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 ${\displaystyle d(S/N)/dt=-\beta SI/N^{2}}$, where ${\displaystyle N}$ is the total population, ${\displaystyle \beta }$ 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 ${\displaystyle SI/N^{2}}$ 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 in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants.^[12])

Between I and R, the transition rate is assumed to be proportional to the number of infectious individuals which is ${\displaystyle \gamma I}$. If an individual is infectious for an average time period ${\displaystyle D}$, then ${\displaystyle \gamma =1/D}$. 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. 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).^[13]

For the special case in which there is no removal from the infectious compartment (${\displaystyle \gamma =0}$), 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 birth and death

A single realization of the SIR epidemic as produced with an implementation of the Gillespie algorithm and the numerical solution of the ordinary differential equation system (dashed)

The dynamics of an epidemic, for example, the flu, 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:^[8][14]

${\displaystyle \left\{{\begin{aligned}&{\frac {dS}{dt}}=-{\frac {\beta IS}{N}},\\[6pt]&{\frac {dI}{dt}}={\frac {\beta IS}{N}}-\gamma I,\\[6pt]&{\frac {dR}{dt}}=\gamma I,\end{aligned}}\right.}$

The SIR model

where ${\displaystyle S}$ is the stock of susceptible population, ${\displaystyle I}$ is the stock of infected, ${\displaystyle R}$ is the stock of removed population (either by death or recovery), and ${\displaystyle N}$ is the sum of these three.

This model was for the first time proposed by William Ogilvy Kermack and Anderson Gray McKendrick as a special case of what we now call Kermack–McKendrick theory, and followed work McKendrick had done with Ronald Ross.

This system is non-linear, however it is possible to derive its analytic solution in implicit form.^[7] Firstly note that from:

${\displaystyle {\frac {dS}{dt}}+{\frac {dI}{dt}}+{\frac {dR}{dt}}=0,}$

it follows that:

${\displaystyle S(t)+I(t)+R(t)={\text{constant}}=N,}$

expressing in mathematical terms the constancy of population ${\displaystyle N}$. 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:

${\displaystyle R_{0}={\frac {\beta }{\gamma }},}$

the so-called basic reproduction number (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.^[15][16] This idea can probably be more readily seen if we say that the typical time between contacts is ${\displaystyle T_{c}=\beta ^{-1}}$, and the typical time until removal is ${\displaystyle T_{r}=\gamma ^{-1}}$. From here it follows that, on average, the number of contacts by an infectious individual with others before the infectious has been removed is: ${\displaystyle T_{r}/T_{c}.}$

By dividing the first differential equation by the third, separating the variables and integrating we get

${\displaystyle S(t)=S(0)e^{-R_{0}(R(t)-R(0))/N},}$

where ${\displaystyle S(0)}$ and ${\displaystyle R(0)}$ are the initial numbers of, respectively, susceptible and removed subjects. Writing ${\displaystyle s_{0}=S(0)/N}$ for the initial proportion of susceptible individuals, and ${\displaystyle s_{\infty }=S(\infty )/N}$ and ${\displaystyle r_{\infty }=R(\infty )/N}$ for the proportion of susceptible and removed individuals respectively in the limit ${\displaystyle t\to \infty ,}$ one has

${\displaystyle s_{\infty }=1-r_{\infty }=s_{0}e^{-R_{0}(r_{\infty }-r_{0})}}$

(note that the infectious compartment empties in this limit). This transcendental equation has a solution in terms of the Lambert W function,^[17] namely

${\displaystyle s_{\infty }=1-r_{\infty }=-R_{0}^{-1}\,W(-s_{0}R_{0}e^{-R_{0}(1-r_{0})}).}$

This shows that at the end of an epidemic that conforms to the simple assumptions of the SIR model, unless ${\displaystyle s_{0}=0}$, 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 and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows:

${\displaystyle {\frac {dI}{dt}}=\left(R_{0}{\frac {S}{N}}-1\right)\gamma I,}$

it yields that if:

${\displaystyle R_{0}\cdot S(0)>N,}$

then:

${\displaystyle {\frac {dI}{dt}}(0)>0,}$

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

${\displaystyle R_{0}\cdot S(0)<N,}$

then

${\displaystyle {\frac {dI}{dt}}(0)<0,}$

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 and the initial susceptibility are extremely important.

