Epidemic model
Encyclopedia
An Epidemic model is a simplified means of describing the transmission of communicable disease
through individuals.
or isolation
plans and may have a significant effect on the mortality rate
of a particular epidemic
. The modeling of infectious diseases is a tool which has been used to study the mechanisms by which diseases spread, to predict the future course of an outbreak and to evaluate strategies to control an epidemic (Daley & Gani, 2005).
The first scientist who systematically tried to quantify causes of death was John Graunt
in his book Natural and Political Observations made upon the Bills of Mortality, in 1662. The bills he studied were listings of numbers and causes of deaths published weekly. Graunt’s analysis of causes of death is considered the beginning of the “theory of competing risks” which according to Daley and Gani (Daley & Gani, 2005, p. 2) is “a theory that is now well established among modern epidemiologists”.
The earliest account of mathematical modeling of spread of disease was carried out in 1766 by Daniel Bernoulli
. Trained as a physician, Bernoulli created a mathematical model to defend the practice of inoculating against smallpox
(Hethcote, 2000). The calculations from this model showed that universal inoculation against smallpox would increase the life expectancy
from 26 years 7 months to 29 years 9 months (Bernoulli & Blower, 2004).
Following Bernoulli, other physicians contributed to modern mathematical epidemiology. Among the most acclaimed of these were A. G. McKendrick
and W. O. Kermack, whose paper A Contribution to the Mathematical Theory of Epidemics was published in 1927. A simple deterministic (compartmental) model was formulated in this paper. The model was successful in predicting the behavior of outbreaks very similar to that observed in many recorded epidemics (Brauer & Castillo-Chavez, 2001).
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. In other words, the changes in population of a compartment can be calculated using only the history used to develop the model (Brauer & Castillo-Chavez, 2001).
Another approach is through discrete analysis on a lattice (such as a two-dimensional square grid), where the updating is done through asynchronous single-site updates (Kinetic Monte Carlo) or synchronous updating (Cellular Automata). The lattice approach enables inhomogeneities and clustering to be taken into account. Lattice systems are usually studied through computer simulation, and are discussed in the Wikipedia page Epidemic models on lattices
.
, infected, 
, and recovered, 
.The compartments used for this model consist of three classes:
The flow of this model may be considered as follows:
Using a fixed population,
, Kermack and McKendrick derived the following equations:
Several assumptions were made in the formulation of these equations: 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 
others per unit time and the fraction of contacts by an infected with a susceptible is 
. The number of new infections in unit time per infective then is 
, giving the rate of new infections (or those leaving the susceptible category) as 
(Brauer & Castillo-Chavez, 2001). For the second and third 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 rate, or 
the mean infective period) of infectives are leaving this class per unit time to enter the removed class. These processes which occur simultaneously are referred to as the Law of Mass Action, a widely accepted idea that the rate of contact between two groups in a population is proportional to the size of each of the groups concerned (Daley & Gani, 2005). 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.
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:
The only difference is that it allows members of the recovered class to be free of infection and rejoin the susceptible class.
In this model an infection does not leave a long lasting immunity thus individuals that have recovered return to being susceptible again, moving back into the S(t) compartment. The following differential equations describe this model:
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)
To indicate this mathematically, an additional compartment is added, M(t), which results in the following differential equations:
This value quantifies the transmission potential of a disease. If the basic reproduction number falls below one (R0 < 1), i.e. the infective may not pass the infection on during the infectious period, the infection dies out. If R0 > 1 there is an epidemic in the population. In cases where R0 = 1, the disease becomes endemic, meaning the disease remains in the population at a consistent rate, as one infected individual transmits the disease to one susceptible (Trottier & Philippe, 2001).
In cases of diseases with varying latent periods, the basic reproduction number can be calculated as the sum of the reproduction number for each transition time into the disease. An example of this is tuberculosis. Blower et al. (1995) calculated from a simple model of TB the following reproduction number:
In their model, it is assumed that the infected individuals can develop active TB by either direct progression (the disease develops immediately after infection) considered above as FAST tuberculosis or endogenous reactivation (the disease develops years after the infection) considered above as SLOW tuberculosis.
Disease
A disease is an abnormal condition affecting the body of an organism. It is often construed to be a medical condition associated with specific symptoms and signs. It may be caused by external factors, such as infectious disease, or it may be caused by internal dysfunctions, such as autoimmune...
through individuals.
Introduction
The outbreak and spread of disease has been questioned and studied for many years. The ability to make predictions about diseases could enable scientists to evaluate inoculationInoculation
Inoculation is the placement of something that will grow or reproduce, and is most commonly used in respect of the introduction of a serum, vaccine, or antigenic substance into the body of a human or animal, especially to produce or boost immunity to a specific disease...
or isolation
Quarantine
Quarantine is compulsory isolation, typically to contain the spread of something considered dangerous, often but not always disease. The word comes from the Italian quarantena, meaning forty-day period....
plans and may have a significant effect on the mortality rate
Mortality rate
Mortality rate is a measure of the number of deaths in a population, scaled to the size of that population, per unit time...
of a particular epidemic
Epidemic
In epidemiology, an epidemic , occurs when new cases of a certain disease, in a given human population, and during a given period, substantially exceed what is expected based on recent experience...
