模型假设
-
被研究人群是封闭的, 总人为
NN
N
，病人、健康人和移出者 (有免疫能力, 移出感染系统的人, 病人治愈后为移出者) 的比例分别为
i(
t
)
,
s
(
t
)
,
r
(
t
)
i (t ), s (t ), r (t )
i
(
t
)
,
s
(
t
)
,
r
(
t
)
-
病人的日接触率
λ\lambda
λ
, 日治愈率
μ\mu
μ
, 接触数
σ=
λ
μ
\sigma = \displaystyle{\frac{\lambda}{\mu}}
σ
=
μ
λ
模型建立
s
(
t
)
+
i
(
t
)
+
r
(
t
)
=
1
,
(
1
)
s (t ) + i(t ) + r (t ) = 1, \quad (1)
s
(
t
)
+
i
(
t
)
+
r
(
t
)
=
1
,
(
1
)
N
[
i
(
t
+
Δ
t
)
−
i
(
t
)
]
=
λ
N
s
(
t
)
i
(
t
)
Δ
t
−
μ
N
i
(
t
)
Δ
t
,
(
2
)
N[i(t+\Delta t)-i(t)]=\lambda N s(t) i(t) \Delta t-\mu N i(t) \Delta t, \quad (2)
N
[
i
(
t
+
Δ
t
)
−
i
(
t
)
]
=
λ
N
s
(
t
)
i
(
t
)
Δ
t
−
μ
N
i
(
t
)
Δ
t
,
(
2
)
N
[
s
(
t
+
Δ
t
)
−
s
(
t
)
]
=
−
λ
N
s
(
t
)
i
(
t
)
Δ
t
,
(
3
)
N[s(t+\Delta t)-s(t)]=-\lambda N s(t) i(t) \Delta t, \quad (3)
N
[
s
(
t
+
Δ
t
)
−
s
(
t
)
]
=
−
λ
N
s
(
t
)
i
(
t
)
Δ
t
,
(
3
)
⟹
\Longrightarrow
⟹
{
d
i
d
t
=
λ
s
i
−
μ
i
d
s
d
t
=
−
λ
s
i
i
(
0
)
=
i
0
,
s
(
0
)
=
s
0
i
0
+
s
0
≈
1
(
通常
r
(
0
)
=
r
0
很小
)
\begin{cases} {\frac{d i}{d t}}=\lambda s i-\mu i \\ {\frac{d s}{d t}}=-\lambda s i \\ i(0)=i_{0}, \ s(0)=s_{0} \\ i_{0}+s_{0} \approx 1\left(\text { 通常 } r(0)=r_{0} \text { 很小 }\right) \end{cases}
⎩
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎧
d
t
d
i
=
λ
s
i
−
μ
i
d
t
d
s
=
−
λ
s
i
i
(
0
)
=
i
0
,
s
(
0
)
=
s
0
i
0
+
s
0
≈
1
(
通常
r
(
0
)
=
r
0
很小
)
无法求出
i
(
t
)
,
s
(
t
)
i(t ), s(t )
i
(
t
)
,
s
(
t
)
的解析解, 在相平面
s
−
t
s-t
s
−
t
上研究解的性质.
相轨线
i
(
s
)
i ( s )
i
(
s
)
及其分析
消去
d
t
d t
d
t
,
σ
=
λ
μ
\sigma=\displaystyle{\frac{\lambda}{\mu}}
σ
=
μ
λ
,
⟹
\Longrightarrow
⟹
{
d
i
d
s
=
1
σ
⋅
s
−
1
i
∣
s
=
s
0
=
i
0
\begin{cases} \displaystyle{\frac{d i}{d s}=\frac{1}{\sigma \cdot s}-1} \\ \left.i\right|_{s=s_{0}}=i_{0} \end{cases}
⎩
⎨
⎧
d
s
d
i
=
σ
⋅
s
1
−
1
i
∣
s
=
s
0
=
i
0
⟹
\Longrightarrow
⟹
相轨线:
i
(
s
)
=
(
s
0
+
i
0
)
−
s
+
1
σ
ln
s
s
0
i(s)=\left(s_{0}+i_{0}\right)-s+\frac{1}{\sigma} \ln \frac{s}{s_{0}}
i
(
s
)
=
(
s
0
+
i
0
)
−
s
+
σ
1
ln
s
0
s
定义域
D
=
{
(
s
,
i
)
∣
s
≥
0
,
i
≥
0
,
s
+
i
≤
1
}
D=\{(s, i) \mid s \geq 0, i \geq 0, s+i \leq 1\}
D
=
{
(
s
,
i
)
∣
s
≥
0
,
i
≥
0
,
s
+
i
≤
1
}
s
(
t
)
s(t)
s
(
t
)
单调减.
