a person comes in contact with another person in one time-step and that that contact results in disease transmission.
625:{\displaystyle {\begin{aligned}I_{t+1}&=\sum _{k=0}^{S_{t}}{\mathcal {B}}(1-(1-p)^{I_{t}}),\\S_{t+1}&=S_{t}-I_{t+1}\end{aligned}}}
54:
in their studies. It was not until 1950 that mathematical formulation was published and turned into a television program entitled
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84:
This is an example of a "chain binomial" model, a simplified, iterative model of how an epidemic will behave over time.
868:{\displaystyle {\begin{aligned}I_{t+1}&=S_{t}\,(1-(1-p)^{I_{t}}),\\S_{t+1}&=S_{t}\,(1-p)^{I_{t}}\end{aligned}}}
1205:"A box, a trough and marbles: How the Reed-Frost epidemic theory shaped epidemiological reasoning in the 20th century"
160:
1178:
157:– in a large population when the initial number of infecteds is small, an infected individual is expected to cause
154:
42:
1262:"A Unified Analysis of the Final Size and Severity Distribution in Collective Reed-Frost Epidemic Processes"
1038:"A Unified Analysis of the Final Size and Severity Distribution in Collective Reed-Frost Epidemic Processes"
1382:
331:
65:
989:"A note on chain-binomial models of epidemic spread: What is wrong with the Reed-Frost formulation?"
447:. Making use of the random-variable multiplication convention, we can write the Reed–Frost model as
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115:
The individuals are wholly segregated from others outside the group. (It is a closed population.)
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The deterministic limit is (found by replacing the random variables with their expectations),
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281:. Assume all cases recover or are removed in exactly one time-step. Let
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977:, retrieved 21 March 2021. Johns Hopkins Science Review, Baltimore, MD
101:
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1179:"Lowell Reed | Johns Hopkins Bloomberg School of Public Health"
149:
The probability of adequate contact corresponds roughly with R
1131:"Epidemics: the fitting of the first dynamic models to data"
634:
with initial number of susceptible and infected individuals
513:
337:
167:
93:
The Reed–Frost model is based on the following assumptions:
146:, to see how these affect the progression of the epidemic.
1084:"The Interpretation of Periodicity in Disease Prevalence"
1310:(2011). "Epidemics and vaccination on weighted graphs".
308:
represent the number of susceptible individuals at time
928:"An examination of the Reed-Frost theory of epidemics"
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These conditions remain constant during the epidemic.
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individual in the group, after such contact with an
261:represent the number of cases of infection at time
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64:The model is an extension of what was proposed by
219:{\displaystyle {\mathcal {R}}_{0}=\ln(1/(1-p))}
1135:Journal of Contemporary Mathematical Analysis
8:
906:Schwabe CW, Riemann HP, Franti CE. (1977).
889:Mathematical modelling of infectious disease
361:be a Bernoulli random variable that returns
122:The following parameters are set initially:
1209:History and Philosophy of the Life Sciences
1025:– via Elsevier Science Publishing Co.
1260:Picard, Philippe; Lefevre, Claude (1990).
1036:Picard, Philippe; Lefevre, Claude (1990).
1366:. Ohio Supercomputer Center. 29 May 2012.
1323:
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1048:(2). Applied Probability Trust: 269–294.
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131:Number of individuals already immune
1203:Engelmann, Lukas (30 August 2021).
908:Epidemiology in Veterinary Practice
134:Number of cases (usually set at 1)
14:
910:. Lea & Febiger. pp. 258–260
354:{\displaystyle {\mathcal {B}}(x)}
1266:Advances in Applied Probability
1042:Advances in Applied Probability
137:Probability of adequate contact
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673:{\displaystyle (S_{0},I_{0})}
1005:10.1016/0025-5564(87)90034-4
971:Epidemic Theory: What Is It?
56:Epidemic theory: What is it?
1364:"Reed–Frost Epidemic Model"
1129:Dietz, Klaus (3 May 2009).
1399:
1221:10.1007/s40656-021-00445-z
33:put forth in the 1920s by
1334:10.1016/j.mbs.2011.04.003
1148:10.3103/S1068362309020034
987:Jacquez, John A. (1987).
884:Kermack–McKendrick theory
155:basic reproduction number
16:Stochastic epidemic model
1312:Mathematical Biosciences
993:Mathematical Biosciences
43:Johns Hopkins University
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301:{\displaystyle S_{t}}
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254:{\displaystyle I_{t}}
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39:Wade Hampton Frost
27:mathematical model
693:{\displaystyle p}
421:with probability
414:{\displaystyle 0}
394:{\displaystyle x}
381:with probability
374:{\displaystyle 1}
321:{\displaystyle t}
274:{\displaystyle t}
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1383:Epidemiology
1358:
1318:(1): 57–65.
1315:
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1208:
1198:
1186:. Retrieved
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1173:
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1094:(1): 34–73.
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1065:. Retrieved
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230:Mathematics
226:new cases.
80:Description
35:Lowell Reed
1215:(3): 105.
1188:29 October
895:References
106:infectious
66:H.E. Soper
1325:1101.4154
1286:0001-8678
1229:1742-6316
1165:162120980
1157:1934-9416
1141:(2): 97.
1108:0952-8385
1054:0001-8678
1023:0025-5564
999:: 73–82.
944:0018-7143
924:Abbey, H.
839:−
766:−
757:−
600:−
534:−
525:−
485:∑
432:−
205:−
185:
31:epidemics
1377:Category
1342:21536052
1247:34462807
1067:9 August
952:12990130
926:(1952).
878:See also
100:Any non-
1350:1744357
1294:1427536
1238:8404547
1116:2341437
1062:1427536
975:Youtube
49:History
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112:group.
102:immune
1346:S2CID
1320:arXiv
1290:JSTOR
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1058:JSTOR
41:, of
25:is a
1338:PMID
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1225:ISSN
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1019:ISSN
948:PMID
940:ISSN
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