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Choose a horizon of 100 days, which is longer than all extinction times in the runs reported here. This independent implementation also avoids executing a sampled event beyond the horizon:

import numpy as np
import matplotlib.pyplot as plt

EPIDEMIC_RNG = np.random.default_rng(2345)

def sir_sim(t_max,X0,beta,gamma,rng=None):
    if rng is None:rng=EPIDEMIC_RNG
    state=np.array(X0,dtype=int);population=state.sum();time=0.
    times=[time];states=[state.copy()]
    while time<t_max and state[1]>0:
        infection=beta*state[0]*state[1]/population;recovery=gamma*state[1]
        rate=infection+recovery
        if rate==0:
            times.append(float(t_max));states.append(state.copy());break
        next_time=time+rng.exponential(1/rate)
        if next_time>t_max:
            times.append(float(t_max));states.append(state.copy());break
        if rng.random()<infection/rate:state+=np.array([-1,1,0])
        else:state+=np.array([0,-1,1])
        time=next_time;times.append(time);states.append(state.copy())
    return np.array(times),np.array(states)

rng = np.random.default_rng(2345)
trajectories = [sir_sim(100, [99,1,0], 1, .25, rng) for _ in range(10)]
for times, states in trajectories:
    plt.step(times, states[:,1], where="post", alpha=.8)
plt.xlabel("Time (days)")
plt.ylabel("Infected individuals")
plt.show()

A step plot represents the piecewise constant infected population between events. The trajectories show both substantial outbreaks and early epidemic extinction despite the large basic reproduction number

R_{0} = β / γ = 4

Figure 1.
Ten original stochastic SIR trajectories, with frequency-dependent transmission and one initial infection
. **Details**.

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Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 52 / 3 / Solution Created 2026-10-03 Updated 2026-10-14

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A microparasite typically multiplies within an infected host on a much shorter time scale than between-host transmission. A useful first model classifies hosts by infection status. A macroparasite, such as an adult parasitic worm, instead requires a count of parasites per host: two infected hosts can carry very different within-host parasite burdens. These are modelling distinctions rather than a rule determined only by an organism's physical size.

For a microparasite with no lasting immunity, a homogeneous SIS model is

\dot{I} = β \frac{S I}{N} - γ I, S + I = N .

(1)

The basic reproduction number is

β / γ

, and the prevalence

I / N

describes the infected-host fraction. A SIR model adds an immune class, and a SEIR model adds a latent class. Contact structure, age, susceptibility, recovery rate and infectiousness can require several such classes. Then a next-generation matrix or operator, rather than a single averaged contact rate, sets the invasion threshold. Stochastic epidemic models are relevant to early extinction and finite populations; network or spatial models represent nonuniform contact opportunities.

For a simple macroparasite burden model, let

p_{j}

be the fraction of hosts carrying

j

adult worms. Assume acquisitions at rate

λ = β L

, independent adult-worm losses at rate

μ j

, and a number

L

of infective stages in a well-mixed environmental reservoir. The immigration-death worm-burden model has

\overset{p}{˙}_{j} = λ p_{j - 1} + μ (j + 1) p_{j + 1} - (λ + μ j) p_{j}, p_{- 1} = 0.

(2)

This birth-death process needs the burden distribution or its probability generating function, not just an infected/not-infected partition. Its mean

m = \sum_{j} j p_{j}

satisfies

\overset{m}{˙} = β L - μ m

. With host number

N

fixed and each adult producing infective stages at rate

σ

, one possible environmental closure is

\dot{L} = σ N m - (d_{L} + βN) L .

(3)

For this asexual, unsaturated transmission approximation, one worm's basic reproduction number is

(σ / μ) βN / (d_{L} + βN)

. Growth above the invasion threshold requires additional density regulation if a finite endemic burden is sought. Sex-dependent mating would change this closure.

With constant external exposure

λ

, the equilibrium burden follows a Poisson distribution with mean

m = λ / μ

, giving infected-host prevalence

1 - e^{- m}

. If exposure rates vary across hosts as a gamma distribution, the Poisson-gamma mixture gives a negative binomial distribution with mean

m

and aggregation parameter

k

Var (j) = m + \frac{m ^{2}}{k}, Pr (j > 0) = 1 - (1 + \frac{m}{k})^{- k} .

