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Compartmental models are a very general modelling technique. They are often applied to the
mathematical modelling of infectious disease Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic (including in plants) and help inform public health and plant health interventions. Models use basic assumptions or collected statistics alo ...
s. The population is assigned to compartments with labels – for example, S, I, or R, (Susceptible, Infectious, or Recovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again. The origin of such models is the early 20th century, with important works being that of
Ross Ross or ROSS may refer to: People * Clan Ross, a Highland Scottish clan * Ross (name), including a list of people with the surname or given name Ross, as well as the meaning * Earl of Ross, a peerage of Scotland Places * RoSS, the Republic of Sou ...
in 1916, Ross and
Hudson Hudson may refer to: People * Hudson (given name) * Hudson (surname) * Henry Hudson, English explorer * Hudson (footballer, born 1986), Hudson Fernando Tobias de Carvalho, Brazilian football right-back * Hudson (footballer, born 1988), Hudso ...
in 1917, Kermack and McKendrick in 1927 and Kendall in 1956. The Reed-Frost model was also a significant and widely-overlooked ancestor of modern epidemiological modelling approaches. The models are most often run with ordinary differential equations (which are deterministic), but can also be used with a stochastic (random) framework, which is more realistic but much more complicated to analyze. Models try to predict things such as how a disease spreads, or the total number infected, or the duration of an epidemic, and to estimate various epidemiological parameters such as the
reproductive number In epidemiology, the basic reproduction number, or basic reproductive number (sometimes called basic reproduction ratio or basic reproductive rate), denoted R_0 (pronounced ''R nought'' or ''R zero''), of an infection is the expected number of ...
. Such models can show how different
public health intervention A public health intervention is any effort or policy that attempts to improve mental and physical health on a population level. Public health interventions may be run by a variety of organizations, including governmental health departments and n ...
s may affect the outcome of the epidemic, e.g., what the most efficient technique is for issuing a limited number of
vaccine A vaccine is a biological Dosage form, preparation that provides active acquired immunity to a particular infectious disease, infectious or cancer, malignant disease. The safety and effectiveness of vaccines has been widely studied and verifie ...
s in a given population.


The SIR model

The SIR model is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments:- :S: The number of susceptible individuals. When a susceptible and an infectious individual come into "infectious contact", the susceptible individual contracts the disease and transitions to the infectious compartment. :I: The number of infectious individuals. These are individuals who have been infected and are capable of infecting susceptible individuals. :R for the number of removed (and immune) or deceased individuals. These are individuals who have been infected and have either recovered from the disease and entered the removed compartment, or died. It is assumed that the number of deaths is negligible with respect to the total population. This compartment may also be called "recovered" or "resistant". This model is reasonably predictive for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance, such as
measles Measles is a highly contagious infectious disease caused by measles virus. Symptoms usually develop 10–12 days after exposure to an infected person and last 7–10 days. Initial symptoms typically include fever, often greater than , cough, ...
,
mumps MUMPS ("Massachusetts General Hospital Utility Multi-Programming System"), or M, is an imperative, high-level programming language with an integrated transaction processing key–value database. It was originally developed at Massachusetts Gener ...
and
rubella Rubella, also known as German measles or three-day measles, is an infection caused by the rubella virus. This disease is often mild, with half of people not realizing that they are infected. A rash may start around two weeks after exposure and ...
. These variables (S, I, and R) represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of ''t'' (time): S(''t''), I(''t'') and R(''t''). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control. As implied by the variable function of ''t'', the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an
endemic Endemism is the state of a species being found in a single defined geographic location, such as an island, state, nation, country or other defined zone; organisms that are indigenous to a place are not endemic to it if they are also found elsew ...
disease with a short infectious period, such as
measles Measles is a highly contagious infectious disease caused by measles virus. Symptoms usually develop 10–12 days after exposure to an infected person and last 7–10 days. Initial symptoms typically include fever, often greater than , cough, ...
in the UK prior to the introduction of a
vaccine A vaccine is a biological Dosage form, preparation that provides active acquired immunity to a particular infectious disease, infectious or cancer, malignant disease. The safety and effectiveness of vaccines has been widely studied and verifie ...
in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(''t'')) over time. During an
epidemic An epidemic (from Ancient Greek, Greek ἐπί ''epi'' "upon or above" and δῆμος ''demos'' "people") is the rapid spread of disease to a large number of patients among a given population within an area in a short period of time. Epidemics ...
, the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment. Each member of the population typically progresses from susceptible to infectious to recovered. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments, i.e.


Transition rates

For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between ''S'' and ''I'', the transition rate is assumed to be ''d(S/N)/dt = -βSI/N2'', where ''N'' is the total population, β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and ''SI/N2'' is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the
law of mass action In chemistry, the law of mass action is the proposition that the rate of the chemical reaction is directly proportional to the product of the activities or concentrations of the reactants. It explains and predicts behaviors of solutions in dyna ...
in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants). Between ''I'' and ''R'', the transition rate is assumed to be proportional to the number of infectious individuals which is γ''I''. This is equivalent to assuming that the probability of an infectious individual recovering in any time interval ''dt'' is simply γ''dt''. If an individual is infectious for an average time period ''D'', then γ = 1/''D''. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an
exponential distribution In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g. the
Erlang distribution The Erlang distribution is a two-parameter family of continuous probability distributions with support x \in independent exponential distribution">exponential variables with mean 1/\lambda each. Equivalently, it is the distribution of the time ...
). For the special case in which there is no removal from the infectious compartment (γ=0), the SIR model reduces to a very simple SI model, which has a logistic solution, in which every individual eventually becomes infected.


