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Figure 2: Solutions to system (5) - (7) with initial conditions s (0) = 0.999, i (0) = 0.001, and r (0) = 0 for R 0 = 1.5, 2, 3, and 4 obtained with Euler ’ s method. The corresponding (nondimensional) times τ* of the maximum infectious proportion are 11.45, 6.825, 3.900, and 2.775. There are also the times with s passes through the herd immunity threshold s * = 1/ R 0 . The larger R 0 is, the steeper the initial increase of the infectious proportion (red curves), the larger its maximum, and the earli er the time τ* that the maximum occurs. The dotted black lines indicate that τ* also corresponds to the time at which the susceptible proportion passes through the herd immunity threshold s *.

ingly more difficult to find people who still remain susceptible.

continue to become infected, their number is now small enough that the outbreak dies out because it cannot sustain the transmission chains necessary to grow. This threshold is often seen from the recov ered point of view with no active cases so that r = 1 - s , which we see from s + i + r = 1 with i = 0. The herd immunity threshold is then said to be when r > 1 - s * = 1 - (1/ R 0 ), which is the second most com mon aspect of modeling to appear in media written for a general audience. This can be achieved through natural immunity from prior infections, but it also gives a target proportion to vaccinate re quired to stem the tide of an outbreak. While peo ple will still become infected above this threshold, the epidemic phase has ended and the number of people with active infections will begin to de crease.

To see when this maximum occurs, it is necessary to solve the system. In Figure 2, we show approxi mate solutions using Euler ’ s method for different values of R 0 . Since our system gives the derivatives of our unknown functions, if we start with initial conditions with 0.1% of the population infectious ( i (0) = 0.001) and the rest of the population suscepti ble ( s (0) = 0.999 and r (0) = 0), we may use the derivates to construct their initial tangent lines and use these to advance them one step forward in time. The maximum of the infectious proportion coin cides with the time that the susceptible proportion passes through s * = 1/ R 0 . We also see that as R 0 increases, the time at which the maximum occurs decreases, and the size of the maximum increases. This slowing of an epidemic through reduced sus ceptibility is called herd immunity . While the com munity still has susceptible people and people still

Conclusions

While the techniques needed to solve the model (that is, approximately solve) may lie beyond a

Virginia Mathematics Teacher vol. 47, no. 1

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