some mathematical models in epidemiologyhome.iitk.ac.in/~peeyush/mth426/epidemiology.pdf · peeyush...

Preliminary Definitions and AssumptionsMathematical Models and their analysis

Some Mathematical Models in Epidemiology

by

Peeyush Chandra

Department of Mathematics and StatisticsIndian Institute of Technology Kanpur, 208016

Email: [email protected]

Peeyush Chandra Some Mathematical Models in Epidemiology


Definition (Epidemiology)

It is a discipline, which deals with the study of infectious diseasesin a population. It is concerned with all aspects of epidemic, e.g.spread, control, vaccination strategy etc.

WHY Mathematical Models ? The aim of epidemic modeling isto understand and if possible control the spread of the disease. Inthis context following questions may arise:

• How fast the disease spreads ?

• How much of the total population is infected or will be infected ?

• Control measures !

• Effects of Migration/ Environment/ Ecology, etc.

• Persistence of the disease.



Infectious diseases are basically of two types:Acute (Fast Infectious):

• Stay for a short period (days/weeks) e.g. Influenza, Chickenpoxetc.Chronic Infectious Disease:

• Stay for larger period (month/year) e.g. hepatitis.

In general the spread of an infectious disease depends upon:

• Susceptible population,

• Infective population,

• The immune class, and

• The mode of transmission.


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Exposed population


Assumptions

We shall make some general assumptions, which are common to allthe models and then look at some simple models before takingspecific problems.

• The disease is transmitted by contact (direct or indirect)between an infected individual and a susceptible individual.

• There is no latent period for the disease, i.e., the disease istransmitted instantaneously when the contact takes place.

• All susceptible individuals are equally susceptible and all infectedones are equally infectious.

• The population size is large enough to take care of thefluctuations in the spread of the disease, so a deterministic modelis considered.

• The population, under consideration is closed, i.e. has a fixedsize.



• X (t)- population of susceptible, Y (t)- population of infectedones.

• If B is the average contact number with susceptible which leadsto new infection per unit time per infective, then

Y (t + ∆t) = Y (t) + BY (t)∆t

which in the limit ∆t → 0 gives dYdt = BY (t).

S-I Model

• It is observed that B depends upon the susceptible population.So, define β = average number of contact between susceptible andinfective which leads to new infection per unit time per susceptibleper infective, then B = βX (t), which gives,

dY

dt= βX (t)Y (t) = β(N − Y )Y .

where N = X + Y is the total population, which is fixed as per theabove assumption.



• Pathogen causes illness for a period of time followed by immunity.

S(Susseptible) −→ I (Infected)

• S: Previously unexposed to the pathogen.

• I: Currently colonized by pathogen.

• Proportion of populations: S = X/N, I = Y /N.

• We ignore demography of population (death/birth & migration).


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SIS (No Immunity) [Ross, 1916 for Malaria]

• Recovery rate α 1Infectious period is taken to be constant though

infection period (the time spent in the infectious class) isdistributed about a mean value & can be estimated from theclinical data. In the above model, it is assumed that the wholepopulation is divided into two classes susceptible and infective; andthat if one is infected then it remains in that class. However, this isnot the case, as an infected person may recover from the disease.

• If γ is the rate of recovery from infective class to susceptibleclass, then {

dSdt = γI − βSI ,dIdt = βSI − γI , (3)

dI

dt= β

[(1− 1

R0

)− I

]I , R0 =

β

γ.

S∗ = 1R0

and I ∗ = 1− 1R0

, feasible if R0 > 1.



SIR Model (Kermack & McKendrick 1927)

• Pathogen causes illness for a period of time followed by immunity.

S(Susseptible) −→ I (Infected) −→ R(Recovered)

• S: Previously unexposed to the pathogen.

• I: Currently colonized by pathogen.

• R: Successfully cleared from the infection.

• Proportion of populations: S = X/N, I = Y /N, R = Z/N.

• We ignore demography of population (death/birth & migration).



In view of the above assumption the SIR model is given by

dSdt = −βSI ,

dIdt = βSI − γI ,

dRdt = γI ,

(4)

with initial conditions: S(0) > 0, I (0) > 0 & R(0) = 0.

• 1γ : Average infectious period.

• Note that if S(0) < γβ , then initially dI

dt < 0 and the infectiondies out.

• Threshold phenomenon.

• Thus, for an infection to invade, S(0) > γβ or if γ

β : removal rateis small enough then the disease will spread.

• The inverse of removal rate is defined as ’basic reproductiveratio’ R0 = γ

β .



Basic Reproduction Ratio

It is defined as the average number of secondary cases arising froman average primary case in an entirely susceptible population i.e.the rate at which new infections are produces by an infectiousindividual in an entirely susceptible population.

• It measures the maximum reproductive potential for an infectiousdisease.

• If S(0) = 1, then disease will spread if R0 > 1. If the disease istransmitted to more than one host then it will spread.

• R0 depends on disease and host population e.g.