(4)

The same mean burden can therefore coexist with very different prevalence and highly infected subgroups. Aggregation of macroparasite burdens affects treatment coverage, burden-dependent host mortality, immunity and parasite fecundity. For sexually reproducing worms, hosts need both sexes for fertile output; a mean alone may not determine it. With independent equally likely sexes and Poisson burden, the probability of both sexes is

(1 - e^{- m /2})^{2}

, illustrating mating-limited macroparasite transmission and its low-burden bottleneck.

Both microparasite and macroparasite models need heterogeneous exposure and host responses. The special extra difficulty for macroparasites is translating a distributed worm burden, and possibly worm sex or developmental stage, into transmission and host harm. Conversely, complicated within-host microparasite processes can also require more than a binary host state when the simple fast-within-host approximation fails.

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Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 52 / 4 / a / Solution Created 2026-10-03 Updated 2026-10-14

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Let

i = I / N

be the infectious proportion in a closed population. With homogeneous frequency-dependent transmission rate

β

and recovery rate

γ > 0

, the SIS model is

\dot{i} = β i (1 - i) - γi .

(1)

The basic reproduction number is the expected number of secondary infections produced by a typical newly infected individual in an otherwise susceptible population under the specified contact conditions. Here the infectious duration has mean

1/ γ

, so

R_{0} = β / γ

For

R_{0} > 1

the attracting endemic equilibrium has

i_{*} = 1 - γ / β

. The homogeneous SIS susceptible fraction is therefore

s_{*} = 1 - i_{*} = \frac{1}{R _{0}} .

(2)

For

R_{0} \leq 1

the physical equilibrium is disease-free,

i_{*} = 0

s_{*} = 1

. At equality approach to zero is algebraic. Thus

s_{*} = 1/ R_{0}

is an endemic relation, not a formula demanding more than one susceptible person per person below the invasion threshold.

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Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 52 / 4 / b / Solution Created 2026-10-03 Updated 2026-10-14

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Use an assortatively mixed two-risk-group SIS model with equal per-person recovery rate

γ

. Let

x, y

be infectious proportions in groups of sizes

pN, (1 - p) N

. Let

c_{H} ≫ c_{L}

be their fixed contact rates and

τ

the transmission probability per contact. Suppose a small fixed fraction

ϵ < 1/2

of high-risk contacts is with the low-risk group. Contact reciprocity in a two-group epidemic requires

pN c_{H} ϵ = (1 - p) N c_{L} ϵ_{L}, ϵ_{L} = \frac{p c _{H} ϵ}{( 1 - p ) c _{L}} .

(1)

For sufficiently small

p

, both groups make most contacts within their own group, and the total cross-group contacts are of order

pN

. Define

A = τ c_{H} (1 - ϵ)

B = τ c_{H} ϵ

D = τ c_{L}

and

C (p) = Bp / (1 - p)

. The forces of infection give

\overset{x}{˙} \overset{y}{˙} = (1 - x) (A x + B y) - γ x, = (1 - y) [C (p) x + (D - C (p)) y] - γ y .

(2)

Here

C (p) < D

and

ϵ_{L} < 1/2

hold in the small-

p

regime. High-risk per-person cross contact is order one but its group size is order

p

; low-risk per-person cross contact is order

p

. These distinctions are needed to count each cross-group partnership consistently.

Linearize about the disease-free state. In infectious fractions the transmission matrix is

M (p) = (A C (p) B D - C (p)) .

(3)

In infected-individual counts the next-generation matrix is instead

K = \frac{1}{γ} (A B C (p) D - C (p)) .

(4)

The count and fraction formulations are related by a diagonal change of coordinates, so their eigenvalues coincide. The basic reproduction number is the spectral radius

R_{0} = ρ (M) / γ

, not an average of the two group reproduction numbers. The larger real eigenvalue is

λ_{+} (p) = \frac{A + D - C ( p ) + [ A - D + C ( p ) ] ^{2} + 4 BC ( p )}{2} .

(5)

This model makes a particular reciprocal mixing convention explicit; different defensible contact laws can change the first-order coefficients.

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Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 57 / 3 / a / Solution Created 2026-10-03 Updated 2026-10-14

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Use fractions of all farms, so

S_{n} + I_{n} + R_{n} = P_{n}

and

\sum_{n} P_{n} = 1

. Assume farm size is fixed, contacts mix randomly, infectious periods have size-independent exponential mean

1/ γ

, and the farm-size distribution has finite moments

μ = \sum_{n} n P_{n}

and

μ_{2} = \sum_{n} n^{2} P_{n}

. Write

I = \sum_{n} I_{n}

. The farm-size stratified SIR model with susceptibility proportional to size is

\dot{S}_{n} = - β n S_{n} I, \dot{I}_{n} = β n S_{n} I - γ I_{n}, \dot{R}_{n} = γ I_{n} .

