1. health and fitness
2. disease

PHYS 414 Midterm Exam
Problem 1: Fokker-Planck equation for the time evolution of system energy
A small system has energy E at time t, where we assume E is a continuous variable in the
range 0 ≤ E < ∞. The system interacts with its external environment, exchanging energy, so
E varies with time. Over the small time interval t to t + δt, the system’s energy can jump from E
to E + ξ, where ξ is a random energy change with some distribution W (ξ). In the limit δt → 0,
this distribution would approach a Dirac delta function δ(ξ), since if δt = 0, there is no time for
the system and environment to interact, and no energy exchange can occur. For small δt > 0, we
will assume the following simple Gaussian form:
W (ξ) = √
1
2
e−(ξ−µδt) /(2νδt)
2πνδt
(1)
where ν and µ are some unknown constantsR which reflect properties of the system-environment
∞
interaction. Note that ν > 0 to ensure that −∞ dξ W (ξ) = 1. The above form for W (ξ) has the
nice property that it goes to δ(ξ) as δt → 0.
a) Derive the Fokker-Planck equation for the probability density P (E, t) that the system has
energy E at time t. Write the equation in a form where the coefficients depend only on the
physical parameters ν and µ, and keep only the lowest order terms in δt.
b) Check that the Fokker-Planck equation you derived in part a has a stationary solution of the
form Ps (E) = Z −1 e−βE , defined for 0 ≤ E < ∞, where Z and β are constants. Find Z and β
in terms of µ and ν. Does the fact that Ps (E) must be a normalized distribution put a constraint
on the sign of µ? What is the physical meaning of this sign constraint in terms of the energy
exchange between the system and environment? How is it possible for µ to be nonzero (implying
that the system on average loses/gains energy from the environment over a time interval δt) and
yet at the same time the system achieves a stationary distribution?
c) Show that the transitions E → E + ξ and E + ξ → E over a time interval δt obey detailed
balance in the stationary state Ps . Hence this is an equilibrium stationary state (the familiar
canonical ensemble).
Note: this problem is an alternative approach to deriving the equilibrium canonical distribution Ps (E), applicable to the case of a continuous energy variable E.
Problem 2: Mean lifetimes of disease outbreaks
Consider a simple model of disease spreading where any individual can either be susceptible
(healthy, but a potential target of the disease) or infected. Label these two conditions S and I.
Let us assume that the disease is similar to the common cold: it is generally not deadly, and the
cause of the disease (i.e. a virus family) is sufficiently diverse and quickly mutating that you
do not acquire immunity after recovering from the illness. Thus any individual can make many
transitions between states S and I.
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If an individual is in state I, let r be the Poisson rate for the transition back to state S. In
other words, during every small time interval δt after the start of the infection, the probability
of returning to S is rδt. During the infection the individual is contagious, and we will model
the reproduction of the disease in the simplest fashion: an individual in state I will increase the
population of I by one new case with a Poisson rate g, so long as the remaining population of S is
nonzero. So during every small time interval δt after the start of an individual’s infection, he/she
will directly cause a new case of I with probability gδt (assuming there are susceptible people
available to get sick), up until the moment of that individual’s recovery.
a) Let us focus first on a single infected individual, embedded in an infinite population of susceptibles (so there are always available targets for the disease to spread). On average, how many
new cases of infection will be generated by an infected individual during a single bout of disease?
(Count only the cases started directly by the individual.) Show that the answer is R0 = g/r. This
is an important quantity in epidemiology known as the
reproduction number of the disease1 .
R ∞basic
n
Hint: Two identities may P
be useful: i) the integral 0 t exp(−at) = n!/an+1 , for a > 0 and
n
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integer n ≥ 0; ii) the sum ∞
n=0 nx = x/(1 − x) for |x| < 1. Additionally, remember from the
midterm review that for a Poisson process with rate k, the probability of seeing exactly k events
during a time interval t is (kt)n exp(−kt)/n!.
b) Now consider a finite population of N people. Label the state of the system by the number n
of infected individuals. The transition rate from state n to n − 1 is rn = nr for n ≥ 1, and the
transition rate from state n to n + 1 is gn = ng for n ≥ 0. Is this an ergodic system? Does it have
a stationary state? If so, what is the probability distribution psn of the stationary state, and is it
an equilibrium stationary state?
c) For the population model in part b, assume a disease outbreak starts at t = 0 with one infected individual (n = 1). Find an expression for τ1 , the mean duration of the disease outbreak
in the population, i.e. the mean first passage time to reach the disease-free state n = 0? Hint:
think about the approach used to solve the protein folding problem in PS#3, and the similarities/differences with the disease model. Your final expression can be in the form of a sum.
d) Show that in the limit N → ∞ the result in part c reduces to τ1 = −g −1 log(1 − R0 ) for the
case where R0 < 1. For R0 ≥ 1, the result for τ1 diverges as N → ∞. In the latter situation, if
you have an arbitrarily large population of susceptibles, a significant proportion of the time you
will never return to a disease-free state, even though you started with just one infected individual.
Show that for R0 > 1, the leading order term in τ1 as N → ∞ is τ1 ≈ (gN )−1 exp(N ln R0 ) so
the divergence is exponential in N .
e) Let us return to the case where N is finite. In the above population model, the rate g0 = 0,
so once the disease dies out, it will not recur. Assume instead that g0 > 0, while all the other
rates are as described in part b. This would correspond to there being a nonzero probability of
the disease entering the population again from an external source. Is this system ergodic? Does
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The common cold has an R0 ≈ 6, while for measles R0 ≈ 15, at the high end of the contagiousness scale
among all known diseases. Ebola, which does not really fit our simple model because it has a high fatality rate, has
an R0 ≈ 2. Food-borne illnesses like salmonella are on the low end of the spectrum, with R0 < 1.
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it have a stationary state? If so, what is the probability distribution psn of the stationary state, and
is it an equilibrium stationary state?
f) Using the results of part e, show that in the limit N → ∞ the probability of being in the diseasefree state n = 0 as t → ∞ is equal to ps0 = (1 + g0 τ1 )−1 for R0 < 1, where τ1 = −g −1 log(1 − R0 )
is the mean outbreak duration from part d. As R0 approaches 1 from below, τ1 diverges and
ps0 → 0.
Note: What does this problem tell us about disease dynamics? For any disease with R0 ≥ 1,
if you want outbreaks of short mean duration, you need to keep the population of potential
targets N small. Though varying N as a function of the disease spreading is outside the scope of
the simple model, we can imagine various ways by which N is limited: quarantine, vaccination
(if available), or by having a fatal disease that depletes the population. From a statistical physics
perspective, the qualitatively different behaviors of the system for R0 < 1 and R0 ≥ 1 as N → ∞
is the hallmark of a phase transition.
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