The SIR model without vital dynamics

The dynamics of an epidemic, for example, the
flu Influenza, commonly known as "the flu", is an infectious disease caused by influenza viruses. Symptoms range from mild to severe and often include fever, runny nose, sore throat, muscle pain, headache, coughing, and fatigue. These symptom ...
, are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following system of ordinary
differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
: : \left\{\begin{aligned} & \frac{dS}{dt} = - \frac{\beta I S}{N}, \\ pt& \frac{dI}{dt} = \frac{\beta I S}{N}- \gamma I, \\ pt& \frac{dR}{dt} = \gamma I, \end{aligned}\right. where S is the stock of susceptible population, I is the stock of infected, R is the stock of removed population (either by death or recovery), and N is the sum of these three. This model was for the first time proposed by
William Ogilvy Kermack William Ogilvy Kermack FRS FRSE FRIC (26 April 1898 – 20 July 1970) was a Scottish biochemist. He made mathematical studies of epidemic spread and established links between environmental factors and specified diseases. He is noteworthy for ...
and
Anderson Gray McKendrick Lt Col Anderson Gray McKendrick DSc FRSE (8 September 1876 – 30 May 1943) was a Scottish military physician and epidemiologist who pioneered the use of mathematical methods in epidemiology. Irwin (see below) commented on the quality of his wor ...
as a special case of what we now call
Kermack–McKendrick theory Kermack–McKendrick theory is a hypothesis that predicts the number and distribution of cases of an infectious disease as it is transmitted through a population over time. Building on the research of Ronald Ross and Hilda Hudson, A. G. McKendrick ...
, and followed work McKendrick had done with
Ronald Ross Sir Ronald Ross (13 May 1857 – 16 September 1932) was a British medical doctor who received the Nobel Prize for Physiology or Medicine in 1902 for his work on the transmission of malaria, becoming the first British Nobel laureate, and the f ...
. This system is
non-linear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
, however it is possible to derive its analytic solution in implicit form. Firstly note that from: : \frac{dS}{dt} + \frac{dI}{dt} + \frac{dR}{dt} = 0, it follows that: : S(t) + I(t) + R(t) = \text{constant} = N, expressing in mathematical terms the constancy of population N . Note that the above relationship implies that one need only study the equation for two of the three variables. Secondly, we note that the dynamics of the infectious class depends on the following ratio: : R_0 = \frac{\beta}{\gamma}, the so-called
basic reproduction number In epidemiology, the basic reproduction number, or basic reproductive number (sometimes called basic reproduction ratio or basic reproductive rate), denoted R_0 (pronounced ''R nought'' or ''R zero''), of an infection is the expected number of ...
(also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible. This idea can probably be more readily seen if we say that the typical time between contacts is T_{c} = \beta^{-1}, and the typical time until removal is T_{r} = \gamma^{-1}. From here it follows that, on average, the number of contacts by an infectious individual with others ''before'' the infectious has been removed is: T_{r}/T_{c}. By dividing the first differential equation by the third, separating the variables and integrating we get : S(t) = S(0) e^{-R_0(R(t) - R(0))/N}, where S(0) and R(0) are the initial numbers of, respectively, susceptible and removed subjects. Writing s_0 = S(0) / N for the initial proportion of susceptible individuals, and s_\infty = S(\infty) / N and r_\infty = R(\infty) / N for the proportion of susceptible and removed individuals respectively in the limit t \to \infty, one has :s_\infty = 1 - r_\infty = s_0 e^{-R_0(r_\infty - r_0)} (note that the infectious compartment empties in this limit). This
transcendental equation In applied mathematics, a transcendental equation is an equation over the real number, real (or complex number, complex) numbers that is not algebraic equation, algebraic, that is, if at least one of its sides describes a transcendental function. ...
has a solution in terms of the Lambert function, namely :s_\infty = 1-r_\infty = - R_0^{-1}\, W(-s_0 R_0 e^{-R_0(1-r_0)}). This shows that at the end of an epidemic that conforms to the simple assumptions of the SIR model, unless s_0=0, not all individuals of the population have been removed, so some must remain susceptible. A driving force leading to the end of an epidemic is a decline in the number of infectious individuals. The epidemic does not typically end because of a complete lack of susceptible individuals. The role of both the
basic reproduction number In epidemiology, the basic reproduction number, or basic reproductive number (sometimes called basic reproduction ratio or basic reproductive rate), denoted R_0 (pronounced ''R nought'' or ''R zero''), of an infection is the expected number of ...
and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows: : \frac{dI}{dt} = \left(R_0 \frac{S}{N} - 1\right) \gamma I, it yields that if: : R_{0} \cdot S(0) > N, then: : \frac{dI}{dt}(0) >0 , i.e., there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if : R_{0} \cdot S(0) < N, then : \frac{dI}{dt}(0) <0 , i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that both the
basic reproduction number In epidemiology, the basic reproduction number, or basic reproductive number (sometimes called basic reproduction ratio or basic reproductive rate), denoted R_0 (pronounced ''R nought'' or ''R zero''), of an infection is the expected number of ...
and the initial susceptibility are extremely important.


The force of infection

Note that in the above model the function: : F = \beta I, models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the
force of infection In epidemiology, force of infection (denoted \lambda) is the rate at which susceptible individuals acquire an infectious disease. Because it takes account of susceptibility it can be used to compare the rate of transmission between different group ...
. However, for large classes of communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population N): : F = \beta \frac{I}{N} . Capasso and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process.


