EPIDEMIOLOGICAL MODELLING

Presented by,

SUMIT KUMAR DAS

INTRODUCTION An epidemiological modeling is a simplified means of

describing the transmission of communicable disease through individuals.

Models are mainly two types stochastic and deterministic. There are Three basic types of deterministic models for

infectious communicable diseases. These simplest models are formulated as initial value problems

for system of ordinary differential equations are formulated mathematically.

Parameter are estimated for various diseases and are used to compare the vaccination levels necessary for herd immunity for these disease.

Here the models considered here are suitable for disease which transmitted directly from person to person.

WHY DO WE NEED EPIDEMIOLOGICAL MODELING Even though vaccines are available for many

infectious disease, these disease cause suffering and mortality in the world.

In developed countries chronic disease of death such as cancer and heart disease received more attention than infectious diseases.

Recently some infectious disease like HIV which can lead to AIDS has become an important infectious disease for both developing and developed countries.

CONTD… The transmission interactions in a population are

very complex so that it is difficult to comprehend the large scale dynamics of disease spread without the formal structure of mathematical model.

An epidemiological model uses a microscopic description (The role of an infectious individual) to predict the macroscopic behavior of disease spread through a population.

CONTD… Experiments with infectious disease spread in human

populations are often impossible, unethical or expensive that is why epidemiological modeling become a need.

Modeling can often be used to compare different diseases in the same population, the same disease in different populations, or the same disease at different time.

Epidemiological models are useful in comparing the effects of prevention or control procedures.

TYPES OF EPIDEMIC MODELS Stochastic:-"Stochastic" means being or having a random

variable. Stochastic models depend on the chance variations in risk of exposure, disease and other illness dynamics.

Deterministic:-When dealing with large populations, as in the case of tuberculosis, deterministic or compartmental mathematical models are used. The transition rates from one class to another are mathematically expressed as derivatives, hence the model is formulated using differential equations. While building such models, it must be assumed that the population size in a compartment is differentiable with respect to time and that the epidemic process is deterministic.

DETERMINISTIC VERSUS STOCHASTIC EPIDEMIC MODELS It is important to stress that the deterministic

models presented here are valid only in case of sufficiently large populations.

In some cases deterministic models should cautiously be used. For example in case of seasonally varying contact rates the number of infectious subjects may reduce to infinitesimal values, thus maybe invalidating some results that are obtained in the field of chaotic epidemics.

NOTATION The population under consideration is divided into

disjoint classes which change with time t. The susceptible class consists of those individuals

who can incur the disease but are not yet infective, this fraction of population denoted as S(t).

The infective class consists of those who are transmitting the disease to others, this class denoted as I(t).

The removed class denoted by R(t), consists of those who are removed from the susceptible-infective interaction by recovery with immunity, isolation, or death.

ASSUMPTIONS

The population considered has constant size N which is sufficiently large so that the sizes of each class can be considered as continuous variables.

If the model is to include vital dynamics, then it is assumed that births and natural deaths occur at equal rates and that all newborns are susceptible.

The population is homogenously mixing, and the type of direct or indirect contact adequate for transmission depends on the specific disease.

THE BASIC MODELS SIS model:- If the recovery does not give immunity

then the model is called an SIS model. It is appropriate for some bacterial agent disease such as meningitis, plague.

SIR model:-If the individual recovers with permanent immunity, then the model is called SIR model. It is appropriate for viral agent such as measles, small pox, mumps.

SIRS model:- If individuals recover with temporary immunity so that they eventually become susceptible again, then it is appropriate for SIRS model.

Some others models with more compartments like SEIS, SEIR, MSIR, MSEIR, MSEIRS.

THE SIR MODEL The flow of this model may be considered as

follows: S I R Using a fixed population, N = S(t) + I(t) + R(t) Kermack and McKendrick derived the following

equations: S(t)= ds/dt= -SI …………….(i)

I(t)= dI/dt= SI - I……………(ii)

R(t)= dR/dt= I…………………(iii) Several assumptions were made in the formulation

of these equations.

CONTD… First, an individual in the population must be

considered as having an equal probability as every other individual of contracting the disease with a rate of , which is considered the contact or infection rate of the disease.

Therefore, an infected individual makes contact and is able to transmit the disease with N others per unit time and the fraction of contacts by an infected with a susceptible is S/N .

The number of new infections in unit time per infective then is N(S/N), giving the rate of new infections as N(S/N)I = SI

CONTD… For the first and second equations, consider the

population leaving the susceptible class as equal to the number entering the infected class.

