Around November $2019$, there was an outbreak of COVID-$19$ in Wuhan, China, caused by the coron

Question

Around November $2019$, there was an outbreak of COVID-$19$ in Wuhan, China, caused by the coronavirus SARS-CoV-$2$ virus, which subsequently spread to the rest of the world. On $23$ January $2020$, there was $1$ confirmed case in Singapore. $70$ days later, there were $1049$ confirmed cases. A student is interested to study the spread of the virus in Singapore and uses the model $\frac{\mathrm{d}N}{\mathrm{d}t} = rN$, where $r$ is a positive constant, $N$ denotes the total number of confirmed cases, and $t$ is the number of days after $23$ January $2020$. Solve the differential equation to express $N$ in terms of $t$.[4] Use the model to estimate the number of days it will take to reach $1000000$ cases.[1] To contain the spread of the virus, Singapore implemented strict circuit breaker lockdown measures starting on $7$ April $2020$, when the number of confirmed cases stood at $1481$. By the time the measures were lifted $55$ days later, the number of confirmed cases had surged to $35292$. To study the spread of the virus during this period, the student uses a second model $\frac{\mathrm{d}N}{\mathrm{d}u} = sN$, where $s$ is a positive constant, and $u$ is the number of days after $7$ April $2020$. Find the value of $s$.[2] By comparing the values of $r$ and $s$, comment on whether the lockdown measures were effective in containing the spread of the virus.[1] Give a reason why neither model can estimate the actual number of confirmed cases accurately.[1] On $30$ December $2020$, Singapore became the first country in Asia to start its COVID-$19$ vaccination campaign. The student now uses a third model $\frac{\mathrm{d}N}{\mathrm{d}v} = aN \left( 1 - \frac{N}{b} \right)$, where $a$ and $b$ are positive constants, and $v$ is the number of days after $30$ December $2020$. Find the maximum rate of change of $N$ in terms of $a$ and $b$.[3]

Accepted Answer

(a)$N=e^{0.0994t}$ (b)140 days (c)0.0577 (f)$\frac{ab}{4}$

2025 YIJC P1 Q13

2025 YIJC P1 Q13