(1)

Here

β

includes the common contact-rate constant; using frequency-dependent mixing avoids introducing an extra total-farm factor. An infectious farm of any size generates

β n P_{n} / γ

infections in initially susceptible size-

n

farms. The next-generation matrix

K_{nm} = β n P_{n} / γ

has rank one, giving the basic reproduction number

R_{0} = β μ / γ .

(2)

The comparison SIR model replaces every size by

μ

, so it has the same basic reproduction number and the same initial exponential growth rate

β μ - γ

As the epidemic develops, put

Λ (t) = β \int_{0}^{t} I (u) d u

. For negligible initial infection,

S_{n} = P_{n} e^{- n Λ}

. Large farms leave the susceptible class faster. If

μ_{S} = \sum_{n} n S_{n} / S

with

S = \sum_{n} S_{n}

, differentiation gives

\overset{μ}{˙}_{S} = - β I Var_{S} (n) \leq 0.

(3)

The mean size of newly infected farms is

μ_{inc} = \sum_{n} n^{2} S_{n} / \sum_{n} n S_{n}

, larger than the susceptible mean unless all sizes agree. If

μ_{I} = \sum_{n} n I_{n} / I

, then

\overset{μ}{˙}_{I} = \frac{β I \sum _{n} n S _{n}}{I} (μ_{inc} - μ_{I}) .

(4)

Recovery cancels from this equation because its rate is size-independent. Starting with representative infectives,

μ_{I}

initially rises toward the size-biased mean

μ_{2} / μ

; later depletion of large susceptible farms lowers

μ_{inc}

and usually lowers

μ_{I}

. It is incorrect to assert that the mean size of infectives must decrease from every possible initial condition. In the growing linearized mode,

I_{n} \propto n P_{n}

, so its mean is already

μ_{2} / μ

The aggregate growth equation is

\dot{I} = I (βS μ_{S} - γ)

. The falling susceptible mean adds a depletion effect absent from the homogeneous SIR model. The final size relation for an epidemic, with vanishing initial seed and attack fraction

A

, is

A = 1 - \sum_{n} P_{n} e^{- (β / γ) n A} .

(5)

By Jensen's inequality, the right side is at most

1 - e^{- R_{0} A}

, so

A

is no larger than the homogeneous attack fraction. For a common small representative seed the illustrated epidemic is flatter and infects fewer farms; exact pointwise ordering of infected fractions for every time and arbitrary initial seed is not a consequence of equality of

R_{0}

Figure 1.
Farm-size heterogeneity: epidemic prevalence and mean livestock counts for the three transmission assumptions
. **Details**.

The original sketch uses equally many farms of sizes 1 and 3, recovery rate

γ = 1

, and a representative infected seed of

0.001

. Each homogeneous comparison has

R_{0} = 2

. The left, middle and right columns correspond to parts (a), (b) and (c). The top row shows infected farm fractions, and the bottom row distinguishes the mean sizes of susceptible and infected farms. These are illustrative trajectories under the stated model, not universal empirical outbreak curves.

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Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 57 / 3 / b / Solution Created 2026-10-03 Updated 2026-10-14

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With susceptibility independent of size and infectivity proportional to size, define

J = \sum_{n} n I_{n}

. The farm-size stratified SIR model is

\dot{S}_{n} = - β S_{n} J, \dot{I}_{n} = β S_{n} J - γ I_{n}, \dot{R}_{n} = γ I_{n} .

(1)

An infectious size-

m

farm has next-generation column

K_{nm} = (β / γ) P_{n} m

. Its sole nonzero eigenvalue gives the basic reproduction number

R_{0} = β μ / γ,

(2)

again equal to the comparison SIR model.

Since every susceptible farm experiences the same force of infection, all ratios

S_{n} / S_{m}

remain fixed. For a representative initial seed,

S_{n} (0) = (1 - i_{0}) P_{n}

and

I_{n} (0) = i_{0} P_{n}

, uniqueness of the differential equations gives

S_{n} = P_{n} S

I_{n} = P_{n} I

R_{n} = P_{n} R

for all time. Consequently

J = μ I

and

\dot{S} = - β μ S I, \dot{I} = β μ S I - γ I, \dot{R} = γ I .