Exact analytical solutions to the SIR model

In 2014, Harko and coauthors derived an exact so-called analytical solution (involving an integral that can only be calculated numerically) to the SIR model. In the case without vital dynamics setup, for \mathcal{S}(u)=S(t), etc., it corresponds to the following time parametrization :\mathcal{S}(u)= S(0)u :\mathcal{I}(u)= N -\mathcal{R}(u)-\mathcal{S}(u) :\mathcal{R}(u)=R(0) -\rho \ln(u) for :t= \frac{N}{\beta}\int_u^1 \frac{du^*}{u^*\mathcal{I}(u^*)} , \quad \rho=\frac{\gamma N}{\beta}, with initial conditions :(\mathcal{S}(1),\mathcal{I}(1),\mathcal{R}(1))=(S(0),N -R(0)-S(0),R(0)), \quad u_T where u_T satisfies \mathcal{I}(u_T)=0. By the transcendental equation for R_{\infty} above, it follows that u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0), if S(0) \neq 0) and I_{\infty}=0. An equivalent so-called analytical solution (involving an integral that can only be calculated numerically) found by Miller yields : \begin{align} S(t) & = S(0) e^{-\xi(t)} \\ ptI(t) & = N-S(t)-R(t) \\ ptR(t) & = R(0) + \rho \xi(t) \\ pt\xi(t) & = \frac{\beta}{N}\int_0^t I(t^*) \, dt^* \end{align} Here \xi(t) can be interpreted as the expected number of transmissions an individual has received by time t. The two solutions are related by e^{-\xi(t)} = u. Effectively the same result can be found in the original work by Kermack and McKendrick. These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to I. The equations may thus be divided through by I, and the time rescaled so that the differential operator on the left-hand side becomes simply d/d\tau, where d\tau=I dt, i.e. \tau=\int I dt. The differential equations are now all linear, and the third equation, of the form dR/d\tau = const., shows that \tau and R (and \xi above) are simply linearly related. A highly accurate analytic approximant of the SIR model as well as exact analytic expressions for the final values S_{\infty}, I_{\infty}, and R_{\infty} were provided by Kröger and Schlickeiser, so that there is no need to perform a numerical integration to solve the SIR model (a simplified example practice on
COVID-19 Coronavirus disease 2019 (COVID-19) is a contagious disease caused by a virus, the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The first known case was COVID-19 pandemic in Hubei, identified in Wuhan, China, in December ...
numerical simulation using
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can be found here ), to obtain its parameters from existing data, or to predict the future dynamics of an epidemics modeled by the SIR model. The approximant involves the Lambert function which is part of all basic data visualization software such as
Microsoft Excel Microsoft Excel is a spreadsheet developed by Microsoft for Microsoft Windows, Windows, macOS, Android (operating system), Android and iOS. It features calculation or computation capabilities, graphing tools, pivot tables, and a macro (comp ...
,
MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation ...
, and
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimizat ...
. While Kendall considered the so-called all-time SIR model where the initial conditions S(0), I(0), and R(0) are coupled through the above relations, Kermack and McKendrick proposed to study the more general semi-time case, for which S(0) and I(0) are both arbitrary. This latter version, denoted as semi-time SIR model, makes predictions only for future times t>0. An analytic approximant and exact expressions for the final values are available for the semi-time SIR model as well.


Numerical solutions to the SIR model with approximations

Numerical solutions to the SIR model can be found in the literature. An example is using the model to analyze
COVID-19 Coronavirus disease 2019 (COVID-19) is a contagious disease caused by a virus, the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The first known case was COVID-19 pandemic in Hubei, identified in Wuhan, China, in December ...
spreading data. Three reproduction numbers can be pulled out from the data analyzed with numerical approximation, :the
basic reproduction number In epidemiology, the basic reproduction number, or basic reproductive number (sometimes called basic reproduction ratio or basic reproductive rate), denoted R_0 (pronounced ''R nought'' or ''R zero''), of an infection is the expected number of ...
: ::R_0=\frac{\beta_0}{\gamma_0} :the real-time reproduction number: ::R_t=\frac{\beta_t}{\gamma_t} :and the real-time effective reproduction number: ::R_e=\frac{\beta_tS}{\gamma_tN} R_0 represents the speed of reproduction rate at the beginning of the spreading when all populations are assumed susceptible, e.g. if \beta_0 = 0.4 day^{-1} and \gamma_0 = 0.2 day^{-1} meaning one infectious person on average infects 0.4 susceptible people per day and recovers in 1/0.2=5 days. Thus when this person recovered, there are two people still infectious directly got from this person and R_0 = 2, i.e. the number of infectious people doubled in one cycle of 5 days. The data simulated by the model with R_0 = 2 or real data fitted will yield a doubling of the number of infectious people faster than 5 days because the two infected people are infecting people. From the SIR model, we can tell that \beta is determined by the nature of the disease and also a function of the interactive frequency between the infectious person I with the susceptible people S and also the intensity/duration of the interaction like how close they interact for how long and whether or not they both wear masks, thus, it changes over time when the average behavior of the carriers and susceptible people changes. The model use SI to represent these factors but it indeed is referenced to the initial stage when no action is taken to prevent the spread and all population is susceptible, thus all changes are absorbed by the change of \beta. \gamma is usually more stable over time assuming when the infectious person shows symptoms, she/he will seek medical attention or be self-isolated. So if we find R_t changes, most probably the behaviors of people in the community have changed from their normal patterns before the outbreak, or the disease has mutated to a new form. Costive massive detection and isolation of susceptible close contacts have effects on reducing 1/\gamma but whose efficiencies are under debate. This debate is largely on the uncertainty of the number of days reduced from after infectious or detectable whichever comes first to before a symptom shows up for an infected susceptible person. If the person is infectious after symptoms show up, or detection only works for a person with symptoms, then these prevention methods are not necessary, and self-isolation and/or medical attention is the best way to cut the 1/\gamma values. The typical onset of the
COVID-19 Coronavirus disease 2019 (COVID-19) is a contagious disease caused by a virus, the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The first known case was COVID-19 pandemic in Hubei, identified in Wuhan, China, in December ...
infectious period is in the order of one day from the symptoms showing up, making massive detection with typical frequency in a few days useless. R_t does not tell us whether or not the spreading will speed up or slow down in the latter stages when the fraction of susceptible people in the community has dropped significantly after recovery or vaccination. R_e corrects this dilution effect by multiplying the fraction of the susceptible population over the total population. It corrects the effective/transmissible interaction between an infectious person and the rest of the community when many of the interaction is immune in the middle to late stages of the disease spreading. Thus, when R_e > 1, we will see an exponential-like outbreak; when R_e = 1, a steady state reached and no number of infectious people changes over time; and when R_e < 1, the disease decays and fades away over time. Using the differential equations of the SIR model and converting them to numerical discrete forms, one can set up the recursive equations and calculate the S, I, and R populations with any given initial conditions but accumulate errors over a long calculation time from the reference point. Sometimes a convergence test is needed to estimate the errors. Given a set of initial conditions and the disease-spreading data, one can also fit the data with the SIR model and pull out the three reproduction numbers when the errors are usually negligible due to the short time step from the reference point. Any point of the time can be used as the initial condition to predict the future after it using this numerical model with assumption of time-evolved parameters such as population, R_t, and \gamma. However, away from this reference point, errors will accumulate over time thus convergence test is needed to find an optimal time step for more accurate results. Among these three reproduction numbers, R_0 is very useful to judge the control pressure, e.g., a large value meaning the disease will spread very fast and is very difficult to control. R_t is most useful in predicting future trends, for example, if we know the social interactions have reduced 50% frequently from that before the outbreak and the interaction intensities among people are the same, then we can set R_t = 0.5R_0. If social distancing and masks add another 50% cut in infection efficiency, we can set R_t = 0.25R_0. R_e will perfectly correlate with the waves of the spreading and whenever R_e>1, the spreading accelerates, and when R_e<1, the spreading slows down thus useful to set a prediction on the short term trends. Also, it can be used to directly calculate the threshold population of vaccination/immunization for the
herd immunity Herd immunity (also called herd effect, community immunity, population immunity, or mass immunity) is a form of indirect protection that applies only to contagious diseases. It occurs when a sufficient percentage of a population has become imm ...
stage by setting R_t = R_0.


