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POPULATIONS 



differing from one another with respect to 

 mode of transmission, incubation period, 

 period of infectivity, immunity, case fatal- 

 ity, etc. The studies on infectious disease 

 have taken two forms: one, the theoretical 

 analysis of epidemic spread; and the other, 

 the experimental investigation of controlled 

 epidemics among populations of laboratory 

 animals ..." It is the analysis of epi- 

 demic spread that is reported here. 



Considerable insight can be gained into 

 the development of a single epidemic 

 wave by a general, theoretical treatment of 

 the dissemination of an infectious disease, 

 provided certain simpUfying assumptions 

 are made. This can be approached, as did 

 Frost (see Zinsser and Wilson, 1932), by 

 an arithmetical method involving finite dif- 

 ferences, or by methods based on the cal- 

 culus, as did Soper (1929). We review 

 here Frost's method as presented by Jor- 

 dan and Burrows. 



If C = the number of cases reported for 

 a particular disease, S = the number of 

 susceptible hosts, and N = the contacts 

 per twenty-four hour period, then the con- 

 tact rate per day, r, is given by the formula 



N = rCS 



CS 



The following assumptions are made: 

 (1) that each case is infectious; (2) that 

 one exposure contact produces the disease 

 in an individual who is susceptible, and 

 (3) that the twenty- four hour unit of time 

 is short enough so that S and C do not 

 change markedly during this interval. 

 Granting these not unreasonable assump- 

 tions, the number of contacts per unit of 

 time, t, is 



Nt = rCSt 



From this it follows that the probability of 

 contact, p, is 



Nt ^,^ 

 p = -g- = rCt 



and the probability of avoiding contact, q, 

 is 



q=i -P=i - 



There are 1/t units of time for the entire 

 period, and therefore the chance of avoid- 

 ing contact over this period, Q, is 



1 

 Q = (1 - rCt)* = e-'C 



and the number of new cases infected dur- 

 ing the twenty-four hour period, PS, is 



PS = (1 -e-'C)S 



Jordan and Burrows construct a hypo- 

 thetical epidemic wave by substituting 

 certain values in the last equation. First, 

 they assmne that the incubation period of 

 the disease is twenty-four hours, or, in 

 other words, "the contact of one day is the 

 case of the next." They start with an illus- 

 trative population of 10,000 susceptibles, 

 one case, and a contact rate, r, of 0.0002. 

 For the first day the formula takes this 

 form: 



(1 - e-0002) 10,000 = 2 (new cases) 



For the second day: (1 -e-'"'"'')9998 = 6. 

 For the third day: (1 -e-""*)9992 = 18, 

 and so on for the course of the complete 

 epidemic wave. It is possible to introduce 

 various modifications into this treatment 

 e.g., the introduction of case fataHty and 

 the development of immunity, the exten- 

 sion of the incubation period, and so forth. 



The significant point is that such a hypo- 

 thetical epidemic shows "a remarkable 

 similarity to observed epidemics of disease, 

 and, although the factors entering into the 

 determination of the value r are highly 

 complex, it is evident that the probabiHty 

 of chance contact is a factor of primary 

 importance in the evolution of the epidemic 

 wave" (Jordan and Burrows). McKen- 

 drick (1940) has shown that if the host 

 population consists largely or entirely of 

 susceptibles, this probabiHty of chance con- 

 tact is high and the disease spreads rapidly. 

 As the number of susceptibles is reduced 

 through conversion to actual cases, fatali- 

 ties, and immunes, the probability dimin- 

 ishes and the epidemic subsides. 



A schematized representation of the 

 course of an epidemic wave adopted from 

 Jordan and Burrows is presented as Figure 

 134, in which the ordinate depicts num- 

 bers; the abscissa, time; the upper curve, 

 numbers of susceptibles; and the lower 

 curve, nvmiber of cases. The points made 

 by this diagram are self-evident. They af- 

 ford both an extension of our arithmetical 

 example as well as a summary of this short 

 discussion. 



We now present several illustrations 

 of host-parasite interactions among insect 

 populations. 



