Download Modeling the Spread of Infectious Diseases

Of Viruses, Mosquitoes & Zombies
How Mathematical Models can Help Control Epidemics
Adnan Khan
Department of Mathematics
Lahore University of Management Sciences
Of Viruses……..
• Literature Review
– Hethcote, H. The Mathematics of Infectious
Diseases, SIAM Review 2000
– Imran A, Malik T, Rqfique H & Khan, A. A model of
bi-mode transmission dynamics of hepatitis C with
optimal control, Theory in Biosciences 2013
…….Mosquitoes
• Literature Review
– Wearing, H. & Rohani P. Ecological and Immunological
Determinants of Dengue Epidemics, PNAS 2006
– Khan, A. Hassan M. & Imran M. Estimating the Basic
Reproduction Number for Single-Strain Dengue Fever
Epidemics, Journal of Biological Dynamics 2014
and…….Zombies
• Literature Review
– P. Munz, I. Hudea, J. Imad and R.J. Smith? When zombies
attack!: Mathematical modelling of an outbreak of zombie
infection (Infectious Disease Modelling Research Progress
2009, in: J.M. Tchuenche and C. Chiyaka, eds, pp133-150).
– White Zombie - 1932
– Revolt of the Zombies- 1936
– Zombie of Mora Tau – 1957
– El Muerto - 1961
– Night of the Living Dead - 1968
– Curse of the Living Dead - 1973
– Day of the Dead – 1985
– 28 Days Later – 2007
Zombie Attack!!
https://code.google.com/p/simzombie/
Zombie Attack!!
• There has been an outbreak of Zombies
• How to model this?
• How to suggest control measure?
• What about their efficacy?
Influenza Data Timeline from Google
googleflu.mov
T. Malik, A. Gumel, L. Thompson, T. Strome, S. Mahmud; "Google Flu Trends" and Emergency
Department Triage Data Predicted the 2009 Pandemic H1N1 Waves in Winnipeg, Manitoba.
Canadian Journal of Public Health. 102(4):294-97.
Influenza
• An Army base has a total staff of 8342. An
individual who has just returned from leave
becomes ill and is diagnosed with Jade fever -- an
exotic, dangerous and highly contagious variety
of flu.
• Will there be a flu epidemic on the base?
• If so, does the base hospital have enough beds?
• How many of the base staff will get the flu?
Dengue Fever Epidemic
• A suitable model
• Must include vector dynamics
• Control Strategies
• Retrospective analysis
Mathematical Epidemiology
• What are epidemics?
• Mathematical Models
• Statistical
– Finding ‘trends’ in available data
• Mechanistic
– Incorporating the mechanics of transmission
• Retrospective Analysis
• Data analysis
Dynamic / Mechanistic Models
• Deterministic Models
– Differential Equations
– Difference Equations
– Delay Differential Equations
• Stochastic Models
– Markov Chains
– Stochastic Differential Equations
Why Dynamic Models
• Models allow you to predict (estimate) when you
don’t KNOW
– What are the costs and benefits of different
control strategies?
– When should there be quarantines?
– Who should receive vaccinations?
– When should wildlife or domestic animals be
killed?
– Which human populations are most vulnerable?
– How many people are likely to be infected? To get
sick? To die?
A Basic SIR Model
• Susceptible Population (S)
– people who are able to catch the disease
• Infectious Population (I)
– people who have the disease and can transmit it
• Removed Population (R)
– people who have recovered and are no longer
susceptible
– People who are naturally immune or isolated etc.
SIR Model Schematic
The Model Assumptions
• Susceptible individuals become infected upon coming in
contact with infectious individuals.
• Each infected individual has a fixed number, r, of contacts
per day that are sufficient to spread the disease
– The parameter r contains information about the number
of contacts and probability of infection
• Infected individuals recover from the disease at rate a
– 1/a is the average recovery time
The Model Assumptions
• The incubation period of the disease is
short enough to be neglected
• All population classes are well mixed
• Births, deaths, immigration, and migration
can be ignored on the timescale of
interest
The Model Equations

dS
 rSI
dt
S(0)  S0
dI
 rSI  aI
dt

I(0)  I0
dR
 aI
dt 
R(0)  0

The rate of transmission of the disease is proportional
to the rate of encounter of susceptible and infected

individual.