The SIR model with vital dynamics and constant population

Consider a population characterized by a death rate \mu and birth rate \Lambda, and where a communicable disease is spreading. The model with mass-action transmission is: : \begin{align} \frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\ pt\frac{dI}{dt} & = \frac{\beta I S}{N} - \gamma I -\mu I \\ pt\frac{dR}{dt} & = \gamma I - \mu R \end{align} for which the disease-free equilibrium (DFE) is: :\left(S(t),I(t),R(t)\right) =\left(\frac{\Lambda}{\mu},0,0\right). In this case, we can derive a
basic reproduction number In epidemiology, the basic reproduction number, or basic reproductive number (sometimes called basic reproduction ratio or basic reproductive rate), denoted R_0 (pronounced ''R nought'' or ''R zero''), of an infection is the expected number of ...
: : R_0 = \frac{ \beta}{\mu+\gamma}, which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that: : R_0 \le 1 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{DFE} = \left(\frac{\Lambda}{\mu},0,0\right) : R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{EE} = \left(\frac{\gamma+\mu}{\beta},\frac{\mu}{\beta}\left(R_0-1\right), \frac{\gamma}{\beta} \left(R_0-1\right)\right). The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that R_{0} may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.


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, S(t); infected, I(t); and recovered, R(t). The compartments used for this model consist of three classes: * S(t) is used to represent the individuals not yet infected with the disease at time t, or those susceptible to the disease of the population. * I(t) denotes the individuals of the population who have been infected with the disease and are capable of spreading the disease to those in the susceptible category. * R(t) is the compartment used for the individuals of the population who have been infected and then removed from the disease, either due to immunization or due to death. 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: : {\color{blue}{\mathcal{S} \rightarrow \mathcal{I} \rightarrow \mathcal{R} Using a fixed population, N = S(t) + I(t) + R(t) in the three functions resolves that the value N should remain constant within the simulation, if a simulation is used to solve the SIR model. Alternatively, the analytic approximant can be used without performing a simulation. The model is started with values of S(t=0), I(t=0) and R(t=0). These are the number of people in the susceptible, infected and removed categories at time equals zero. If the SIR model is assumed to hold at all times, these initial conditions are not independent. Subsequently, the flow model updates the three variables for every time point with set values for \beta and \gamma. The simulation first updates the infected from the susceptible and then the removed category is updated from the infected category for the next time point (t=1). This describes the flow persons between the three categories. During an epidemic the susceptible category is not shifted with this model, \beta changes over the course of the epidemic and so does \gamma. These variables determine the length of the epidemic and would have to be updated with each cycle. : \frac{dS}{dt} = - \frac{\beta S I}{N} : \frac{dI}{dt} = \frac{\beta S I}{N} - \gamma I : \frac{dR}{dt} = \gamma I 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 a and an equal fraction b of people that an individual makes contact with per unit time. Then, let \beta be the multiplication of a and b. This is the transmission probability times the contact rate. Besides, an infected individual makes contact with b persons per unit time whereas only a fraction, S/N of them are susceptible.Thus, we have every infective can infect a b S = \beta S susceptible persons, and therefore, the whole number of susceptibles infected by infectives per unit time is \beta S I . 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 \gamma (which represents the mean recovery/death rate, or 1/\gamma 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. 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.


Steady-state solutions

The expected duration of susceptibility will be \operatorname E min(T_L\mid T_S)/math> where T_L reflects the time alive (life expectancy) and T_S reflects the time in the susceptible state before becoming infected, which can be simplified to: : \operatorname E min(T_L\mid T_S)\int_0^\infty e^{-(\mu+\delta) x} \, dx= \frac{1}{\mu+\delta}, such that the number of susceptible persons is the number entering the susceptible compartment \mu N times the duration of susceptibility: :S = \frac{\mu N}{\mu + \lambda}. Analogously, the steady-state number of infected persons is the number entering the infected state from the susceptible state (number susceptible, times rate of infection) \lambda = \tfrac{\beta I}{N}, times the duration of infectiousness \tfrac{1}{\mu+v}: :I = \frac{\mu N}{\mu + \lambda} \lambda \frac{1}{\mu + v}.


Other compartmental models

There are many modifications of the SIR model, including those that include births and deaths, where upon recovery there is no immunity (SIS model), where immunity lasts only for a short period of time (SIRS), where there is a latent period of the disease where the person is not infectious (SEIS and SEIR), and where infants can be born with immunity (MSIR).