The SIR model with vital dynamics and constant population

Consider a population characterized by a death rate ${\displaystyle \mu }$ and birth rate ${\displaystyle \Lambda }$, and where a communicable disease is spreading.^[8] The model with mass-action transmission is:

${\displaystyle {\begin{aligned}{\frac {dS}{dt}}&=\Lambda -\mu S-{\frac {\beta IS}{N}}\\[8pt]{\frac {dI}{dt}}&={\frac {\beta IS}{N}}-\gamma I-\mu I\\[8pt]{\frac {dR}{dt}}&=\gamma I-\mu R\end{aligned}}}$

for which the disease-free equilibrium (DFE) is:

${\displaystyle \left(S(t),I(t),R(t)\right)=\left({\frac {\Lambda }{\mu }},0,0\right).}$

In this case, we can derive a basic reproduction number:

${\displaystyle R_{0}={\frac {\beta }{\mu +\gamma }},}$

which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that:

${\displaystyle R_{0}\leq 1\Rightarrow \lim _{t\to \infty }(S(t),I(t),R(t))={\textrm {DFE}}=\left({\frac {\Lambda }{\mu }},0,0\right)}$

${\displaystyle R_{0}>1,I(0)>0\Rightarrow \lim _{t\to \infty }(S(t),I(t),R(t))={\textrm {EE}}=\left({\frac {\gamma +\mu }{\beta }},{\frac {\mu }{\beta }}\left(R_{0}-1\right),{\frac {\gamma }{\beta }}\left(R_{0}-1\right)\right).}$

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 ${\displaystyle R_{0}}$ 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

Diagram of the SIR model with initial values ${\textstyle S(0)=997,I(0)=3,R(0)=0}$, and rates for infection ${\textstyle \beta =0.4}$ and for recovery ${\textstyle \gamma =0.04}$

Animation of the SIR model with initial values ${\textstyle S(0)=997,I(0)=3,R(0)=0}$, and rate of recovery ${\textstyle \gamma =0.04}$. The animation shows the effect of reducing the rate of infection from ${\textstyle \beta =0.5}$ to ${\textstyle \beta =0.12}$. If there is no medicine or vaccination available, it is only possible to reduce the infection rate (often referred to as "flattening the curve") by appropriate measures such as social distancing.

In 1927, W. O. Kermack and A. G. McKendrick created a model in which they considered a fixed population with only three compartments: susceptible, ${\displaystyle S(t)}$; infected, ${\displaystyle I(t)}$; and recovered, ${\displaystyle R(t)}$. The compartments used for this model consist of three classes:^[4]

• ${\displaystyle S(t)}$ is used to represent the individuals not yet infected with the disease at time t, or those susceptible to the disease of the population.
• ${\displaystyle I(t)}$ 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.
• ${\displaystyle R(t)}$ 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:

${\displaystyle {\color {blue}{{\mathcal {S}}\rightarrow {\mathcal {I}}\rightarrow {\mathcal {R}}}}}$

Using a fixed population, ${\displaystyle N=S(t)+I(t)+R(t)}$ in the three functions resolves that the value ${\displaystyle N}$ should remain constant within the simulation, if a simulation is used to solve the SIR model. Alternatively, the analytic approximant^[9] can be used without performing a simulation. The model is started with values of ${\displaystyle S(t=0)}$, ${\displaystyle I(t=0)}$ and ${\displaystyle R(t=0)}$. 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.^[9] Subsequently, the flow model updates the three variables for every time point with set values for ${\displaystyle \beta }$ and ${\displaystyle \gamma }$. 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, ${\displaystyle \beta }$ changes over the course of the epidemic and so does ${\displaystyle \gamma }$. These variables determine the length of the epidemic and would have to be updated with each cycle.