(3)

For this representative seed, the heterogeneous and homogeneous epidemics coincide exactly, and both susceptible and infected farm means stay at $μ$ . This is the overlapping middle-column sketch above, not merely an approximation based on equal reproduction numbers.

With a nonrepresentative seed the susceptible mean remains its initial value, but the infected mean need not. The influx of new infectives has the constant susceptible mean, so

\overset{μ}{˙}_{I} = \frac{βS J}{I} (μ_{S} - μ_{I}) .

(4)

Thus it is pulled toward

μ_{S}

while incidence continues. Infectivity variation alone does not selectively remove large susceptible farms. The exact homogeneous reduction depends on the representative initial condition; specifying only

P_{n}

and

R_{0}

does not determine a trajectory for every possible seed.

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Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 57 / 3 / c / Solution Created 2026-10-03 Updated 2026-10-14

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Both weights now equal farm size, giving the farm-size stratified SIR model

\dot{S}_{n} = - β n S_{n} J, \dot{I}_{n} = β n S_{n} J - γ I_{n}, J = \sum_{n} n I_{n} .

(1)

The next-generation matrix is

K_{nm} = (β / γ) n P_{n} m

, whose nonzero eigenvalue is

R_{0} = \frac{β μ _{2}}{γ}, R_{0, hom} = \frac{β μ ^{2}}{γ}, \frac{R _{0}}{R _{0, hom}} = 1 + \frac{Var ( n )}{μ ^{2}} .

(2)

The variance matters because large farms are both more likely to acquire infection and more effective sources afterward. This positive correlation increases the basic reproduction number relative to the homogeneous SIR model. In particular the heterogeneous population can permit invasion when the homogeneous comparison is subcritical.

Put

Λ = β \int_{0}^{t} J (u) d u

. Apart from the negligible initial seed,

S_{n} = P_{n} e^{- n Λ}

. The susceptible mean obeys

\overset{μ}{˙}_{S} = - β J Var_{S} (n)

. For infectives, the incidence mean is again

μ_{inc} = \sum_{n} n^{2} S_{n} / \sum_{n} n S_{n}

, and

\overset{μ}{˙}_{I} = \frac{β J \sum _{n} n S _{n}}{I} (μ_{inc} - μ_{I}) .

(3)

The growing mode

I_{n} \propto n P_{n}

has mean

μ_{2} / μ

, not

μ

. As the larger susceptible farms are depleted, the mean size of incoming infectives falls; the current infected mean follows it with a delay. These equations give the direction of change for any initial condition without imposing false monotonicity of the infected mean.

The weighted infected total satisfies

\dot{J} = (β \sum_{n} n^{2} S_{n} - γ) J

, so the effective reproduction number declines from

β μ_{2} / γ

as the susceptible size distribution changes. The final size relation for an epidemic becomes

Λ_{\infty} = \frac{β}{γ} \sum_{n} n P_{n} (1 - e^{- n Λ_{\infty}}), A = 1 - \sum_{n} P_{n} e^{- n Λ_{\infty}} .

(4)

The right-column sketch illustrates the larger initial growth rate and preferential depletion of large farms. The homogeneous comparison has size

μ

in both transmission factors; it must not be assigned the heterogeneous second moment. The general conclusions are enhanced invasion and selective removal of large susceptible farms. The final unweighted fraction of farms infected has no universal ordering relative to the homogeneous comparison: higher transmission and stronger heterogeneity-induced depletion compete, and the displayed final-size equations determine the answer for a specified distribution. For example, with

P_{1} = P_{3} = 1/2

, the infinitesimal-seed attack fractions are approximately

0.367

versus

0.176

when the homogeneous reproduction number is

1.1

, but

0.740

versus

0.797

when it is

2

; the two comparisons have opposite ordering.

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Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 57 / 4 / Solution Created 2026-10-03 Updated 2026-10-14

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The basic reproduction number is the expected value of secondary infections caused by one typical infectious individual introduced into an otherwise susceptible population. For a heterogeneous model it is the dominant eigenvalue of the next-generation operator, so the appropriate infected-type distribution is part of “typical.” A small introduction grows when

R_{0} > 1

and decays when

R_{0} < 1

, under the usual linear invasion assumptions.

The asymptotic susceptible level depends on the model. In a homogeneous SIS model with transmission rate

β

and recovery rate

γ

, the positive endemic equilibrium has

S_{*} = 1/ R_{0}

when

R_{0} = β / γ > 1

. In a closed SIR model, however, the final size relation for an epidemic is

lo g (S_{\infty} / S_{0}) = - R_{0} (R_{\infty} - R_{0}^{recovered}) .