. The modeling of infectious diseases is a tool which has been used to study the mechanisms by which diseases spread, to predict the future course of an outbreak and to evaluate strategies to control an epidemic (Daley & Gani, 2005).
The first scientist who systematically tried to quantify causes of death was John Graunt
John Graunt
John Graunt was one of the first demographers, though by profession he was a haberdasher. Born in London, the eldest of seven or eight children of Henry and Mary Graunt. His father was a draper who had moved to London from Hampshire...
in his book Natural and Political Observations made upon the Bills of Mortality, in 1662. The bills he studied were listings of numbers and causes of deaths published weekly. Graunt’s analysis of causes of death is considered the beginning of the “theory of competing risks” which according to Daley and Gani (Daley & Gani, 2005, p. 2) is “a theory that is now well established among modern epidemiologists”.
The earliest account of mathematical modeling of spread of disease was carried out in 1766 by Daniel Bernoulli
Daniel Bernoulli
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...
. Trained as a physician, Bernoulli created a mathematical model to defend the practice of inoculating against smallpox
Smallpox
Smallpox was an infectious disease unique to humans, caused by either of two virus variants, Variola major and Variola minor. The disease is also known by the Latin names Variola or Variola vera, which is a derivative of the Latin varius, meaning "spotted", or varus, meaning "pimple"...
(Hethcote, 2000). The calculations from this model showed that universal inoculation against smallpox would increase the life expectancy
Life expectancy
Life expectancy is the expected number of years of life remaining at a given age. It is denoted by ex, which means the average number of subsequent years of life for someone now aged x, according to a particular mortality experience...
from 26 years 7 months to 29 years 9 months (Bernoulli & Blower, 2004).
Following Bernoulli, other physicians contributed to modern mathematical epidemiology. Among the most acclaimed of these were A. G. McKendrick
Anderson Gray McKendrick
Anderson Gray McKendrick was a Scottish physician and epidemiologist pioneered the use of mathematical methods in epidemiology...
and W. O. Kermack, whose paper A Contribution to the Mathematical Theory of Epidemics was published in 1927. A simple deterministic (compartmental) model was formulated in this paper. The model was successful in predicting the behavior of outbreaks very similar to that observed in many recorded epidemics (Brauer & Castillo-Chavez, 2001).
Stochastic
"Stochastic" means being or having a random variable. A stochastic model is a tool for estimating probability distributions of potential outcomes by allowing for random variation in one or more inputs over time. Stochastic models depend on the chance variations in risk of exposure, disease and other illness dynamics. They are used when these fluctuations are important, as in small populations (Trottier & Philippe, 2001).Deterministic
When dealing with large populations, as in the case of tuberculosis, deterministic or compartmental mathematical models are used. In the deterministic model, individuals in the population are assigned to different subgroups or compartments, each representing a specific stage of the epidemic. Letters such as M, S, E, I, and R are often used to represent different stages.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. In other words, the changes in population of a compartment can be calculated using only the history used to develop the model (Brauer & Castillo-Chavez, 2001).
Another approach is through discrete analysis on a lattice (such as a two-dimensional square grid), where the updating is done through asynchronous single-site updates (Kinetic Monte Carlo) or synchronous updating (Cellular Automata). The lattice approach enables inhomogeneities and clustering to be taken into account. Lattice systems are usually studied through computer simulation, and are discussed in the Wikipedia page Epidemic models on lattices
Epidemic models on lattices
Classic epidemic models of disease transmission are described in Epidemic model and Compartmental models in epidemiology. Here we discuss the behavior when such models are simulated on a lattice.-Introduction:...
.
Terminology
The following is a summary of the notation used in this and the next sections.-
-
-
: Passively Immune Infants -
: Susceptibles -
: Exposed Individuals in the Latent Period -
: Infectives -
: Removed with Immunity -
: Contact Rate -
: Average Death Rate -
: Average Birth Rate -
: Average Latent Period -
: Average Infectious Period -
: Basic Reproduction Number -
: Total Population -
: Average Loss of Immunity Rate of Recovered Individuals -
: Average Temporary Immunity Period
-
-
The SIR Model
In 1927, W. O. Kermack and A. G. McKendrick created a model in which they considered a fixed population with only three compartments, susceptible:, infected, , and recovered, .The compartments used for this model consist of three classes:-
is used to represent the number of individuals not yet infected with the disease at time t, or those susceptible to the disease -
denotes the number of individuals who have been infected with the disease and are capable of spreading the disease to those in the susceptible category -
is the compartment used for those individuals who have been infected and then recovered from the disease. Those in this category are not able to be infected again or to transmit the infection to others.
