This question is about modelling the spread of a computer virus to predict the number of computers in a city which will be infected by the virus.

A systems analyst defines the following variables in a model:

• $t$ is the number of days since the first computer was infected by the virus.
• $Q\left(t\right)$ is the total number of computers that have been infected up to and including day $t$.

The following data were collected:

A model for the early stage of the spread of the computer virus suggests that

$Q\text{'}\left(t\right)=\beta NQ\left(t\right)$

where $N$ is the total number of computers in a city and $\beta$ is a measure of how easily the virus is spreading between computers. Both $N$ and $\beta$ are assumed to be constant.

The data above are taken from city X which is estimated to have $2.6$ million computers.
The analyst looks at data for another city, Y. These data indicate a value of $\beta =9.64\times {10}^{-8}$.

An estimate for $Q\prime \left(t\right), t\ge 5$, can be found by using the formula:

$Q\text{'}\left(t\right)\approx \frac{Q\left(t+5\right)-Q\left(t-5\right)}{10}$.

The following table shows estimates of $Q\text{'}\left(t\right)$ for city X at different values of $t$.

An improved model for $Q\left(t\right)$, which is valid for large values of $t$, is the logistic differential equation

$Q\text{'}\left(t\right)=kQ\left(t\right)\left(1-\frac{Q\left(t\right)}{L}\right)$

where $k$ and $L$ are constants.

Based on this differential equation, the graph of $\frac{Q\text{'}\left(t\right)}{Q\left(t\right)}$ against $Q\left(t\right)$ is predicted to be a straight line.

In this question you will explore possible models for the spread of an infectious disease

An infectious disease has begun spreading in a country. The National Disease Control Centre (NDCC) has compiled the following data after receiving alerts from hospitals.

A graph of $n$ against $d$ is shown below.

The NDCC want to find a model to predict the total number of people infected, so they can plan for medicine and hospital facilities. After looking at the data, they think an exponential function in the form $n=a{b}^d$ could be used as a model.

Use your answer to part (a) to predict

The NDCC want to verify the accuracy of these predictions. They decide to perform a ${\mathrm{\chi <!--\; \chi \; -->}}^2$ goodness of fit test.

The predictions given by the model for the first five days are shown in the table.

In fact, the first day when the total number of people infected is greater than 1000 is day 14, when a total of 1015 people are infected.

Based on this new data, the NDCC decide to try a logistic model in the form $n=\frac{L}{1+c{e}^{-<!--\; -\; -->kd}}$.

Use the data from days 1–5, together with day 14, to find the value of

An estate manager is responsible for stocking a small lake with fish. He begins by introducing $1000$ fish into the lake and monitors their population growth to determine the likely carrying capacity of the lake.

After one year an accurate assessment of the number of fish in the lake is taken and it is found to be $1200$.

Let $N$ be the number of fish $t$ years after the fish have been introduced to the lake.

Initially it is assumed that the rate of increase of $N$ will be constant.

When $t=8$ the estate manager again decides to estimate the number of fish in the lake. To do this he first catches $300$ fish and marks them, so they can be recognized if caught again. These fish are then released back into the lake. A few days later he catches another $300$ fish, releasing each fish after it has been checked, and finds $45$ of them are marked.

Let $X$ be the number of marked fish caught in the second sample, where $X$ is considered to be distributed as $\text{B}\left(n, p\right)$. Assume the number of fish in the lake is $2000$.

The estate manager decides that he needs bounds for the total number of fish in the lake.

The estate manager feels confident that the proportion of marked fish in the lake will be within $1.5$ standard deviations of the proportion of marked fish in the sample and decides these will form the upper and lower bounds of his estimate.

The estate manager now believes the population of fish will follow the logistic model $N\left(t\right)=\frac{L}{1+C{e}^{-kt}}$ where $L$ is the carrying capacity and $C, k>0$.

The estate manager would like to know if the population of fish in the lake will eventually reach $5000$.

Consider the functions $f$, $g$ : $\mathbb{R}\times <!--\; \times \; -->\mathbb{R}\to <!--\; \to \; -->\mathbb{R}\times <!--\; \times \; -->\mathbb{R}$ defined by

$f\left(\left(x\text{,}y\right)\right)=\left(x+y,x-<!--\; -\; -->y\right)$ and $g\left(\left(x\text{,}y\right)\right)=\left(xy,x+y\right)$.

A suitable site for the landing of a spacecraft on the planet Mars is identified at a point, $\mathbf{\text{A}}$. The shortest time from sunrise to sunset at point $\mathbf{\text{A}}$ must be found.

Radians should be used throughout this question. All values given in the question should be treated as exact.

Mars completes a full orbit of the Sun in $669$ Martian days, which is one Martian year.

On day $t$, where $t\in \mathrm{\mathbb{Z}}$, the length of time, in hours, from the start of the Martian day until sunrise at point $\text{A}$ can be modelled by a function, $R\left(t\right)$, where

$R\left(t\right)=a \sin \left(bt\right)+c, t\in \mathrm{ℝ}$.

The graph of $R$ is shown for one Martian year.

Mars completes a full rotation on its axis in $24$ hours and $40$ minutes.

The time of sunrise on Mars depends on the angle, $\delta$, at which it tilts towards the Sun. During a Martian year, $\delta$ varies from $-0.440$ to $0.440$ radians.

The angle, $\omega$, through which Mars rotates on its axis from the start of a Martian day to the moment of sunrise, at point $\text{A}$, is given by $\cos \omega =0.839 \tan \delta$, $0\le \omega \le \pi$.

Use your answers to parts (b) and (c) to find

Let $S\left(t\right)$ be the length of time, in hours, from the start of the Martian day until sunset at point $\text{A}$ on day $t$. $S\left(t\right)$ can be modelled by the function

$S\left(t\right)=1.5 \sin (0.00939t+2.83)+18.65$.

The length of time between sunrise and sunset at point $\text{A}$, $L\left(t\right)$, can be modelled by the function

$L\left(t\right)=1.5 \sin (0.00939t+2.83)-1.6 \sin (0.00939t)+d$.

Let $f\left(t\right)=1.5 \sin (0.00939t+2.83)-1.6 \sin (0.00939t)$ and hence $L\left(t\right)=f\left(t\right)+d$.

$f\left(t\right)$ can be written in the form $\text{Im}(z_1-z_2)$ , where $z_1$ and $z_2$ are complex functions of $t$.

This question explores methods to determine the area bounded by an unknown curve.

The curve $y=f\left(x\right)$ is shown in the graph, for $0⩽<!--\; ⩽\; -->x⩽<!--\; ⩽\; -->4.4$.

The curve $y=f\left(x\right)$ passes through the following points.

It is required to find the area bounded by the curve, the $x$-axis, the $y$-axis and the line $x=4.4$.

One possible model for the curve $y=f\left(x\right)$ is a cubic function.

A second possible model for the curve $y=f\left(x\right)$ is an exponential function, $y=p{\text{e}}^{qx}$, where $p\text{,}q\in <!--\; \in \; -->\mathbb{R}$.