(1)

For an infinitesimal seed in a wholly susceptible population this becomes

- lo g S_{\infty} = R_{0} (1 - S_{\infty})

. The epidemic peak occurs at

S = 1/ R_{0}

, and a nontrivial outbreak leaves

S_{\infty} < 1/ R_{0}

. Thus $1/ R_{0}$ is an endemic or instantaneous invasion threshold, not a universal final susceptible fraction.

For cooperative infections, let

s, x, y, z

be the fractions carrying neither infection, only

A

, only

B

, and both, with sum one. Let

λ_{A} = β_{A} (x + z)

λ_{B} = β_{B} (y + z)

, assume each infection independently recovers at rate

γ

, and multiply susceptibility to the second infection by

q > 1

. One suitable pair of SIS models is

\overset{x}{˙} \overset{y}{˙} \overset{z}{˙} \overset{s}{˙} = λ_{A} s - q λ_{B} x - γ x + γ z, = λ_{B} s - q λ_{A} y - γ y + γ z, = q λ_{B} x + q λ_{A} y - 2 γ z, = - (λ_{A} + λ_{B}) s + γ (x + y) .

(2)

Recovery of one infection from

z

leads to the other single-infection class, not directly to

s

. The probability simplex is invariant. At the disease-free equilibrium the eigenvalues are

β_{A} - γ

β_{B} - γ

and

- 2 γ

. Therefore failure of both isolated and low-density invasion requires

R_{A} = β_{A} / γ < 1

R_{B} = β_{B} / γ < 1

. Enhancement of secondary susceptibility does not alter these linear thresholds because coinfection is second order at introduction.

For completeness, a joint positive equilibrium must have

x = \frac{R _{B}^{- 1} - s}{q}, y = \frac{R _{A}^{- 1} - s}{q}, z = 1 - s - x - y,

(3)

and

(q - 1) s^{2} - q s + \frac{1}{R _{A} R _{B}} = 0.

(4)

To obtain these, divide the equilibrium equations for total

A

and total

B

prevalence by their positive prevalences, yielding

s + q y = R_{A}^{- 1}

and

s + q x = R_{B}^{- 1}

. Substitution into the

s

balance gives the quadratic. Its discriminant must be nonnegative and the resulting

x, y, z

positive. These conditions test equilibrium feasibility for unequal rates; stability must also be checked rather than inferred from the mere presence of a root.

A fully explicit Allee effect results in the symmetric permitted case

β_{A} = β_{B} = β

R = β / γ

. The joint equilibria have

s_{\pm} = \frac{q \pm q ^{2} - 4 ( q - 1 ) / R ^{2}}{2 ( q - 1 )}, x = y = \frac{R ^{- 1} - s _{\pm}}{q} .

(5)

Two feasible positive equilibria coexist with the stable disease-free state exactly in the strict range

q > 2, \frac{2 q - 1}{q} < R < 1.

(6)

Indeed at the fold the two roots coalesce, and at

R = 1

the smaller-prevalence branch reaches the disease-free boundary. For

q \leq 2

, roots at

R < 1

do not give positive singly infected fractions. Strict inequalities exclude these marginal endpoints.

Here is the stability check underlying the cooperative SIS coinfection threshold. On the symmetric subspace put

v = x + z = y + z

and

c = z

. Then

\overset{v}{˙} = γ v {R [1 + (q - 2) v - (q - 1) c] - 1}, \overset{c}{˙} = 2 γ [q R v (v - c) - c] .

(7)

At a positive equilibrium

c = q R v^{2} / (1 + q R v)

. Its two-dimensional Jacobian matrix has

tr J = - γ [R v (q + 2) + 2], det J = \frac{2 γ ^{2} R v}{1 + q R v} [(1 + q R v)^{2} - (q - 1)] .

(8)

The higher-prevalence branch has positive determinant and negative trace, while the lower-prevalence branch is a saddle equilibrium. The remaining antisymmetric mode, from

x - y

, has eigenvalue

- qβ v < 0

. Thus the smaller susceptible root

s_{-}

is a stable joint endemic equilibrium, separated from the stable zero state by a finite seeding threshold. Neither infection can persist alone in this range. Strong cooperation can sustain both infections after a sufficiently large joint introduction even though neither invades from rarity. The numerical condition belongs to the specified secondary-susceptibility model; alternative cooperation mechanisms need their own threshold calculation.