The flow of this model may be considered as follows:
Using a fixed population,
, Kermack and McKendrick derived the following equations:
Several assumptions were made in the formulation of these equations: 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 others per unit time and the fraction of contacts by an infected with a susceptible is . The number of new infections in unit time per infective then is , giving the rate of new infections (or those leaving the susceptible category) as (Brauer & Castillo-Chavez, 2001). For the second and third 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 rate, or the mean infective period) of infectives are leaving this class per unit time to enter the removed class. These processes which occur simultaneously are referred to as the Law of Mass Action, a widely accepted idea that the rate of contact between two groups in a population is proportional to the size of each of the groups concerned (Daley & Gani, 2005). 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:The SIS Model with Births and Deaths
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.
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:
The SIRS Model
This model is simply an extension of the SIR model as we will see from its construction.
The only difference is that it allows members of the recovered class to be free of infection and rejoin the susceptible class.
The SEIS Model
The SEIS model takes into consideration the exposed or latent period of the disease, giving an additional compartment, E(t).
In this model an infection does not leave a long lasting immunity thus individuals that have recovered return to being susceptible again, moving back into the S(t) compartment. The following differential equations describe this model:
-
= B - βSI - μS + γI
= βSI - (ε + μ)E
= εE - (γ + μ)I
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.
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)
-
= B - βSI - μS
= βSI - (ε + μ)E
= εE - (γ + μ)I
= γI - μR
The MSIR Model
There are several diseases where an individual is born with a passive immunity from its mother.
To indicate this mathematically, an additional compartment is added, M(t), which results in the following differential equations:
-
= B - δMS - μM
= δMS - βSI - μS
= βSI - γI - μI
= γI - μR
The MSEIR Model
For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model.-
-
= B - δMS - μM
= δMS - βSI - μS
= βSI - (ε + μ)E
= εE - (γ + μ)I
= γI - μR
-
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.
Reproduction Number
There is a threshold quantity which determines whether an epidemic occurs or the disease simply dies out. This quantity is called the basic reproduction number, denoted by R0, which can be defined as the number of secondary infections caused by a single infective introduced into a population made up entirely of susceptible individuals (S(0) ≈ N) over the course of the infection of this single infective. This infective individual makes βN contacts per unit time producing new infections with a mean infectious period of 1/γ. Therefore, the basic reproduction number is-
- R0 = (βN)/γ
This value quantifies the transmission potential of a disease. If the basic reproduction number falls below one (R0 < 1), i.e. the infective may not pass the infection on during the infectious period, the infection dies out. If R0 > 1 there is an epidemic in the population. In cases where R0 = 1, the disease becomes endemic, meaning the disease remains in the population at a consistent rate, as one infected individual transmits the disease to one susceptible (Trottier & Philippe, 2001).
In cases of diseases with varying latent periods, the basic reproduction number can be calculated as the sum of the reproduction number for each transition time into the disease. An example of this is tuberculosis. Blower et al. (1995) calculated from a simple model of TB the following reproduction number:
-
- R0 = R0FAST + R0SLOW
In their model, it is assumed that the infected individuals can develop active TB by either direct progression (the disease develops immediately after infection) considered above as FAST tuberculosis or endogenous reactivation (the disease develops years after the infection) considered above as SLOW tuberculosis.
Vertical Transmission
In the case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is called Vertical Transmission. The influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment (Brauer & Castillo-Chavez, 2001).Vector Transmission
Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for direct transmission. For more information on this type of model see the reference Population Dynamics of Infectious Diseases: Theory and Applications, by R. M. Anderson (Brauer & Castillo-Chavez, 2001).Others
Other occurrences (taken from Mathematical Models in Population Biology and Epidemiology by Fred Brauer and Carlos Castillo-Chávez ) which may need to be considered when modeling an epidemic include things such as the following:- • Nonhomogeneous mixing
- • Age-Structured populations
- • Variable infectivity
- • Distributions that are spatially non-uniform
- • Diseases caused by macroparasites
- • Acquired immunity through vaccinations
See also
- Mathematical modelling of infectious disease
- Compartmental models in epidemiologyCompartmental models in epidemiologyIn order to model the progress of an epidemic in a large population, comprising many different individuals in various fields, the population diversity must be reduced to a few key characteristics which are relevant to the infection under consideration...
- EpidemicEpidemicIn epidemiology, an epidemic , occurs when new cases of a certain disease, in a given human population, and during a given period, substantially exceed what is expected based on recent experience...
- Endemic (epidemiology)Endemic (epidemiology)In epidemiology, an infection is said to be endemic in a population when that infection is maintained in the population without the need for external inputs. For example, chickenpox is endemic in the UK, but malaria is not...
- Transmission risks and ratesTransmission risks and ratesTransmission of an infection requires three conditions:*an infectious individual*a susceptible individual*an effective contact between themAn effective contact is defined as any kind of contact between two individuals such that, if one individual is infectious and the other susceptible, then the...
- Epidemic models on latticesEpidemic models on latticesClassic epidemic models of disease transmission are described in Epidemic model and Compartmental models in epidemiology. Here we discuss the behavior when such models are simulated on a lattice.-Introduction:...
.
Further reading
- An Introduction to Infectious Disease Modelling by Emilia Vynnycky and Richard G White. An introductory book on infectious disease modelling and its applications.