Page - 227 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

8.3 SpreadingofDisease:ASystemofFirstOrderODEs 227 and I categories. The probability that people meet in pairs during a time T is (by the empirical frequency definition of probability) equal to m/n, i.e., the number of successes divided by the number of possible outcomes. From such statistics we normallyderivequantitiesexpressedperunit time, i.e.,herewewant theprobability perunit time,μ, which is foundfromdividingbyT :μ=m/(nT). Given the probabilityμ, the expected number of meetings per time interval of SI possible pairs of people is (from basic statistics) μSI. During a time interval Δt, there will be μSIΔt expected number of meetings between susceptibles and infected people such that the virus may spread. Only a fraction of the μΔtSI meetings are effective in the sense that the susceptible actually becomes infected. Counting thatmpeopleget infected inn suchpairwisemeetings(say5are infected from1000meetings),wecanestimatetheprobabilityofbeinginfectedasp=m/n. Theexpectednumberofindividualsin theScategorythat inatimeintervalΔt catch the virus and get infected is thenpμΔtSI. Introducinga new constantβ =pμ to savesomewriting,we arriveat the formulaβΔtSI. The value of β must be known in order to predict the future with the disease model.Onepossibility is toestimatep andμ fromtheirmeanings in the derivation above. Alternatively,we can observe an “experiment” where there areS0 suscepti- blesandI0 infectedat somepoint in time.Duringa timeintervalT wecount thatN susceptibleshavebecomeinfected.Using (8.9)as a roughapproximationof howS has developedduring timeT (and nowT is not necessarily small, but we use (8.9) anyway),weget N =βTS0I0 ⇒ β= N TS0I0 . (8.11) We need an additional equation to describe the evolution of I(t). Such an equation is easy to establish by noting that the loss in the S category is a correspondinggain in the I category.Moreprecisely, In+1 −In=βΔtSnIn . (8.12) However, there is also a loss in the I category because people recover from the disease. Suppose that we can measure thatmout ofn individuals recover in a time periodT (say 10 of 40 sick people recoverduringa day:m= 10,n= 40,T = 24 h). Now,γ =m/(nT) is the probability that one individual recovers in a unit time interval. Then (on average)γΔtI infected will recover in a time intervalΔt. This quantity represents a loss in the I category and a gain in the R category. We can thereforewrite the total change in the I categoryas In+1 −In=βΔtSnIn−γΔtIn . (8.13) The change in the R category is simple: there is always an increasegot fromthe Icategory: Rn+1 −Rn=γΔtIn . (8.14)