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Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 207 / 3 / d / Solution Created 2026-10-03 Updated 2026-10-06

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The infectious count in the SIR model obeys

\dot{I} = I (βS - γ)

. At the initial state,

\dot{I} (0) = βN - γ

, so initial epidemic growth requires the epidemic threshold for the closed SIR model

N > \frac{γ}{β}, R_{0} = \frac{βN}{γ} > 1.

(1)

Here

R_{0}

is the basic reproduction number in this mass-action infection convention. Necessity also follows globally:

S (t)

is nonincreasing, so if

βN \leq γ

, then

βS (t) - γ \leq 0

for all time and

I (t)

never increases. When

β = 0

, growth is impossible.

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Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 4 / 6B / Solution Created 2026-09-24 Updated 2026-10-05

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At an endemic equilibrium,

I_{*} > 0

, so

β S_{*} = μ + ν

. Substitution into the susceptible equation gives

S_{*} = \frac{N}{R _{0}}, I_{*} = \frac{μ N}{μ + ν} (1 - \frac{1}{R _{0}}) .

(1)

These values are physically positive exactly when

R_{0} > 1

. The Jacobian matrix at this equilibrium is

J = (- μ R_{0} μ (R_{0} - 1) - (μ + ν) 0), tr J = - μ R_{0} < 0, det J = μ (μ + ν) (R_{0} - 1) > 0.

(2)

Both eigenvalues have negative real parts, so the endemic equilibrium is locally asymptotically stable. Its decay exponents are

λ_{\pm} = - \frac{μ R _{0}}{2} \pm \frac{μ ^{2} R _{0}^{2}}{4} - μ (μ + ν) (R_{0} - 1) .

(3)

The regime

ν ≫ μ

means recovery is rapid compared with demographic replacement. For fixed

R_{0} > 1

away from the threshold, the exponents form a complex conjugate pair with frequency approximately

μν (R_{0} - 1)

. Hence

T ≃ \frac{2 π}{μν ( R _{0} - 1 )} .

(4)

Strictly, oscillations require

μ (μ + ν) (R_{0} - 1) > μ^{2} R_{0}^{2} /4

;

ν ≫ μ

alone does not ensure this uniformly for every

R_{0}

arbitrarily close to one or arbitrarily large.

Vaccinating a fraction

v

of newborns replaces the susceptible input by

μ N (1 - v)

. The effective basic reproduction number is

R_{v} = (1 - v) R_{0}

, and the same calculation replaces

R_{0}

R_{v}

. Provided

R_{v} > 1

and the oscillatory approximation remains applicable, vaccination lengthens the oscillation period, because

R_{v} - 1

is smaller. Extremely near eradication the exponents become real and there is no oscillation period to extrapolate indefinitely.

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Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 6C / Solution Created 2026-09-24 Updated 2026-10-03

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Adding the three equations of the SIRS model gives

\frac{d}{d t} (S + I + R) = 0,

(1)

S (t) + I (t) + R (t) = N .

(2)

At a fixed point,

I (b S - c) = 0

. The disease-free equilibrium is

(S, I, R) = (N, 0, 0) .

(3)

For an endemic equilibrium

I^{*} > 0

, one has

S^{*} = c / b

and

a R^{*} = c I^{*}

. Conservation then gives

S^{*} = \frac{c}{b}, I^{*} = \frac{a ( b N - c )}{b ( a + c )}, R^{*} = \frac{c ( b N - c )}{b ( a + c )} .

(4)

It exists with positive infected population exactly when

b N > c,

(5)

equivalently when the basic reproduction number

b N / c

exceeds one.

Eliminating

R = N - S - I

yields

\dot{S} = a (N - S - I) - b S I, \dot{I} = (b S - c) I .

(6)

At the endemic equilibrium its Jacobian matrix is

J = (- a - b I^{*} b I^{*} - (a + c) 0) .

(7)

Its trace and determinant are

T = - \frac{a ( a + b N )}{a + c} < 0, D = a (b N - c) > 0.

(8)

Therefore the endemic equilibrium is locally asymptotically stable. It is a stable node when

\frac{a ^{2} ( a + b N ) ^{2}}{( a + c ) ^{2}} \geq 4 a (b N - c),

(9)

and a stable focus when the inequality is reversed. Near a node, perturbations are sums of two nonoscillatory decaying exponentials. Near a focus, they execute damped oscillations with decay rate

- T /2

and angular frequency

4 D - T^{2} /2

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Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 6B / ii / Solution Created 2026-09-24 Updated 2026-09-29

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At an endemic equilibrium,

I^{*} > 0

, so

0 = \dot{I} = I^{*} (β S^{*} - γ)

forces

S^{*} = \frac{γ}{β} .

