1 . 在一个传染病流行的群体中,通常有3类人群:
在一个600人的封闭环境中,设第n天S类,I类,R类人群人数分别为
,
,
.其中第1天
,
,
.为了简化模型,我们约定各类人群每天转化的比例参数恒定:
(1)已知对于传染病A有
,
,
.求
,
;
(2)已知对于传染病B有
,
,
.
(Ⅰ)证明:存在常数p,q,使得
是等比数列;
(Ⅱ)已知防止传染病大规模传播的关键途径至少包含:①控制感染人数;②保护易感人群.请选择一项,通过相关计算说明:实际生活中,相较于传染病A需要投入更大力量防控传染病B.
类别 | 特征 |
S类(Susceptible) | 易感染者,体内缺乏有关抗体,与I类人群接触后易变为I类人群. |
I类(Infectious) | 感染者,可以接触S类人群,并把传染病传染给S类人群;康复后成为R类人群. |
R类(Recovered) | 康复者,自愈或者经治疗后康复且体内存在相关抗体的I类人群;若抗体存在时间有限,可能重新转化为S类人群. |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d72f2cbabcb955a433e99bf0ee8ec020.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e15526f7c892333030073b85fc3baee6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bf71cf37199d1275ecab9bec0854191.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58cb9d3dc465add3927c6413c4e67921.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b3cf37e52060e71bb710f4a54addeb7.png)
S类→I类占当天S类比例 | I类→R类占当天I类比例 | R类→S类占当天R类比例 |
![]() | ![]() | ![]() |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3b41a1596f5bafd4126cdfa27cab1b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11048760a97aaf9b0e9bb3c1f7d82f65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf09e3b9c967b0910007878ee1ed861e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d1a0fd1ad044a9ecfcba672779bd678.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e15526f7c892333030073b85fc3baee6.png)
(2)已知对于传染病B有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/553db32eb0934af3938b00f1391a62e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7cb43dbf4bfeaa09725d4747b5220d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/162f2427b62ce62e0af1dcf7803df960.png)
(Ⅰ)证明:存在常数p,q,使得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aba455b290aa6f2ebd42f764b532c815.png)
(Ⅱ)已知防止传染病大规模传播的关键途径至少包含:①控制感染人数;②保护易感人群.请选择一项,通过相关计算说明:实际生活中,相较于传染病A需要投入更大力量防控传染病B.
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