◦ R0 = 2.6 for TB in cattle.

◦ R0 = 3− 4 for Influenza in humans.

◦ R0 = 3.5− 6 for small pox in humans.

◦ R0 = 16− 18 for Measles in humans.Peeyush Chandra Some Mathematical Models in Epidemiology

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Note

• For an infectious disease with an average infectious period 1γ and

transmission rate β, R0 = βγ .

• For a closed population, infection with specified R0, can invadeonly if threshold fraction of susceptible population is greater than1R0

.

• Vaccination can be used to reduce the susceptible populationbelow 1

R0.

Also we see that

dS

dR= −R0S ⇒ S(t) = S(0)e−R0R(t).

• This implies as the epidemic builds up S(t) ↓ and so R(t) ↑.• There will always be same susceptible in the population asS(t) > 0.


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• The epidemic breaks down when I = 0, we can say

S(∞) = 1− R(∞) = S(0)e−R0R(∞).

or1− R(∞)− S(0)e−R0R(∞) = 0. (5)

• R∞ is the final proportion of individuals recovered = totalnumber of infected proportion.

• We can assume that for given S(0) > R0, the equation (2) has aroots between 0 & 1.

[R∞ = 0⇒ 1− S0 > 0 and R∞ = 1⇒ −S(0)e−R0 < 0].

•dR

dt= γ(1− S − R) = γ[1− S(0)e−R0R − R].

• Solve it (i) Numerically.(ii) For small R0R, i.e. when R0 << 1 or at the start of theepidemic.


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SIR model with demography

Assumptions

• Population size does not change - good in developed countries.

• New born are with passive immunity and hence susceptible.

• This is good enough if the average age at which immunity is lost<< typical age of infection.(e.g. in developed countries t1 = 6months << t2 = 4− 5 years.)

• Vertical transmission is ignored.dSdt = ν − βSI − µS ,dIdt = βSI − γI − µI ,dRdt = γI − µR,

(6)



• µ is the rate at which individual suffer natural mortality i.e. 1µ is

natural host life span.

• R0 = βγ+µ .

• 1γ+µ is the average time which an individual spends in each class.

Equilibrium States

(i) (S , I ,R) = (1, 0, 0), disease free.(ii) (S∗, I ∗,R∗) = ( 1

R0, µβ (R0 − 1), (1− 1

R0)(1− µ

γ )), for µ < γ.

• Clearly, R0 > 1 for endemic equilibrium.

• It can be seen that endemic equilibrium is stable if R0 > 1,otherwise disease free equilibrium is stable.


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infected


• The characteristic equation for Jacobian at endemic equilibrium

(λ+ µ)[λ2 + µR0λ+ (µ+ γ)µ(R0 − 1)] = 0. (7)

λ1 = −µ and λ2,3 =−µR0

2±

√µR0

2 − 4AG

2,

where, A = 1µ(R0−1) , denotes mean age of infection.

G = 1µ+γ : typical period of infectivity.



SIRS (Waning Immunity)dSdt = ν + ωR − βSI − µS ,dIdt = βSI − (γ + µ)I ,dRdt = γI − µR − ωR,

(8)

ω : the rate at which immunity is lost.

R0 =β

γ + µ.

SEIR Model

Transmission of disease starts with a low number of pathogen(bacterial cells etc.) which reproduce rapidly within host.

• At this time the pathogen is present in host but can not transmitdisease to other susceptible.



• Infected but not infectious ⇒ E class.dSdt = ν − (βI + µ)S ,dEdt = βSI − (µ+ σ)EdIdt = σE − (γ + µ)I ,dRdt = γI − µR.

(9)

1σ gives latent period.

Carrier dependent

• Hepatitis B: where a proportion of infected persons becomechronic carrier.

• Transmit disease for a long time at low rate.

• Those infected by acutely infectious individuals and thoseinfected by carriers are indistingulus.



• A recently infected person is acutely infectious and then recoverscompletely or moves to carrier class.

S −→ I↗C↘R

dSdt = ν − (βI + εβC )S − µS ,Idt = (βI + εβC )S − (µ+ γ)IdCdt = γqI − δC − µC ,dRdt = γ(1− q)I + δC − µR.

(10)

where ε is reduced transfer rate from chronic carriers.δ is the rate at which individuals leave C class.

R0 =β

µ+ γ+ (

qγ

µ+ γ)(

εβ

µ+ δ).

• qγµ+γ : accounts for people who do not die but become carrier.



Further Considerations:

• The above models are very general in nature. It may be notedthat these models follow a Black Box approach and do notconsider the mechanism of transmission of a disease. In fact thestructure of a disease plays an important role in the modeling.

• It may be noted that a disease can be transmitted directly orindirectly. The direct mode of transmission could be by viralagents (influenza, measles, rubella, chicken pox); by bacteria (TB,meningitis, gonorrhea). While the former ones confer immunityagainst re-infection, there is no such immunity in the later cases.The indirect mode of transmission could be due to vectors orcarriers (malaria).

