Mathematical Epidemiology Lecture 1 - LUTpersonal.lut.fi/wiki/lib/exe/fetch.php/en/jablonska/math_epid.pdf · Matylda Jabłońska-Sabuka Mathematical Epidemiology. What is Mathematical

Mathematical Epidemiology – Lecture 1

Matylda Jabłońska-Sabuka

Lappeenranta University of Technology

Wrocław, Fall 2013

What is Mathematical Epidemiology?Basic model

Simple extensions of SIR modelBasic terminology

Epidemiology is the subject that studies the spread of diseases inpopulations, and primarily the human populations.

XIX century – William Farr’s statisticsXIX century – Pasteur’s germ theory of infection1880 – notice of measles’ periodic behaviour by Arthur Ransome1906 – discrete time epidemic model for transmission of measles byWilliam Hamer1927 – simplest mathematical model by Kermack and McKendrick.

Matylda Jabłońska-Sabuka Mathematical Epidemiology



Mathematical epidemiologyMathematical epidemiology is concerned with quantitative aspects of thesubjects and usually consists of:

model building,estimation of parameters,investigation of the sensitivity of the model to changes in theparameters,simulations.




Mathematical modellingThe diseases which are modeled most often are the so called infectiousdiseases, that is, diseases that are contagious and can be transferred fromone individual to another through contact.

Examples are: measles, rubella, chicken pox, mumps, HIV, hepatitis,tuberculosis, as well as the very well known influenza.

For various diseases different types of contact are needed in order for thedisease to be transmitted.







Disease classesWhen a disease spreads, the population is divided into nonintersectingclasses:

Susceptible individuals (susceptibles)Infective individuals (infectives)Removed/Recovered individuals.

An optional class is also Latent or Exposed.



Simple extensions of SIR model

Parameter EstimationModel with explicit demographyLong term behavior of the solutionBasic reproduction/reproductive number R0

Kermack and McKendrick, 1927

dS

dt= −βSI (1)

dI

dt= βSI − αI (2)

dR

dt= αI (3)





Parameter estimationEstimation of the parameters in ODE models relies on one observation ofhow various class exit rates relate to reality.

To see how that relates to α, the recovery/removal rate, let us assumethat there is no inflow in the infectious class.

Then the differential equation that gives the dynamics of this class is

I ′(t) = −αI, I(0) = I0





SolutionThe Kermack-McKendrick model is a relatively simple epidemiologicalmodel. Quite a bit of analysis is possible to be done "manually", forinstance, we can obtain an implicit solution, which is rarely possible forepidemic models.

I + S − α

βlnS = C (4)

R = N − I − S (5)





The Kermack-McKendrick model is equipped with initial conditions:S(0) = S0 and I(0) = I0. Those are assumed given.

We assume also that limt→∞ I(t) = 0 while S∞ = limt→∞ S(t) gives thefinal number of susceptible individuals after the epidemic is over.

The implicit solution holds both for (S0, I0) and for (S∞, 0). Thus

β

α=

ln S0S∞

S0 + I0 − S∞





The Great Plague of EyamThe village of Eyam near Sheffield, England, suffered an outbreak ofbubonic plague in 1665-1666. The source of that plague was believed tobe the Great Plague of London. The community has persuadedquarantine itself. Detailed records were preserved.

The initial population of Eyam was 350. In mid-May 1666 there were 254susceptibles and 7 infectives.





Plague data

Date 1666 Susceptibles InfectivesMid-May 254 7July 3/4 235 14.5July 19 201 22

August 3/4 153.5 29August 19 121 21

September 3/4 108 8September 19 97 8October 3/4 Unknown UnknownOctober 19 83 0





These data give the following values: S0 = 254, I0 = 7, S∞ = 83.

From these we obtain βα = 0.00628

The infective period of bubonic plague is about 11 days. Since the data isgiven in months, we convert days to months

11 days = 0.3548387 months, α =1

0.3548387= 2.82

β = 0.00628α = 0.0177





Maximum number of infectivesWhen I = 0 we can find the maximal number of infected individuals

Imax = −αβ

+α

βlnα

β+ S0 + I0 −

α

βlnS0

Then for the Eyam plague we get Imax = 27.5, with the real datapointing at 29.





