Math 443/543 – Fall 2020 – Extended Homework

1（15分）我们在课堂上讨论的原始SIR模型假设

𝑆+𝐼→2𝐼，
𝐼 → 𝑅,
𝑆 → 𝑅,
𝑅 → 𝑆.

a） 接种疫苗的原因是什么？解释原因。
b） 写下相应的速率方程。然后用守恒定律

c） 简化问题的稳态是什么？你有什么限制吗

d） 你找到的一个稳态是I=0。在什么条件下是稳定的

e） 你发现的一个稳态是≠0。这是在什么条件下

f） 长期目标是控制感染人数

2（20点）带有“生命动力学”的SIR模型由
𝑑𝑆
𝑑𝑡 = −𝛽𝐼𝑆 + 𝑚(𝑆 + 𝐼 + 𝑅) − 𝑚𝑆
𝑑𝐼
𝑑𝑡 = 𝛽𝐼𝑆 − 𝑔𝐼 − 𝑚𝐼
𝑑𝑅
𝑑𝑡 = 𝑔𝐼 − 𝑚𝑅

a) Describe the physical meaning of each term and parameter in the above system.
b) Show that the system of equations can be derived from the law of mass action. Do
this by finding the reactions that give rise to these equations. (You will have an
𝑆 → 𝑆 equation.) For each reaction, explain what assumptions were made to
produce the given reaction.
c) Find a conservation law for the system and reduce the problem to a system in S
and I only.
d) Nondimensionalize the reduced problem using N0 = S0 + I0 to scale both S and I
(use s and i for the nondimensional variables). The final problem should only
contain three nondimensional parameters: μ = m/(βN0), ρ = I0/N0, and γ = g/(βN0).
Explain why 0 ≤ s ≤ 1 and 0 ≤ i ≤ 1.
e) Find the steady states for the problem. One of them is obtained only if the
parameters satisfy an inequality. Find this inequality.
f) One of the steady states has i = 0. This corresponds to elimination of the disease.
Under what conditions on the parameters is this steady state asymptotically
stable?
g) One of the steady states has i ≠ 0. This is called an epidemic equilibrium. Under
what conditions on the parameters is this steady state asymptotically stable?

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