${\displaystyle {\frac {dS}{dt}}=-{\frac {\beta SI}{N}}}$

${\displaystyle {\frac {dI}{dt}}={\frac {\beta SI}{N}}-\gamma I}$

${\displaystyle {\frac {dR}{dt}}=\gamma I}$

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 ${\displaystyle a}$ and an equal fraction ${\displaystyle b}$ of people that an individual makes contact with per unit time. Then, let ${\displaystyle \beta }$ be the multiplication of ${\displaystyle a}$ and ${\displaystyle b}$. This is the transmission probability times the contact rate. Besides, an infected individual makes contact with ${\displaystyle b}$ persons per unit time whereas only a fraction, ${\displaystyle S/N}$ of them are susceptible. Thus, we have every infective can infect ${\displaystyle abS=\beta S}$ susceptible persons, and therefore, the whole number of susceptibles infected by infectives per unit time is ${\displaystyle \beta SI}$. 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 ${\displaystyle \gamma }$ (which represents the mean recovery/death rate, or ${\displaystyle 1/\gamma }$ 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.^[23]

Steady-state solutions

The expected duration of susceptibility will be ${\displaystyle \operatorname {E} [\min(T_{L}\mid T_{S})]}$ where ${\displaystyle T_{L}}$ reflects the time alive (life expectancy) and ${\displaystyle T_{S}}$ reflects the time in the susceptible state before becoming infected, which can be simplified^[24] to:

${\displaystyle \operatorname {E} [\min(T_{L}\mid T_{S})]=\int _{0}^{\infty }e^{-(\mu +\delta )x}\,dx={\frac {1}{\mu +\delta }},}$

such that the number of susceptible persons is the number entering the susceptible compartment ${\displaystyle \mu N}$ times the duration of susceptibility:

${\displaystyle S={\frac {\mu N}{\mu +\lambda }}.}$

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) ${\displaystyle \lambda ={\tfrac {\beta I}{N}},}$ times the duration of infectiousness ${\displaystyle {\tfrac {1}{\mu +v}}}$:

${\displaystyle I={\frac {\mu N}{\mu +\lambda }}\lambda {\frac {1}{\mu +v}}.}$

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). Compartmental models can also be used to model multiple risk groups, and even the interaction of multiple pathogens.^[25]

The SIS model

Yellow=Susceptible, Maroon=Infected

Some infections, for example, those from the common cold and influenza, 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.

SIS compartmental model

We have the model:

${\displaystyle {\begin{aligned}{\frac {dS}{dt}}&=-{\frac {\beta SI}{N}}+\gamma I\\[6pt]{\frac {dI}{dt}}&={\frac {\beta SI}{N}}-\gamma I\end{aligned}}}$

Note that denoting with N the total population it holds that:

${\displaystyle {\frac {dS}{dt}}+{\frac {dI}{dt}}=0\Rightarrow S(t)+I(t)=N}$.

It follows that:

${\displaystyle {\frac {dI}{dt}}=(\beta -\gamma )I-{\frac {\beta }{N}}I^{2}}$,

i.e. the dynamics of infectious is ruled by a logistic function, so that ${\displaystyle \forall I(0)>0}$:

${\displaystyle {\begin{aligned}&{\frac {\beta }{\gamma }}\leq 1\Rightarrow \lim _{t\to +\infty }I(t)=0,\\[6pt]&{\frac {\beta }{\gamma }}>1\Rightarrow \lim _{t\to +\infty }I(t)=\left(1-{\frac {\gamma }{\beta }}\right)N.\end{aligned}}}$

It is possible to find an analytical solution to this model (by making a transformation of variables: ${\displaystyle I=y^{-1}}$ and substituting this into the mean-field equations),^[26] such that the basic reproduction rate is greater than unity. The solution is given as

${\displaystyle I(t)={\frac {I_{\infty }}{1+Ve^{-\chi t}}}}$.

where ${\displaystyle I_{\infty }=(1-\gamma /\beta )N}$ is the endemic infectious population, ${\displaystyle \chi =\beta -\gamma }$, and ${\displaystyle V=I_{\infty }/I_{0}-1}$. As the system is assumed to be closed, the susceptible population is then ${\displaystyle S(t)=N-I(t)}$.

Whenever the integer nature of the number of agents is evident (populations with fewer than tens of thousands of individuals), inherent fluctuations in the disease spreading process caused by discrete agents result in uncertainties.^[27] In this scenario, the evolution of the disease predicted by compartmental equations deviates significantly from the observed results. These uncertainties may even cause the epidemic to end earlier than predicted by the compartmental equations.