t
→
∞
t \rightarrow \infty
t
→
∞
时,
i
→
0
i \rightarrow 0
i
→
0
.
⟹
\Longrightarrow
⟹
t
→
∞
t \rightarrow \infty
t
→
∞
时
s
0
+
i
0
−
s
∞
+
1
σ
ln
s
∞
s
0
=
0
s_{0}+i_{0}-s_{\infty}+\frac{1}{\sigma} \ln \frac{s_{\infty}}{s_{0}}=0
s
0
+
i
0
−
s
∞
+
σ
1
ln
s
0
s
∞
=
0
P
1
:
s
0
>
1
σ
P_{1}: s_{0}>\displaystyle{\frac{1}{\sigma}}
P
1
:
s
0
>
σ
1
⟹
i
(
t
)
\Longrightarrow i(t)
⟹
i
(
t
)
先升后降至
0
0
0
⟹
\Longrightarrow
⟹
传染病蔓延;
P
2
:
s
0
<
1
σ
P_{2}: s_{0}<\displaystyle{\frac{1}{\sigma}}
P
2
:
s
0
<
σ
1
⟹
i
(
t
)
\Longrightarrow i(t)
⟹
i
(
t
)
单调降至
0
0
0
⟹
\Longrightarrow
⟹
传染病不蔓延.
1
σ
\displaystyle{\frac{1}{\sigma}}
σ
1
—— 阈值
预防传染病蔓延的手段
-
提高阈值
1σ
⟹
\displaystyle{\frac{1}{\sigma }}\Longrightarrow
σ
1
⟹
降低
σ(
=
λ
μ
)
⟹
λ
↓
,
μ
↑
\sigma \left(=\displaystyle{\frac{\lambda}{\mu }} \right) \Longrightarrow \lambda \downarrow, \ \mu \uparrow
σ
(
=
μ
λ
)
⟹
λ
↓
,
μ
↑
-
λ\lambda
λ
(日接触率)
↓⟹
\downarrow \ \Longrightarrow
↓
⟹
卫生水平
↑\uparrow
↑
-
μ\mu
μ
(日治愈率)
↑\uparrow
↑
⟹\Longrightarrow
⟹
医疗水平
↑\uparrow
↑
-
-
降低
s0
s_{0}
s
0
⇒s
0
+
i
0
+
r
0
=
1
\displaystyle{\xRightarrow{s_{0}+i_{0}+r_{0}=1}}
s
0
+
i
0
+
r
0
=
1
提高
r0
r_{0}
r
0
⟹\Longrightarrow
⟹
群体免疫
σ
\sigma
σ
的估计
s
0
+
i
0
−
s
∞
+
1
σ
ln
s
∞
S
0
=
0
\displaystyle{s_{0}+i_{0}-s_{\infty}+\frac{1}{\sigma} \ln \frac{s_{\infty}}{S_{0}}=0}
s
0
+
i
0
−
s
∞
+
σ
1
ln
S
0
s
∞
=
0
, 忽略
i
0
i_0
i
0
⟹
\Longrightarrow
⟹
σ
=
ln
s
0
−
ln
s
∞
s
0
−
s
∞
\sigma=\frac{\ln s_{0}-\ln s_{\infty}}{s_{0}-s_{\infty}}
σ
=
s
0
−
s
∞
ln
s
0
−
ln
s
∞