Variations on the basic SIR model


The SIS model

Some infections, for example, those from the
common cold The common cold or the cold is a viral infectious disease of the upper respiratory tract that primarily affects the respiratory mucosa of the nose, throat, sinuses, and larynx. Signs and symptoms may appear fewer than two days after exposu ...
and
influenza Influenza, commonly known as "the flu", is an infectious disease caused by influenza viruses. Symptoms range from mild to severe and often include fever, runny nose, sore throat, muscle pain, headache, coughing, and fatigue. These symptoms ...
, do not confer any long-lasting immunity. Such infections may give temporary resistance but do not give long-term immunity upon recovery from infection, and individuals become susceptible again. We have the model: : \begin{align} \frac{dS}{dt} & = - \frac{\beta S I}{N} + \gamma I \\ pt\frac{dI}{dt} & = \frac{\beta S I}{N} - \gamma I \end{align} Note that denoting with ''N'' the total population it holds that: :\frac{dS}{dt} + \frac{dI}{dt} = 0 \Rightarrow S(t)+I(t) = N. It follows that: : \frac{dI}{dt} = (\beta - \gamma) I - \frac{\beta}{N} I^2 , i.e. the dynamics of infectious is ruled by a
logistic function A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with equation f(x) = \frac, where For values of x in the domain of real numbers from -\infty to +\infty, the S-curve shown on the right is obtained, with the ...
, so that \forall I(0) > 0: : \begin{align} & \frac{\beta}{\gamma} \le 1 \Rightarrow \lim_{t \to +\infty}I(t)=0, \\ pt& \frac{\beta}{\gamma} > 1 \Rightarrow \lim_{t \to +\infty}I(t) = \left(1 - \frac{\gamma}{\beta} \right) N. \end{align} It is possible to find an analytical solution to this model (by making a transformation of variables: I = y^{-1} and substituting this into the mean-field equations), such that the basic reproduction rate is greater than unity. The solution is given as :I(t) = \frac{I_\infty}{1+V e^{-\chi t. where I_\infty = (1 -\gamma/\beta)N is the endemic infectious population, \chi = \beta-\gamma, and V = I_\infty/I_0 - 1. As the system is assumed to be closed, the susceptible population is then S(t) = N - I(t). As a special case, one obtains the usual logistic function by assuming \gamma=0. This can be also considered in the SIR model with R=0, i.e. no removal will take place. That is the ''SI model''. The differential equation system using S=N-I thus reduces to: : \frac{dI}{dt} \propto I\cdot (N-I). In the long run, in the SI model, all individuals will become infected.


The SIRD model

The ''Susceptible-Infectious-Recovered-Deceased model'' differentiates between ''Recovered'' (meaning specifically individuals having survived the disease and now immune) and ''Deceased''. This model uses the following system of differential equations: : \begin{align} & \frac{dS}{dt} = - \frac{\beta I S}{N}, \\ pt& \frac{dI}{dt} = \frac{\beta I S}{N} - \gamma I - \mu I, \\ pt& \frac{dR}{dt} = \gamma I, \\ pt& \frac{dD}{dt} = \mu I, \end{align} where \beta, \gamma, \mu are the rates of infection, recovery, and mortality, respectively.


The SIRV model

The ''Susceptible-Infectious-Recovered-Vaccinated model'' is an extended SIR model that accounts for vaccination of the susceptible population. This model uses the following system of differential equations: : \begin{align} & \frac{dS}{dt} = - \frac{\beta(t) I S}{N} - v(t) S, \\ pt& \frac{dI}{dt} = \frac{\beta(t) I S}{N} - \gamma(t) I, \\ pt& \frac{dR}{dt} = \gamma(t) I, \\ pt& \frac{dV}{dt} = v(t) S, \end{align} where \beta, \gamma, v are the rates of infection, recovery, and vaccination, respectively. For the semi-time initial conditions S(0)=(1-\eta)N, I(0)=\eta N, R(0)=V(0)=0 and constant ratios k=\gamma(t)/\beta(t) and b=v(t)/\beta(t) the model had been solved approximately. The occurrence of a pandemic outburst requires k+b<1-2\eta and there is a critical reduced vaccination rate b_c beyond which the steady-state size S_\inftyof the susceptible compartment remains relatively close to S(0). Arbitrary initial conditions satisfying S(0)+I(0)+R(0)+V(0)=N can be mapped to the solved special case with R(0)=V(0)=0. The numerical solution of this model to calculate the real-time reproduction number R_t of COVID-19 can be practiced based on information from the different populations in a community. Numerical solution is a commonly used method to analyze complicated kinetic networks when the analytical solution is difficult to obtain or limited by requirements such as boundary conditions or special parameters. It uses recursive equations to calculate the next step by converting the integration into
Riemann sum In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
of discrete time steps e.g., use yesterday's principal and interest rate to calculate today's interest which assumes the interest rate is fixed during the day. The calculation contains projected errors if the analytical corrections on the numerical step size are not included, e.g. when the interest rate of annual collection is simplified to 12 times the monthly rate, a projected error is introduced. Thus the calculated results will carry accumulative errors when the time step is far away from the reference point and a convergence test is needed to estimate the error. However, this error is usually acceptable for data fitting. When fitting a set of data with a close time step, the error is relatively small because the reference point is nearby compared to when predicting a long period of time after a reference point. Once the real-time R_t is pulled out, one can compare it to the
basic reproduction number In epidemiology, the basic reproduction number, or basic reproductive number (sometimes called basic reproduction ratio or basic reproductive rate), denoted R_0 (pronounced ''R nought'' or ''R zero''), of an infection is the expected number of ...
R_0. Before the vaccination, R_t gives the policy maker and general public a measure of the efficiency of social mitigation activities such as social distancing and face masking simply by dividing \frac{R_t}{R_0}. Under massive vaccination, the goal of disease control is to reduce the effective reproduction number R_e = \frac{R_tS}{N} < 1, where S is the number of susceptible population at the time and N is the total population. When R_e < 1, the spreading decays and daily infected cases go down.


