S(0)=51,964,000 then as the recovered exponentially increase

S(0)=51,964,000 I(0)=20 R(0)=0Our t which is time in days is zero because no days have passed it is the start of the epidemic. The number of infected is 20 because that is our trace level, the beginning number of infected, of infection and the number of susceptible is 51,964,000 because that is the total population and finally the number of recovered is 0 because none have recovered. In the terms of our scaled variables these conditions are s(t)=S(t)Ns(0)=S(0)Ns(0)=51,964,00051,964,000s(0)=1i(t)=I(t)Ni(0)=I(0)Ni(0)=20 51,964,000i(0)=.00000038i(0)=3.810-7r(t)=R(t)Nr(0)=R(0)Nr(0)=051,964,000r(0)=0Now our complete model is dsdt=-bs(t)i(t)                  s(0)=1didt=bs(t)i(t)-ki(t)                  i(0)=3.810-7drdt=ki(t)                           r(0)=0We have not yet set our values for b and k so we will provide an example. Let’s say the estimated average period of infectiousness is 5 days so that would mean k=15, basically on average then one fifth of the currently infected population becomes noninfectious each day, either through death or recovery. Then if we say that each infected would make a possibly infecting contact every two days then b=12. The following graph shows the solution curves for these choices of b and k. Now if we substitute these numbers into our equations we get dsdt=-1213.810-7 dsdt=-1.910-7didt=1213.810-7-153.810-7didt=6.4610-14drdt=153.810-7drdt=0.7610-7ConclusionAlgebraically from this we can conclude that at the beginning of the epidemic there would be a relatively low level of infection due to the slow rate of b and that eventually at its peak it would not need to rise that much to a higher level due to the amount of days of infectivity. Like the level of infected the level of recovered will be low at the beginning because no one has been infected then as the days continue gradually increase reaching its max and leveling out. Finally the level of susceptibles will begin at a high level then as the recovered exponentially increase the susceptibles will exponentially decrease at the same time, until both functions reach their max and min and level out.Now we must consider that this was an expirment and that the degree of accuracy is relatively small thus one must inlcude all the possible decimal places to allow for this close range of accuracy.From this model one could predict what might happen in future epidemics and establish situations to observe and possibly prepare for these situations.One could extend this to any disease and just substitue the different values specific to the individual disease.

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