(1)

The equation

\dot{R} = 0

gives

R^{*} = γ I^{*} / α

. The conservation law

S^{*} + I^{*} + R^{*} = N

therefore yields

I^{*} = \frac{α ( βN - γ )}{β ( α + γ )}, R^{*} = \frac{γ ( βN - γ )}{β ( α + γ )} .

(2)

A positive endemic state exists exactly when

βN > γ

, equivalently when the basic reproduction number

R_{0} = βN / γ

exceeds one.

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Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 207 / 3 / g / Solution 2026-09-28

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Dividing the

I

equation by the

S

equation gives

\frac{d I}{d S} = - 1 + \frac{γ}{βS} .

(1)

Therefore

I + S - \frac{γ}{β} lo g S = I (0) + S (0) - \frac{γ}{β} lo g S (0)

(2)

is a first integral. Define the basic reproduction number by

R_{0} = βS (0) / γ

. At the removal peak,

S = S (0) / R_{0}

, and hence

R_{peak} = \frac{S ( 0 )}{R _{0}} lo g R_{0},

(3)

and, using

S + I + R = N

I_{peak} = I (0) + S (0) - \frac{S ( 0 )}{R _{0}} 1 + lo g R_{0} .

(4)

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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 6C / a / Solution Created 2026-09-24 Updated 2026-10-03

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This is the SIR model with waning immunity. The mass-action interaction term

β I S

transfers people from the susceptible compartment to the infected compartment,

ν I

transfers infectives to the recovered compartment, and

f R

returns recovered people to susceptibility as immunity wanes. Adding the equations gives

\frac{d}{d t} (S + I + R) = 0,

(1)

so the total population

N

is conserved.

When

f = 0

and infection is rare,

S ≃ N

and

I^{'} = (βS - ν) I ≃ (βN - ν) I .

(2)

Thus $I$ initially decays if $βN < ν$ . Equivalently, the basic reproduction number is $R_{0} = βN / ν < 1$ , so the epidemic invasion threshold is not crossed and no epidemic occurs.

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 207 / 2 / e / Solution Created 2026-09-24 Updated 2026-09-25

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Under frequency-dependent mass-action kinetics, if

I (t)

of the

N (t)

individuals are infectious, the force of infection is

λ (t) = β \frac{I ( t )}{N ( t )} .

(1)

One newly infectious person in an otherwise susceptible population transmits at total rate approximately

β

for mean infectious duration

1/ γ

. Hence the basic reproduction number is

R_{0} = \frac{β}{γ} = β d_{I},

(2)

where

d_{I}

is the infectious, rather than latent-plus-infectious, duration.

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Past exam of the short options course of the University of Oxford / 2017 / 9 / b / ii / Solution 2026-10-31

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If the seed

v_{0}

is positive, infection initially increases precisely when

R_{0} > 1.

(1)

Indeed,

v^{'} (0) = (R_{0} - 1) v_{0}

. Susceptibility is nonincreasing, so if

R_{0} \leq 1

, the infectious count never starts an increasing phase. When

R_{0} > 1

, it grows until

u = 1/ R_{0}

, the SIR infection peak, and then declines.

The basic reproduction number compares transmission and removal: one infectious individual generates infections at initial rate

β S_{0}

and remains infectious for mean time

1/ γ

. Their product

β S_{0} / γ

is the expected number of secondary cases at that susceptible level. Above one, an infectious generation can replace itself with more cases. An exactly zero infectious seed remains zero regardless of this threshold.

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Past exam of the systems biology course of the University of Cambridge / 2024 / Paper 1 / Section I / 1.3 / Solution 2026-10-20

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A simple model is especially valuable when it overturns a clearly stated expectation and explains why, but surprise is a means to understanding rather than a requirement that every useful model be counterintuitive. The expectation must be explicit enough to test, and the model should identify which assumptions make it fail. A surprising simulation without a mechanism or a plausible connection to observations is not sufficient.

