An Invitation to Mathematical Epidemiology

Universidad Nacional Mayor de San Marcos

Adnan Khan & Mudassar Imran Lahore University of Management Sciences

Mathematical Models

Easy math Unrealistic model

Model Tradeoffs

Too Much Detail Impossible to Analyze

Model Tradeoffs

Deterministic vs. Stochastic • Deterministic Models

• Usually Simpler to Analyze• Simulation is Cheaper• Non Linear Models can Capture Rich Variety of Behavior

• Stochastic Models» Build in Uncertainty in Knowledge » Capture Inability to Account for Details » Simulation is Expensive

» Deterministic Models Approximate Stochastic Models under certain conditions……??

Outline of the WorkshopWeek 1

• Day 1:– Introduction and Basic Epidemiological Models

• Day 2:– Dynamical Systems Analysis and Applications to

Epidemiological Models

• Day 3:– Control Theory and Applications– Estimating Parameters in Deterministic Models

• Day 4:– More on Parameter Estimation– Introduction to Probability and Stochastic Modeling

Outline of the WorkshopWeek 2

• Day 5:– Discrete Time Markov Chains (DTMC)– DTMC Epidemiological Models

• Day 6:– Continuous Time Markov Chains (CTMC)– CTMC Epidemiological Models

• Day 7:– Markov Chain Monte Carlo (MCMC)– Estimating Parameters using MCMC

Spread of SARS- Timeline

http://www.who.int/csr/sars/en

Influenza Data Timeline from Google

http://www.google.org/flutrends/

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.

Mathematical Epidemiology

• What are epidemics?A widespread occurrence of an infectious disease in a community at a particular time: "a flu epidemic”

• Mathematical Models• Statistical

– Finding ‘trends’ in available data

• Mechanistic– Incorporating the mechanics of transmission

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• Constant rates (transmission, removal

rates• All population classes are well mixed• Births, deaths, immigration, and migration

can be ignored on the timescale of interest

The Model Equations

The rate of transmission of the disease is proportional to the rate of encounter of susceptible and infected individual.

dIdt

= rSI − aI

dSdt

= −rSI

dRdt

= aI

S(0) = S0

I(0) = I0

R(0) = 0

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

• Therefore, the I population will increase initially if and the I populationwill decrease initially otherwise. Therefore, this condition sufficient for an epidemic to occur.

dIdt

= rSI − aI At t = 0 ( )arSIdtdI

−= 00

S0 > a r

When Will an Epidemic Occur?

• Now consider the S-equation

• Therefore, if there will not be an epidemic.

• However if, then an epidemic will occur.

dSdt

= −rSI

dSdt

< 0,∀t

S < S0,∀t

So if

S0 < a r We know

dIdt

= I rS − a( )< 0,∀t

S0 < a r

S0 > a r

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

• 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.

dIdt

= rSI − aI

dSdt

= −rSI

dRdt

= aI

Phase Plane-Graphically

• Nullclines– S-Nullclines: S = 0 and I = 0

• Movement is up/down

– I Nullclines: I = 0 and S = a/r

• Note: S0 +I0 = N– So all trajectories originate on this line

No Epidemic For Small Population ( )- Infected monotonically go to zero, given any initial condition

N

Epidemic For Large Population ( )- Infected increase rapidly before going to zero, given any initial condition.

N

Phase Plane -- Analytically

• Recall that R is decoupled

dIdt

= rSI − aI

dSdt

= −rSIrSa

dSdI

+−= 1

I = −S +ar

ln S + C

000 lnln SraSIS

raSI −+=−+

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* = is the number of people who did not catch the disease. rho=a/r.

S* = S0e−

N−S*( )ρ

R* = N − S*

Gaining Insight into the Model

• How many people will catch the disease before it dies out?

*00 SSIItotal −+=

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

• I achieves its max when S = a/r

rawhere

SSSNI

=

+−=

ρ

ρ

ln0

Imax = N − ρ + ρ ln ρS0

Severity of the Epidemic

Basic Reproduction Number

• As we’ve seen is an important parameter.

• It is called the infectious contact number or basic reproduction number of the of the infection.

R0 =S0ra

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.

R0

• The basic reproduction number, R0 , is defined as the expected number of secondary cases produced by a single (typical) infection in a completely susceptible population.

• It is important to note that R0 is a dimensionless number and not a rate, which would have units of time^(−1)

• R0 ∝ (infection /contact) *(contact/time)*(time/contact)

R0

• More specifically: R0 = τ · c¯· d • where τ is the transmissibility (i.e., probability

of infection given contact between a susceptible and infected individual), ¯c is the average rate of contact between susceptible and infected individuals, and d is the duration of infectiousness.

Nu=25 and r=.3

Nu=25 and r=.3, N=50

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

–

– Therefore an epidemic will occur.

ρ =ar

=0.325e−5 = 6,400

R0 =S0ra

=1.3 >1

Answers

• Does the hospital have enough beds?

• Therefore if the hospital has 245 beds, it will need more.

Imax = N − ρ + ρ ln ρS0

= 268

Answers

• How many of the base staff will get the flu?

• S* = 4783, so about 3649 people (43% of the population) will catch the disease.

Itotal = I0 + S0 − S*

Time Course of the Jade Fever Epidemic

S.I.R.S Model

• The model equations are

• This allows for endemic steady states

Stochastic Models of Epidemics

• A more natural setting to study epidemics: Why?

• Several Approaches• Discrete Time Markov Chain• Continuous Time Markov Chain• Stochastic Differential Equation

DTMC SIR Epidemic Model

• The transition probability is given by

with

Evolution Equation for the Probability Distribution

• The probability distribution of the DTMC SIR Model evolves in time according to

• Note: “Solution” of a stochastic process is its probability distribution!!

Sample Paths for DTMC SIR Model• Three sample paths and the corresponding

deterministic path

Final Outbreak Size

SIR Epidemic Model in continuous Time

• The joint probability distribution is given by

Kolmogorov Equations

• The forward equations are given by

where

Asymptotic Results

• For large N and small I(0)=j

Difference between DTMC and CTMC SIR Model

• If the time step is small the DMC is an approximation to the CTMC

• In a DTMC an event occurs every fixed time step with the given probabilities

• In a CTMC an event occurs after an exponentially distributed time and with the given probabilities