The MSIR model

For many infections, including
measles Measles is a highly contagious infectious disease caused by measles virus. Symptoms usually develop 10–12 days after exposure to an infected person and last 7–10 days. Initial symptoms typically include fever, often greater than , cough, ...
, babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the
placenta The placenta is a temporary embryonic and later fetal organ that begins developing from the blastocyst shortly after implantation. It plays critical roles in facilitating nutrient, gas and waste exchange between the physically separate mater ...
and additionally through
colostrum Colostrum, also known as beestings or first milk, is the first form of milk produced by the mammary glands of mammals (including humans) immediately following delivery of the newborn. Colostrum powder is rich in high protein and low in sugar and ...
). This is called
passive immunity Passive immunity is the transfer of active humoral immunity of ready-made antibodies. Passive immunity can occur naturally, when maternal antibodies are transferred to the fetus through the placenta, and it can also be induced artificially, when hi ...
. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model. To indicate this mathematically, an additional compartment is added, ''M''(''t'')''.'' This results in the following differential equations: :: \begin{align} \frac{dM}{dt} & = \Lambda - \delta M - \mu M\\ pt\frac{dS}{dt} & = \delta M - \frac{\beta SI}{N} - \mu S\\ pt\frac{dI}{dt} & = \frac{\beta SI}{N} - \gamma I - \mu I\\ pt\frac{dR}{dt} & = \gamma I - \mu R \end{align}


Carrier state

Some people who have had an infectious disease such as
tuberculosis Tuberculosis (TB) is an infectious disease usually caused by '' Mycobacterium tuberculosis'' (MTB) bacteria. Tuberculosis generally affects the lungs, but it can also affect other parts of the body. Most infections show no symptoms, in ...
never completely recover and continue to
carry Carry or carrying may refer to: People *Carry (name) Finance * Carried interest (or carry), the share of profits in an investment fund paid to the fund manager * Carry (investment), a financial term: the carry of an asset is the gain or cost of h ...
the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably
Mary Mallon Mary Mallon (September 23, 1869 – November 11, 1938), commonly known as Typhoid Mary, was an Irish-born American cook believed to have infected between 51 and 122 people with typhoid fever. The infections caused three confirmed deaths, ...
, who infected 22 people with
typhoid fever Typhoid fever, also known as typhoid, is a disease caused by '' Salmonella'' serotype Typhi bacteria. Symptoms vary from mild to severe, and usually begin six to 30 days after exposure. Often there is a gradual onset of a high fever over several ...
. The carrier compartment is labelled C.


The SEIR model

For many important infections, there is a significant latency period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment ''E'' (for exposed). Assuming that the latency period is a random variable with exponential distribution with parameter a (i.e. the average latency period is a^{-1}), and also assuming the presence of vital dynamics with birth rate \Lambda equal to death rate N\mu (so that the total number N is constant), we have the model: : \begin{align} \frac{dS}{dt} & = \mu N - \mu S - \frac{\beta I S}{N} \\ pt\frac{dE}{dt} & = \frac{\beta I S}{N} - (\mu +a ) E \\ pt\frac{dI}{dt} & = a E - (\gamma +\mu ) I \\ pt\frac{dR}{dt} & = \gamma I - \mu R. \end{align} We have S+E+I+R=N, but this is only constant because of the simplifying assumption that birth and death rates are equal; in general N is a variable. For this model, the basic reproduction number is: :R_0 = \frac{a}{\mu+a}\frac{\beta}{\mu+\gamma}. Similarly to the SIR model, also, in this case, we have a Disease-Free-Equilibrium (''N'',0,0,0) and an Endemic Equilibrium EE, and one can show that, independently from biologically meaningful initial conditions : \left(S(0),E(0),I(0),R(0)\right) \in \left\{(S,E,I,R)\in ,N4 : S \ge 0, E \ge 0, I\ge 0, R\ge 0, S+E+I+R = N \right\} it holds that: : R_0 \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = DFE = (N,0,0,0), : R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = EE. In case of periodically varying contact rate \beta(t) the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients: : \begin{align} \frac{dE_1}{dt} & = \beta(t) I_1 - (\gamma +a ) E_1 \\ pt\frac{dI_1}{dt} & = a E_1 - (\gamma +\mu ) I_1 \end{align} is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).


The SEIS model

The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end. :::{\color{blue}{\mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{S} In this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible, moving back into the ''S''(''t'') compartment. The following differential equations describe this model: :: \begin{align} \frac{dS}{dt} & = \Lambda - \frac{\beta SI}{N} - \mu S + \gamma I \\ pt\frac{dE}{dt} & = \frac{\beta SI}{N} - (\epsilon + \mu)E \\ pt\frac{dI}{dt} & = \varepsilon E - (\gamma + \mu)I \end{align}


The MSEIR model

For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model. ::: \color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R :: \begin{align} \frac{dM}{dt} & = \Lambda - \delta M - \mu M \\ pt\frac{dS}{dt} & = \delta M - \frac{\beta SI}{N} - \mu S \\ pt\frac{dE}{dt} & = \frac{\beta SI}{N} - (\varepsilon + \mu)E \\ pt\frac{dI}{dt} & = \varepsilon E - (\gamma + \mu)I \\ pt\frac{dR}{dt} & = \gamma I - \mu R \end{align}


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. :::{\color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R} \to \mathcal{S}


Variable contact rates

It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic. In addition, Some diseases are seasonal, such as the
common cold The common cold or the cold is a viral infectious disease of the upper respiratory tract that primarily affects the respiratory mucosa of the nose, throat, sinuses, and larynx. Signs and symptoms may appear fewer than two days after exposu ...
viruses, which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases. As a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate : F = \beta(t) \frac{I}{N} , \quad \beta(t+T)=\beta(t) with period T equal to one year. Thus, our model becomes : \begin{align} \frac{dS}{dt} & = \mu N - \mu S - \beta(t) \frac{I}{N} S \\ pt\frac{dI}{dt} & = \beta(t) \frac{I}{N} S - (\gamma +\mu ) I \end{align} (the dynamics of recovered easily follows from R=N-S-I), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if: :\frac 1 T \int_0^T \frac{\beta(t)}{\mu+\gamma} \, dt < 1 \Rightarrow \lim_{t \to +\infty} (S(t),I(t)) = DFE = (N,0), whereas if the integral is greater than one the disease will not die out and there may be such resonances. For example, considering the periodically varying contact rate as the 'input' of the system one has that the output is a periodic function whose period is a multiple of the period of the input. This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic.