One expectation is that a deterministic system with few variables must settle to a simple, predictable state. The logistic map,

x_{t + 1} = r x_{t} (1 - x_{t}), 0 \leq x_{t} \leq 1, 0 \leq r \leq 4,

(1)

provides a compact counterexample. Its positive fixed point is

x_{*} = 1 - 1/ r

, and the derivative there is

f^{'} (x_{*}) = 2 - r

. Thus the fixed point is locally stable for

1 < r < 3

; changing a single parameter can destroy that stability. At

r = 3.2

, repeated iteration approaches a two-cycle near

0.513045

and

0.799455

. For typical initial conditions at

r = 4

, the Lyapunov exponent is

lo g 2 > 0

, so nearby initial states separate exponentially on average until the bounded state space limits their separation. Robert M. May's analysis made the consequences of such simple difference equations particularly influential in population modeling.^[ref]

The broken expectation is not that determinism is false: the rule remains deterministic. Rather, an irregular trajectory need not be caused by a complicated mechanism or external stochasticity, and exact rules need not permit long-range prediction from imperfectly measured initial conditions. This does not show that every irregular population time series is chaotic. Real populations may experience demographic noise, environmental variation or overlapping generations, and the map's parameters and state must have a defensible biological interpretation. Evidence should distinguish those alternatives. Even within the mathematical example, special initial states can be periodic; the claim about

r = 4

concerns typical trajectories, not every point of the interval.

A second expectation is that diffusion can only erase spatial variation. In a coupled reaction-diffusion equation,

\partial_{t} (u v) = (f (u, v) g (u, v)) + (D_{u} 0 0 D_{v}) \nabla^{2} (u v),

(2)

diffusion smooths each species separately, yet can destabilize their coupled uniform equilibrium. Linearizing about that equilibrium gives the mode matrix

J - k^{2} D

. Local stability of

J

therefore does not by itself imply stability of every spatial mode. This is the organizing insight of the Turing instability.^[ref]

For a concrete dimensionless linearization, take

J = (1 - 3 1 - 2), D = diag (1, 10) .

(3)

The well-mixed system has trace

- 1

and determinant

1

, so its perturbations decay. Writing

q = k^{2}

, the spatial mode has trace

- 1 - 11 q

and determinant

det (J - q D) = 1 - 8 q + 10 q^{2} .

(4)

This determinant is negative for

(4 - 6) /10 < q < (4 + 6) /10

, creating one positive eigenvalue and a growing nonuniform mode. Equal diffusivities would instead shift both eigenvalues left and could not create this two-species instability. On a finite domain, at least one of the modes allowed by the boundary conditions must lie in the unstable band. The algebra exposes exactly why the intuition about smoothing fails: differential transport changes the balance of an interacting local system.

A Turing instability establishes a possible pattern-generating mechanism, not proof that a particular animal's markings use it. One must identify candidate interacting species, measure transport and response properties, and test perturbations that alter the predicted wavelength or stability. The linear example specifies onset only; nonlinear dynamics determine whether growth saturates into a stable pattern. Its value is to convert a vague statement about morphogenesis into discriminating questions, while leaving those experimental obligations visible.

Figure 1.
Original mathematical illustrations of changing logistic-map dynamics, sensitivity to initial conditions and diffusion-driven instability
. **Details**.

A third expectation is that an infection with a basic reproduction number above one must produce a major outbreak. Consider the early phase as a continuous-time branching process, with independent per-infective transmission rate

β

and recovery rate

γ

, while susceptible depletion is negligible. If

q

is the probability of eventual extinction starting from one infective, conditioning on the first event gives

q = \frac{γ}{β + γ} + \frac{β}{β + γ} q^{2} .

(5)

Taking the smallest solution in

[0, 1]

yields

q = 1

when

β \leq γ

, and

q = γ / β = 1/ R_{0}

when

R_{0} > 1

. Thus

R_{0} = 4

still gives an early-lineage extinction probability of

1/4

. A deterministic SIR model describes mean-field growth but omits this individual realization uncertainty. In a finite closed stochastic SIR model, every epidemic eventually ends; the branching calculation concerns early extinction before substantial susceptible depletion, not that eventual end or an arbitrary peak-size cutoff.

These examples show different benefits of simplicity: a stability calculation can separate predictable from sensitive dynamics, a mode calculation can uncover a new mechanism, and a probability recursion can expose what an average trajectory hides. The same discipline also explains why the word “should” needs qualification. A simple exponential-decay model that confirms a familiar expectation can still calibrate a rate, serve as a control, or expose a faulty measurement. A useful mathematical model must earn its simplifications through the question it answers. Breaking an expectation is a powerful route to discovery when the contradiction is precise, robust to reasonable assumptions and experimentally consequential; novelty alone is not a validation criterion.

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