• Thus, modeling for most of the communicable diseases, requirefurther considerations. In the following, we consider some suchfactors.



(1) Heterogeneous Mixing-Sexually transmitted diseases (STD),e.g. AIDS, the members may have different level of mixing, e.g.preference for mixing with a particular activity.(2) Age structured population- Many of the diseases spread withthe contact in the group of a given age structure, e.g. measles,chicken pox and other childhood diseases spread mainly by contactbetween children of similar age group.(3) The incubation period- It was assumed that the latent periodof disease is very small and that there is no incubation time for thedisease to show. This is not true with disease like Typhoid. Thereis latent period. Such consideration will give rise to time delaymodels.(4) Variable infectivity- In diseases like HIV/AIDS the infectivemembers are highly infective in the initial stages of gettinginfection. Thereafter, the they have a relatively low infectivity for along period before again becoming highly infective beforedeveloping full blown AIDS.



(5) Spatial non-uniformity- It is well known that movement ofbubonic plague from place to place was carried by rats. Thus, thespatial spread becomes important in such cases, which will lead toagain a system of PDEs.(6) Vertical Transmission- In some diseases, the offspring ofinfected members may be born infective (AIDS). So the birth ininfective class needs to be considered. This is called VerticalTransmission.(7) Models with Compartments-In many cases the populationcan be placed in different compartments, whereinter-compartmental interaction takes place, e.g. in Malaria.(8) Stochastic Models- So far we have considered onlydeterministic models, and random effects have not been takencare. This is fine if the population size is large, however, forpopulation size small the stochastic models will be required.



Spread of Malaria- Environmental Effects.

• Malaria is a parasitic disease transmitted by mosquitoes. It iscaused by parasites (such as : Plasmodium (P) falciparum, P.Vivax, P. malariae and P. ovale). It spreads by bites of femalemosquito of Anopheline specie. (Anopheles (An) stephensi, An.dirus, An. gambiae, An. freeborni).

• It is during the course of blood feeding that the protozoanparasites of malaria is ingested by mosquitoes and later transmittedafter reproductions to humans. The phases of the cycle of malariaparasite in mosquitoes (definitive hosts) and humans (intermediatehosts) represent a complex series of processes.



• The production of malarial parasite in human host begins in livercells and undergoes asexual multiplication in red blood cells wherethe merozoites are produced. These merozoites eventually lyse theinfected cells and invade other erythrocytes. During this processcertain merozoites develop into sexual forms- the male and femalegametocytes.

• Mosquitoes become infected when they feed and ingest humanblood containing mature gametocytes. In the mosquito midgut,the macro (female) - micro (male) - gametocytes shed the redblood cell membranes that surround them and develop intogametes. These gametes mature into macrogametes, which acceptmicrogametes to become zygotes.



• The zygote then elongates to become ookinete. The ookinetepenetrates the midgut epithelium and oocyst development takesplace in about 24 hours after the blood meal. The length of timerequired to go from ookinetes to mature sporozoites in oocystdepends upon the type of malaria parasite, mosquito species, theambient temperature, the density of infection, etc. Later,sporozoites escape from oocysts to find their way to mosquitosalivary glands. Only from there the sporozoites are transmitted tohuman host when the female mosquito next feeds.

• It may be noted that to be a good vector the Anophelinemosquito must have appropriate environment (natural, manmadeas well as physiological) so that it can survive long enough for theparasite to develop in its gut and be ready to infect humans duringthe next blood meal. The prevailing environment in its habitatshould also be conducive for fertilization and breeding to increaseits population and there by infecting more persons due toenhanced contacts.



In view of the fact that mosquito feed on water, nectars, sugarsolutions, blood etc. and breed on store/ stagnant water,vegetation and grasses in water, household wastes, etc, thefollowing factors may be important for the spread of malaria:(i) Natural environmental and ecological factors conducive to thesurvival and growth of mosquito population: Examples are rain,temperature, humidity, vegetation and grasses in water, etc.(ii) Human population density related factors (human madeenvironment) conducive to the survival and growth of mosquitopopulation. Examples are open drainage of sewage water, waterponds, water tanks, household wastes, hedges and damp parks, etc.(iii) Demographic factors: Examples are growth of humanpopulation due to immigration, living conditions etc.Although there have been several experimental studies related tosurveys of malaria in different regions, there are not manymathematical models of the spread of disease considering thefactors mentioned above.



REFERENCES

1: Brauer F., Castillo-Chavez C.: Mathematical models inpopulation biology and epidemiology, Springer, New York,2000 .

2: Wai-Yuan Tan, Hulin Wu: Deterministic and stochasticmodels of AIDS Epidemics and HIV Infections withIntervention, World Scientific, London 2005.

3: Herbert W. Hethcote: The Mathematics of InfectiousDiseases. SIAM Review, Vol. 42, No. 4. (Dec., 2000), pp.599-653.



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