However, a number equal to the fraction ( which represents the mean recovery/death rate, or 1/ the mean infective period) of infective are leaving this class per unit time to enter the removed class.

Finally, it is assumed that the rate of infection and recovery is much faster than the time scale of births and deaths and therefore, these factors are ignored in this model.

THE SIR MODEL WITH BIRTHS AND DEATHS Using the case of measles, for example, there is an

arrival of new susceptible individuals into the population. For this type of situation births and deaths must be included in the model.

The following differential equations represent this model, assuming a death rate and birth rate equal to the death rate:

S(t)= dS/dt= -SI + (N – S)

I(t)= dI/dt= SI - I - I R(t)= dR/dt= I - R

THE SIS MODEL The SIS model can be easily derived from the SIR

model by simply considering that the individuals recover with no immunity to the disease, that is, individuals are immediately susceptible once they have recovered.

S I S Removing the equation representing the recovered

population from the SIR model and adding those removed from the infected population into the susceptible population gives the following differential equations:

S(t)= ds/dt= -SI + (N – S) + I I(t)= dI/dt= SI - I - I

THE SIRS MODEL This model is simply an extension of the SIR

model as we will see from its construction. S I R S The only difference is that it allows members of

the recovered class to be free of infection and rejoin the susceptible class.

S(t)= dS/dt= -SI + (N – S) + R I(t)= dI/dt= SI - I - I R(t)= dR/dt= I - R - RWhere is average loss of immunity rate of

recovered individuals.

MODELS WITH MORE COMPARTMENTS Some notations related to the following models:

M(t) : Passively immune infants

E(t) : Exposed individuals in the latent period

1/ : Average latent period

1/ : Average infectious period

B : Average birth rate

 : Average temporary immunity period

Some models with more compartments are discussed below

CONTD… The SEIS model:- The SEIS model takes into

consideration the exposed or latent period of the disease, giving an additional compartment, E(t).

S E I S In this model an infection does not leave any

immunity thus individuals that have recovered return to being susceptible again, moving back into the S(t) compartment.

CONTD… The SEIR model:- The SIR model discussed above

takes into account only those diseases which cause an individual to be able to infect others immediately upon their infection. Many diseases have what is termed a latent or exposed phase, during which the individual is said to be infected but not infectious.

S E I R In this model the host population (N) is broken into

four compartments: susceptible, exposed, infectious, and recovered, with the numbers of individuals in a compartment, or their densities denoted respectively by S(t), E(t), I(t), R(t), that is N = S(t) + E(t) + I(t) + R(t)

CONTD… The MSIR model:- There are several diseases

where an individual is born with a passive immunity from its mother.

M S I R To indicate this mathematically, an additional

compartment is added, M(t). The MSEIR model:- For the case of a disease,

with the factors of passive immunity, and a latency period there is the MSEIR model.

M S E I R

CONTD… The MSEIRS model:- An MSEIRS model is

similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended.

M S E I R S

HERD IMMUNITY AND VACCINATION A population is said to have herd immunity for

disease if enough people are immune so that the disease would not spread if it were suddenly introduced somewhere in the population.

In presence of a communicable diseases, one of main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program.

In order to prevent the spread of infection from an infective, enough people must be immune so that replacement number satisfies S < 1.

CONTD… The susceptible fraction must be small enough so

that the average infective infects less than one person during the infectious period.

Herd immunity in a population is achieved by vaccination of susceptible in the population.

If R is fraction of the population which is immune due to vaccination, then since S = 1-R when I = 0, herd immunity is achieved if (1-R) < 1 or R > 1-1/

For example, if the contact number is 5, at least 80% must be immune to have herd immunity.

CONCLUSION The SIR model with vital dynamics and SIS model

have two intuitively appealing features.

(a) The disease dies out if the contact number satisfies 1 and disease remains endemic if 1.

(b) At an endemic equilibrium, the replacement number is 1; i.e., the average infective replaces itself with one new infective during the infectious period.

The SIR model without vital dynamics might be appropriate for describing an epidemic outbreak during a short time period, whereas the SIR with vital dynamics would be appropriate over longer time period.

CONTD… Although the models discussed here do provide

some insights and useful comparisons, most model now being applied to specific diseases are more complicated.

The other models leading to periodic solutions have features such as a delay corresponding to temporary immunity, nonlinear incidence, variable population size or cross immunity with age structure.

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