2. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. 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.
8. 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.
9. 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.
10. 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.
11. 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.
12. 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
13. 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.
14. 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
15. 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
16. 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 - R
Where is average loss of immunity rate of
recovered individuals.
17. 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
18. 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.
19. 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)
20. 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
21. 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
22. 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.
23. 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.
24. 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.
25. 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.
