MAT009/MA4609 Healthcare Modelling

1（i）提供以下术语的定义：

•豁免权
•潜伏期
•传染期
•潜伏期[15％]

（ii）考虑一种非致命的传染性儿童疾病，即感染儿童

•有固定数量的儿童，n。
•S表示人口中易感儿童的数量。
•I表示人口中被感染儿童的数量。
•R表示人口中恢复的儿童数量。
•b是感染率（SI接触导致新感染的比例

•r表示每单位时间的恢复率。

[20％]

（iii）陈述传染性儿童期传播的必要条件

[15％]

（iv）现在假设人口不是固定的。让你成为平均值

（但不能解决）S，I和R的变化率。
[20％]

（v）进一步的观察表明，儿童实际上可以反复抓到

S和I的变化率。
[15％]

（vi）建议并简要描述可用于以下目的的替代OR方法

1 (i) Provide definitions for the following terms:

• Immunity
• Latent period
• Infectious period
• Incubation period [15%]

(ii) Consider a non-fatal contagious childhood disease, whereby infected children
will recover from the illness. A differential equation model to track the
dynamics of the spread of such a disease within a population of children is to
be developed with the following assumptions:

• There is a fixed size population of children, n.
• S denotes the number of susceptible children in the population.
• I denotes the number of infected children in the population.
• R denotes the number of recovered children in the population.
• b is the infection rate (the proportion of SI contacts which lead to a new
infection)
• r denotes the recovery rate per unit time.

Clearly stating any assumptions you make, derive (but do not solve) the
necessary equations to capture the rate of change of S, I and R.
[20%]

(iii) State the required conditions for the spread of the contagious childhood
disease to become endemic and briefly describe ways in which health officials
might try to reduce the chances of an endemic.
[15%]

(iv) Now assume that the population is not of a fixed size. Let u be the average
birth/arrival rate of children into the population per unit time and v be the rate
of children leaving the population per unit time. Derive the new equations
(but do not solve) for the rate of change of S, I and R.
[20%]

(v) Further observations suggest that children can in fact repeatedly catch the
illness, thus moving between susceptible and infected states. Building on your
answer to part (iv) above, derive the new equations (but do not solve) for the
rate of change of S and I.
[15%]

(vi) Suggest and briefly describe an alternative OR method that could be used to
more realistically model the spread of a contagious childhood disease.

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