SIR model with diffusion

Spatiotemporal compartmental models describe not the total number, but the density of susceptible/infective/recovered persons. Consequently, they also allow to model the distribution of infected persons in space. In most cases, this is done by combining the SIR model with a diffusion equation : \begin{align} & \partial_t S = D_S \nabla^2 S - \frac{\beta I S}{N}, \\ pt& \partial_t I = D_I \nabla^2 I + \frac{\beta I S}{N}- \gamma I, \\ pt& \partial_t R = D_R \nabla^2 R + \gamma I, \end{align} where D_S, D_I and D_R are diffusion constants. Thereby, one obtains a reaction-diffusion equation. (Note that, for dimensional reasons, the parameter \beta has to be changed compared to the simple SIR model.) Early models of this type have been used to model the spread of the black death in Europe. Extensions of this model have been used to incorporate, e.g., effects of nonpharmaceutical interventions such as social distancing.


Interacting Subpopulation SEIR Model

As social contacts, disease severity and lethality, as well as the efficacy of prophylactic measures may differ substantially between interacting subpopulations, e.g., the elderly versus the young, separate SEIR models for each subgroup may be used that are mutually connected through interaction links. Such Interacting Subpopulation SEIR models have been used for modeling the COVID-19 pandemic at continent scale to develop personalized, accelerated, subpopulation-targeted vaccination strategies that promise a shortening of the pandemic and a reduction of case and death counts in the setting of limited access to vaccines during a wave of virus Variants of Concern.


SIR Model on Networks

The SIR model has been studied on networks of various kinds in order to model a more realistic form of connection than the homogeneous mixing condition which is usually required. A simple model for epidemics on networks in which an individual has a probability p of being infected by each of his infected neighbors in a given time step leads to results similar to giant component formation on Erdos Renyi random graphs.


SIRSS model - combination of SIR with modelling of social stress

Dynamics of epidemics depend on how people's behavior changes in time. For example, at the beginning of the epidemic, people are ignorant and careless, then, after the outbreak of epidemics and alarm, they begin to comply with the various restrictions and the spreading of epidemics may decline. Over time, some people get tired/frustrated by the restrictions and stop following them (exhaustion), especially if the number of new cases drops down. After resting for some time, they can follow the restrictions again. But during this pause the second wave can come and become even stronger than the first one. Social dynamics should be considered. The
social physics Social physics or sociophysics is a field of science which uses mathematical tools inspired by physics to understand the behavior of human crowds. In a modern commercial use, it can also refer to the analysis of social phenomena with big data. Soci ...
models of
social stress Social stress is stress that stems from one's relationships with others and from the social environment in general. Based on the appraisal theory of emotion, stress arises when a person evaluates a situation as personally relevant and perceives t ...
complement the classical epidemics models. The simplest SIR-social stress (SIRSS) model is organised as follows. The susceptible individuals (S) can be split in three subgroups by the types of behavior: ignorant or unaware of the epidemic (Sign), rationally resistant (Sres), and exhausted (Sexh) that do not react on the external stimuli (this is a sort of refractory period). In other words: S(t) = Sign(t) + Sres(t) + Sexh(t). Symbolically, the social stress model can be presented by the "reaction scheme" (where I denotes the infected individuals) * Sign + 2I → Sres + 2I – mobilization reaction (the autocatalytic form here means that the transition rate is proportional to the square of the infected fraction I); * Sres → Sexh – exhaustion process due to fatigue from anti-epidemic restrictions; * Sexh → Sign – slow relaxation to the initial state (end of the refractory period). The main SIR epidemic reaction * S...+I → 2I has different reaction rate constants \beta for Sign, Sres, and Sexh. Presumably, for Sres, \beta is lower than for Sign and Sign. The differences between countries are concentrated in two kinetic constants: the rate of mobilization and the rate of exhaustion calculated for COVID-19 epidemic in 13 countries. These constants for this epidemic in all countries can be extracted by the fitting of the SIRSS model to publicly available data


The KdV-SIR equation

Based on the classical SIR model, a Korteweg-de Vries (KdV)–SIR equation and its analytical solution have been proposed to illustrate the fundamental dynamics of an epidemic wave, the dependence of solutions on parameters, and the dependence of predictability horizons on various types of solutions. The KdV-SIR equation is written as follows: \frac{d^2I}{dt} -\sigma_o^2 I + \frac{3}{2} \frac{\sigma_o^2} {I_{max I^2 = 0 . Here, \sigma_o = \gamma (R_o -1) , R_o = \frac{\beta}{\gamma} \frac{S_o}{N} , and I_{max}=\frac{S_o}{2}\frac{(R_o-1)^2}{R_o^2} . S_o indicates the initial value of the state variable S. Parameters \sigma_o(σ-naught) and R_o (R-naught) are the time-independent relative growth rate and basic reproduction number, respectively. I_{max}presents the maximum of the state variables I(for the number of infected persons). An analytical solution to the KdV-SIR equation is written as follows: I=I_{max}sech^2 \left( \frac{\sigma_o}{2}t \right) , which represents a solitary wave solution.


Modelling vaccination

The SIR model can be modified to model vaccination. Typically these introduce an additional compartment to the SIR model, V, for vaccinated individuals. Below are some examples.


Vaccinating newborns

In presence of a communicable diseases, one of the main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate P \in (0,1): : \begin{align} \frac{dS}{dt} & = \nu N (1-P) - \mu S - \beta \frac{I}{N} S \\ pt\frac{dI}{dt} & = \beta \frac{I}{N} S - (\mu+\gamma) I \\ pt\frac{dV}{dt} & = \nu N P - \mu V \end{align} where V is the class of vaccinated subjects. It is immediate to show that: : \lim_{t \to +\infty} V(t)= N P, thus we shall deal with the long term behavior of S and I, for which it holds that: : R_0 (1-P) \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = DFE = \left(N \left(1-P\right),0\right) : R_0 (1-P) > 1 , \quad I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = EE = \left(\frac{N}{R_0(1-P)},N \left(R_0 (1-P)-1\right)\right). In other words, if : P < P^{*}= 1-\frac{1}{R_0} the vaccination program is not successful in eradicating the disease, on the contrary, it will remain endemic, although at lower levels than the case of absence of vaccinations. This means that the mathematical model suggests that for a disease whose
basic reproduction number In epidemiology, the basic reproduction number, or basic reproductive number (sometimes called basic reproduction ratio or basic reproductive rate), denoted R_0 (pronounced ''R nought'' or ''R zero''), of an infection is the expected number of ...
may be as high as 18 one should vaccinate at least 94.4% of newborns in order to eradicate the disease.


