这个作业是完成SIR数学模型

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
已经恢复或被移除,你不能再变得敏感。然而,许多
疾病只产生暂时的免疫力。在这种情况下,康复的人可以
再次变得敏感。在这个SIRS模型中,我们将探讨疫苗接种的效果
易感人群。在这种情况下,反应方程式变成
𝑆+𝐼→2𝐼,
𝐼 → 𝑅,
𝑆 → 𝑅,
𝑅 → 𝑆.
这里假设S(0)=S0,I(0)=I0,R(0)=0。总人口N保持不变。
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
将系统简化为S和I的系统。
c) 简化问题的稳态是什么?你有什么限制吗
必须申请稳态才有效?
d) 你找到的一个稳态是I=0。在什么条件下是稳定的
状态渐近稳定?
e) 你发现的一个稳态是≠0。这是在什么条件下
稳态渐近稳定?
f) 长期目标是控制感染人数
至少,疫苗接种应附加什么条件(如果有的话)
费率?
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
其中S(0)=S0,I(0)=I0,R(0)=0。在这个模型中,我们包含了死亡率/出生率m。
请注意,我们假设死亡率等于出生率,并且所有新出生的婴儿
出生于易感人群。因此,我们系统中的人数N,
保持不变。
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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