Page - 112 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

112 4 SolvingOrdinaryDifferentialEquations meetings are effective in the sense that the susceptible actually becomes infected. Countingthatmpeopleget infected innsuchpairwisemeetings (say5are infected from 1000meetings), we can estimate the probability of being infected asp D m=n. The expectednumberof individuals in theS category that in a time interval t catch the virus and get infected is thenp tSI. Introducing a new constant ˇDp to savesomewriting,wearriveat the formulaˇ tSI. The value ofˇmust be known in order to predict the futurewith the disease model.Onepossibility is toestimatep and fromtheirmeanings in thederivation above. Alternatively,wecanobservean“experiment”where thereareS0 suscepti- blesandI0 infectedatsomepoint in time.DuringatimeintervalTwecount thatN susceptibleshavebecomeinfected.Using (4.9)asa roughapproximationofhowS hasdevelopedduring timeT (andnowT is not necessarily small, butweuse (4.9) anyway),weget N DˇTS0I0 ) ˇD N TS0I0 : (4.11) Weneedanadditionalequation todescribe theevolutionofI.t/. Suchanequa- tion is easy to establish bynoting that the loss in theScategory is a corresponding gain in the I category.Moreprecisely, InC1 In Dˇ tSnIn : (4.12) However, there is also a loss in the I category because people recover from the disease. Suppose thatwecanmeasure thatmoutofn individuals recover ina time periodT (say 10of 40 sick people recover during a day:m D 10,n D 40,T D 24h). Now, D m=.nT/ is the probability that one individual recovers in a unit time interval. Then (on average) tI infectedwill recover in a time interval t. This quantity represents a loss in the I category and a gain in theR category. We can thereforewrite the total change in the I categoryas InC1 In Dˇ tSnIn tIn : (4.13) The change in theR category is simple: there is always an increase from the I category: RnC1 Rn D tIn : (4.14) Since there is no loss in theRcategory (people are either recovered and immune, ordead),wearedonewith themodelingof this category. In fact,wedonot strictly need theequation(4.14)forR, butextensionsof themodel laterwillneedanequa- tion forR. Dividing by t in (4.13) and (4.14) and letting t ! 0, results in the corre- spondingdifferential equations I 0 DˇSI I; (4.15) and R0 D I : (4.16) Tosummarize,wehavederiveddifferenceequations(4.9)–(4.14),andalternative differential equations (4.15)–(4.16). For reference,we list the complete set of the