Gaining Insight into the Model
• Epidemiological Questions
– Will an epidemic occur?
– If an epidemic does occur
• how severe will it be?
• will the disease eventually die out or persist in the
population?
• how many people will get the disease during the
course of the infection?
Gaining Insight into the Model
• Definition: We will say that an epidemic
occurs if the number of infectious individuals
is greater than the initial number I0 for some
time t
• Epidemiological Question: Given r, S0 , I0, and
a, when will an epidemic occur?
• Mathematical Question: Is I(t) > I0 for any
time, t?
When Will an Epidemic Occur?
• Consider the I-equation first
dI
 rSI  aI
dt
At t = 0 dI  I 0 rS 0  a 
dt
• Therefore, the I population will increase
initially if S0  a r and the I population
will decrease initially otherwise. Therefore,
this condition sufficient for an epidemic to
occur.

When Will an Epidemic Occur?
• Now consider the S-equation
dS
 rSI
dt
dS
S  S0,t
 0,t
dt
dI
 IrS  a  0,t
So if S0  a r
We know
dt

 if S  a r there will not be an
• Therefore,
0

epidemic.

• However if, S0  a r
will occur.

then an epidemic
Gaining Insight into the Model
• Epidemiological Question: If an epidemic
occurs, will the disease eventually die out or
will it persist in the population.
• Mathematical Question: What are the steady
states and their stability.
– More specifically, is I = 0 a steady state and if so is
it stable?
Steady States
dS
 rSI
dt


dI
 rSI  aI
dt
dR
 aI
dt
• Note: I = 0 makes all
three equations zero.
• Therefore I = 0
represents an entire
line (or plane) of steady
states
• Traditional stability
analysis leads to a zero
eigenvalue.
Phase Plane -- Analytically
• Recall that R is decoupled
dS
 rSI
dt
dI
 rSI  aI
dt


dI
a
 1 
dS
rS
a
I  S  ln S  C
r
a
a
I  S  ln S  I 0  S 0  ln S 0
r
r

Gaining Insight into the Model
Equations
• Epidemiological Question: What will be the
final state of the population be after the
infection has run its course?
• Mathematical Question: What are the
steady states for S and R?
Steady States for S and R
S  S0e
*

N S*

R NS
*
*
• 
S* = is the number of people who did not
catch the disease.

Gaining Insight into the Model
• How many people will catch the disease
before it dies out?
I total  I 0  S0  S
*
Gaining Insight into the Model
• Epidemiological Questions: When an
epidemic occurs, how severe will it be?
• Mathematical Question: What is the
maximum number of infectious individuals?
Severity of Epidemic
S
I  N  S   ln
S0
a
where  
r
• I achieves its max when S = a/r
Imax  N     ln

