By A. Manning

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**Example text**

The proof of this theorem is similar to that for the SIR epidemic model. The tetrahedron T is positively invariant, since no direction vectors at a boundary point are outward. The only equilibrium points, which are along the s axis where (e , i) = (0, 0), are neutrally unstable for s > 1/σ and are neutrally stable for s < 1/σ. The equation ds/dt = −βsi ≤ 0 implies that s (t) is non-increasing and s(t) ≥ 0, so that a unique limit s∞ exists as t → ∞. Since dr/dt = γi ≥ 0 and r(t) is bounded above by 1, the limit r(∞) exists.

9in x 6in – (for Lecture Note Series, IMS, NUS) 46 LNSVol16-chapters1-3 H. W. Hethcote [92]. Because of this major measles epidemic, epidemiologists decided in 1989 that the one-dose vaccination program for measles, which had been used for 26 years, should be replaced with a two-dose program with the ﬁrst measles vaccination at age 12–15 months and the second vaccination at 4–6 years, just before children start school [34]. Reported measles cases declined after 1991 until there were only 137, 100, and 86 reported cases in 1997, 1998, and 1999, respectively.

After the latent period ends, the individual enters the class I of infectives, who are infectious in the sense that they are capable of transmitting the infection. Models with the M and E epidemiological states have behaviors that are analogous to the models without these states. Examples of models with diﬀerent epidemiological states are MSEIR, MSEIRS, SEIR, SEIRS, SIR, SIRS, SEI, SEIS, SI, and SIS. Models such as MSEIRS, SEIRS, SIRS, SEIS, and SIS with a ﬂow back into the susceptible class S are always endemic models and never epidemic models, since there is always a threshold above which the disease remains endemic.