Models which do not include explicitly births and deaths in the populationare called epidemic models without explicit demography.

They are useful for epidemic modeling on a short time scale.





In reality many disease models are not that simple. And therefore, findingtheir solutions analytically is not possible.

However, parameter estimation is still possible as long as real data isavailable.

Then the estimation can be done through least squares method, that isoptimizing parameter values in such a way that the sum of squares ofdifferences between the model and real data

SS =

n∑i=1

(f(xi, θ)− yi)2

is minimized.





Suppose the total population changes in time according to the simplestdemographic model which describes logistic growth.

Model with explicit demography

S′(t) = Λ− βIS − µS (6)I ′(t) = βIS − αI − µI (7)R′(t) = αI − µR (8)





We would like to know what will happen to the disease in a long run: willit die out or will it establish itself in the population and become anendemic?

To answer this question we have to investigate the long-term behavior ofthe solution. This behavior depends largely on the equilibrium points, thatis time-independent solutions of the system, i.e. for S′(t) = 0, I ′(t) = 0and R′(t) = 0.





We get the system

0 = Λ− βIS − µS (9)0 = βIS − αI − µI (10)0 = αI − µR (11)

From the last equationR =

α

µI

From the second equation we have either I = 0 or

S =α+ µ

β





Case I = 0

Then R = 0. From the first equation we have S = Λµ . Thus we have the

equilibrium solution (Λ

µ, 0, 0

)

This equilibrium exists for any parameter values. Notice that the numberof infectives in it is zero, which means that if a solution of the ODEsystem approaches this equilibrium, the number of infectives I(t) will beapproaching zero, that is the disease will disappear from the population.That is why it is called the disease-free equilibrium.





Case I 6= 0

Then from the first and second equation we have

Λ = βIS + µS, and S =α+ µ

β

Substituting S and solving for I we have

I =βΛ− µ(α+ µ)

β(α+ µ)

Clearly, I > 0 if and only if

βΛ > µ(α+ µ)





Thus only then we have the equilibrium(α+ µ

β,βΛ− µ(α+ µ)

β(α+ µ),α

µ

βΛ− µ(α+ µ)

β(α+ µ)

)

In this equilibrium solution the number of infected is strictly positive. Soif some of the solutions of the ODE system I(t) approaches time goes toinfinity this equilibrium the number of infectives will remain strictlypositive for a long time and approximately equal to I. Thus the diseaseremains in the population and the solution becomes an endemiceqilibrium.





The condition for existence of an endemic equilibrium can be rewritten inthe form

βΛ

µ(α+ µ)> 1

The expression on the left hand side is denoted by R0

R0 =βΛ

µ(α+ µ)

The parameter R0 is called the reproduction number of the disease.





Epidemiologically, the reproductive number of the disease tells us howmany secondary cases will one infected individual produce in an entirelysusceptible population.

If R0 < 1 then there exists only the disease-free equilibrium. Theequilibrium is attractive so that every solution of the ODE systemapproaches this equilibrium and the disease disappears from thepopulation.If R0 > 1 then there are two equilibria: the disease-free equilibriumand the endemic equilibrium. Here the disease-free equilibrium is notattractive in the sense that solutions of the ODE system that startvery close to it tend to go away. The endemic equilibrium isattractive so that solutions of the ODE system approach it as timegoes to infinity. Thus, in this case the disease remains endemic inthe population.




SIRS

dS

dt= −βSI + ρR (12)

dI

dt= βSI − αI (13)

dR

dt= αI − ρR (14)




SIRS with demography

S′(t) = Λ− βIS + ρR− µS (15)I ′(t) = βIS − αI − µI (16)R′(t) = αI − ρR− µR (17)


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Mathematical Epidemiology Lecture 1 - LUTpersonal.lut.fi/wiki/lib/exe/fetch.php/en/jablonska/math_epid.pdf · Matylda Jabłońska-Sabuka Mathematical Epidemiology. What is Mathematical