S0
Severity of the Epidemic
Basic Reproduction Number
• As we’ve seen R0  S0 r
a
parameter.
is an important
 the infectious contact number
• It is called
or basic reproduction number of the of the
infection.
Basic Reproduction Number:
Breakdown
• A given infective will, on average, be infectious
for 1/a units of time.
• The number of susceptibles infected by one
infectious individual per unit time is rS.
• Therefore, the number of infections produced by
one infective is rS/a.
• If R0>1, then an epidemic will occur
Importance of R0
• R0 is not actually a characteristic of the
disease) but of the virus in a specific
population at a specific time and place. By
altering some or all of the components you
can also alter R0.
Fit of the Model to Data for Influenza
In 1978, a flu epidemic occurred in a boys
boarding school in the north of England. There
were 763 resident boys, including one initial
infective. The school kept records of the
number of residents confined to bed, and we
will assume those to be the infectives. The
data for the two-week can be fit to the SIR
model.
Fit of the Model to Data for Influenza
Time Course of the Influenza Epidemic
Results of the SIR Model
• The occurrence of an epidemic depends
solely on the number susceptibles, the
transmission rate, and recovery rate.
– The initial number of infectives plays no role in
whether or not there is an epidemic.
– That is, no matter how many infectives there
are, an epidemic will not occur unless S0 > a/r.
Modifications of the SIR Model
• Other considerations, such as vital
dynamics (births and deaths), length of
immunity, the incubation period of the
disease, and disease induced mortality can
all have large influences on the course of an
outbreak.
Another Example
• An Army base has a total staff of 8342. An
individual who has just returned from leave
becomes ill and is diagnosed with Jade fever -an exotic, dangerous and highly contagious
variety of flu.
• All individuals getting this flu must be
hospitalized. The base hospital has 240 beds.
The transmission parameter for this flu at this
base is r = 5x10-5 per day and the recovery rate
is a = .32 per day.
Questions For You
• Is the condition for an epidemic satisfied?
• Does the base hospital have enough beds?
• How many of the base staff will get the flu?
Answers
• Is the condition for an epidemic satisfied?
– S0 =8341, I0 = 1, R0 = 0
–
a 0.32
   5  6,400
r 5e
S0 r
R0 
1.3 1
a
– Therefore an epidemic will occur.


Answers
• Does the hospital have enough beds?
Imax  N     ln

S0
 268
• Therefore if the hospital has 245 beds,
it will need more.
Answers
• How many of the base staff will get the
flu?
Itotal  I0  S0  S
*
• S* = 4783, so about 3649 people (43%
of the population) will catch the
 disease.
Time Course of the Jade Fever
Epidemic
S.I.R.S Model
• The model equations are
• This allows for endemic steady states
Zombie Epidemic Model
• The zombie epidemic spreads by zombie to
person contact and resurrection as a zombie
• We consider three classes
– Susceptibles (S)
– Zombies (Z)
– Removed (R)
• Zombies are created by
– Zombies attacking a human successfully
– The Removed Humans being resurrected as Zombies
Mathematical formulation
• The dynamics are given by an SIR type model
Analysis of the Model
• When time period is long the vital dynamics
cannot be ignored
• In this case note
• So that as
• As
this is DOOMSDAY for humanity
Short Time Dynamics
• When the time period is short as compared to
human lifetimes
• There are two equilibria
• Unfortunately the zombie takeover is
asymptotically stable
But what about Latency?
• All zombie flicks teach us that there is latency
• Lets include an ‘infected (I)’ latent class
Any Good News?
• Long term dynamics still spell doomsday
• Short term dynamics have two equilibria
• Unfortunately once again the Zombie free
equilibrium is unstable
Will Quarantine Help?
• Lets quarantine the infected population
• Mathematically we have
Analysis of Quarantine
• In a short outbreak we have two equilibria
• Stability depends on
• The DFE is stable for
• Else the co existence equilibrium is stable
Perhaps!!!
• We obtain the basic reproductive number
• Realistically
• In this case need
to be very large for
• Need a large percentage of infected to be
quarantined for co-existence
Does Salvation Lie in Treatment?
• Suppose a cure becomes available
• The mathematical model is
Analysis
• Besides the DFE we have the coexistence
equilibrium
• Stability analysis show that this ‘endemic’
equilibrium is stable
• With our parameters this results in some
humans and lots of zombies coexisting
A Model for the Spread of Dengue
Fever
• We have developed a model involving two
strains of the DF
• This is relevant in Pakistan where an
overwhelming majority of cases are type II
and type III
• Data available form previous outbreaks will be
used to predict R0
Model Assumptions
• Constant recruitment rate into susceptibles
• Constant natural death rate
• Two strains prevalent in the population
• Rate of infection incorporates Antibody Dependent
Enhancement
• Recovery from a strain of dengue results in
permanent immunity
The Mathematical Model
The Model Variables
The Model Parameters
The Basic Reproductive Number
Data (Lahore 2011)
Estimating
QUESTIONS