Vaccination and information

Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. In order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects: : P=P(I), \quad P'(I)>0. In such a case the eradication condition becomes: : P(0) \ge P^{*}, i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline.


Vaccination of non-newborns

In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows: : \begin{align} \frac{dS}{dt} & = \mu N (1-P) - \mu S - \rho S - \beta \frac{I}{N} S \\ pt\frac{dV}{dt} & = \mu N P + \rho S - \mu V \end{align} leading to the following eradication condition: : P \ge 1- \left(1+\frac{\rho}{\mu}\right)\frac{1}{R_0}


Pulse vaccination strategy

This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects: : \begin{align} \frac{dS}{dt} & = \mu N - \mu S - \beta \frac{I}{N} S, \quad S(n T^+) = (1-p) S(n T^-), & & n=0,1,2,\ldots \\ pt\frac{dV}{dt} & = - \mu V, \quad V(n T^+) = V(n T^-) + p S(n T^-), & & n=0,1,2,\ldots \end{align} It is easy to see that by setting one obtains that the dynamics of the susceptible subjects is given by: : S^*(t) = 1- \frac{p}{1-(1-p)E^{-\mu T E^{-\mu MOD(t,T)} and that the eradication condition is: : R_0 \int_0^T S^*(t) \, dt < 1


The influence of age: age-structured models

Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic classes s(t,a),i(t,a),r(t,a) (to limit ourselves to the susceptible-infectious-removed scheme) such that: :S(t)=\int_0^{a_M} s(t,a)\,da :I(t)=\int_0^{a_M} i(t,a)\,da :R(t)=\int_0^{a_M} r(t,a)\,da (where a_M \le +\infty is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by
integro-differential equation In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function. General first order linear equations The general first-order, linear (only with respect to the term involving derivative ...
s: :\partial_t s(t,a) + \partial_a s(t,a) = -\mu(a) s(a,t) - s(a,t)\int_0^{a_M} k(a,a_1;t)i(a_1,t)\,da_1 :\partial_t i(t,a) + \partial_a i(t,a) = s(a,t)\int_{0}^{a_M}{k(a,a_1;t)i(a_1,t)da_1} -\mu(a) i(a,t) - \gamma(a)i(a,t) :\partial_t r(t,a) + \partial_a r(t,a) = -\mu(a) r(a,t) + \gamma(a)i(a,t) where: :F(a,t,i(\cdot,\cdot))=\int_0^{a_M} k(a,a_1;t)i(a_1,t) \, da_1 is the force of infection, which, of course, will depend, though the contact kernel k(a,a_1;t) on the interactions between the ages. Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed: :i(t,0)=r(t,0)=0 but that are nonlocal for the density of susceptible newborns: :s(t,0)= \int_0^{a_M} \left(\varphi_s(a) s(a,t)+\varphi_i(a) i(a,t)+\varphi_r(a) r(a,t)\right) \, da where \varphi_j(a), j=s,i,r are the fertilities of the adults. Moreover, defining now the density of the total population n(t,a)=s(t,a)+i(t,a)+r(t,a) one obtains: :\partial_t n(t,a) + \partial_a n(t,a) = -\mu(a) n(a,t) In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility \varphi(.) with the mortality \mu(a) must hold: : 1 = \int_0^{a_M} \varphi(a) \exp\left(- \int_0^a{\mu(q)dq} \right) \, da and the demographic equilibrium is :n^*(a)=C \exp\left(- \int_0^a \mu(q) \, dq \right), automatically ensuring the existence of the disease-free solution: :DFS(a)= (n^*(a),0,0). A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.


Other considerations within compartmental epidemic models


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.


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.


Others

Other occurrences which may need to be considered when modeling an epidemic include things such as the following: *Non-homogeneous mixing *Variable infectivity *Distributions that are spatially non-uniform *Diseases caused by macroparasites


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, and as such should be used cautiously. To be more precise, these models are only valid in the
thermodynamic limit In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of particles.S.J. Blundel ...
, where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications. One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on
percolation theory In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnected ...
concepts. Stochastic epidemic models have been studied on different networks and more recently applied to the
COVID-19 pandemic The COVID-19 pandemic, also known as the coronavirus pandemic, is an ongoing global pandemic of coronavirus disease 2019 (COVID-19) caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The novel virus was first identif ...
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See also

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Mathematical modelling in epidemiology Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic (including in plants) and help inform public health and plant health interventions. Models use basic assumptions or collected statistics a ...
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Flatten the curve Flattening the curve was a public health strategy to slow down the spread of the SARS-CoV-2 virus during the early stages of the COVID-19 pandemic. The curve being flattened is the epidemic curve, a visual representation of the number of infect ...
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Modifiable areal unit problem __NOTOC__ The modifiable areal unit problem (MAUP) is a source of statistical bias that can significantly impact the results of statistical hypothesis tests. MAUP affects results when point-based measures of spatial phenomena are aggregated into ...
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Next-generation matrix In epidemiology, the next-generation matrix is used to derive the basic reproduction number, for a compartmental model of the spread of infectious diseases. In population dynamics it is used to compute the basic reproduction number for structured ...
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Risk assessment Broadly speaking, a risk assessment is the combined effort of: # identifying and analyzing potential (future) events that may negatively impact individuals, assets, and/or the environment (i.e. hazard analysis); and # making judgments "on the to ...
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Attack rate In epidemiology, the attack rate is the proportion of an at-risk population that contracts the disease during a specified time interval. It is used in hypothetical predictions and during actual outbreaks of disease. An at-risk population is defined ...
* List of COVID-19 simulation models


References


Further reading

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External links


SIR model: Online experiments with JSXGraph
* {{Computer modeling, state=expanded Epidemiology Scientific models