As a special case, one obtains the usual logistic function by assuming ${\displaystyle \gamma =0}$. This can be also considered in the SIR model with ${\displaystyle R=0}$, i.e. no removal will take place. That is the SI model.^[28] The differential equation system using ${\displaystyle S=N-I}$ thus reduces to:

${\displaystyle {\frac {dI}{dt}}\propto I\cdot (N-I).}$

In the long run, in the SI model, all individuals will become infected.

The SIRD model

Diagram of the SIRD model with initial values ${\displaystyle S(0)=997,I(0)=3,R(0)=0}$ and the rates of infection ${\displaystyle \beta =0.4}$, recovery ${\displaystyle \gamma =0.035}$ and mortality ${\displaystyle \mu =0.005}$

Animation of the SIRD model with initial values ${\textstyle S(0)=997,I(0)=3,R(0)=0}$, and rates of recovery ${\textstyle \gamma =0.035}$ and mortality ${\textstyle \mu =0.005}$. The animation shows the effect of reducing the rate of infection from ${\textstyle \beta =0.5}$ to ${\textstyle \beta =0.12}$. If there is no medicine or vaccination available, it is only possible to reduce the infection rate (often referred to as "flattening the curve") by measures such as "social distancing".

The Susceptible-Infectious-Recovered-Deceased model differentiates between Recovered (meaning specifically individuals having survived the disease and now immune) and Deceased.^[15] The SIRD model has semi analytical solutions based on the four parts method.^[29] This model uses the following system of differential equations:

${\displaystyle {\begin{aligned}&{\frac {dS}{dt}}=-{\frac {\beta IS}{N}},\\[6pt]&{\frac {dI}{dt}}={\frac {\beta IS}{N}}-\gamma I-\mu I,\\[6pt]&{\frac {dR}{dt}}=\gamma I,\\[6pt]&{\frac {dD}{dt}}=\mu I,\end{aligned}}}$

where ${\displaystyle \beta ,\gamma ,\mu }$ are the rates of infection, recovery, and mortality, respectively.^[30]

The SIRV model

The Susceptible-Infectious-Recovered-Vaccinated model is an extended SIR model that accounts for vaccination of the susceptible population.^[31] This model uses the following system of differential equations:

${\displaystyle {\begin{aligned}&{\frac {dS}{dt}}=-{\frac {\beta (t)IS}{N}}-v(t)S,\\[6pt]&{\frac {dI}{dt}}={\frac {\beta (t)IS}{N}}-\gamma (t)I,\\[6pt]&{\frac {dR}{dt}}=\gamma (t)I,\\[6pt]&{\frac {dV}{dt}}=v(t)S,\end{aligned}}}$

A cartoon for the SIRV model

where ${\displaystyle \beta ,\gamma ,v}$ are the rates of infection, recovery, and vaccination, respectively. For the semi-time initial conditions ${\displaystyle S(0)=(1-\eta )N}$, ${\displaystyle I(0)=\eta N}$, ${\displaystyle R(0)=V(0)=0}$ and constant ratios ${\displaystyle k=\gamma (t)/\beta (t)}$ and ${\displaystyle b=v(t)/\beta (t)}$ the model had been solved approximately.^[31] The occurrence of a pandemic outburst requires ${\displaystyle k+b<1-2\eta }$ and there is a critical reduced vaccination rate ${\displaystyle b_{c}}$ beyond which the steady-state size ${\displaystyle S_{\infty }}$of the susceptible compartment remains relatively close to ${\displaystyle S(0)}$. Arbitrary initial conditions satisfying ${\displaystyle S(0)+I(0)+R(0)+V(0)=N}$ can be mapped to the solved special case with ${\displaystyle R(0)=V(0)=0}$.^[31]

The numerical solution of this model to calculate the real-time reproduction number ${\displaystyle R_{t}}$ of COVID-19 can be practiced based on information from the different populations in a community.^[22] 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 numerical integration into Riemann sum 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 ${\displaystyle R_{t}}$ is pulled out, one can compare it to the basic reproduction number ${\displaystyle R_{0}}$. Before the vaccination, ${\displaystyle R_{t}}$ 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 ${\displaystyle {\frac {R_{t}}{R_{0}}}}$. Under massive vaccination, the goal of disease control is to reduce the effective reproduction number ${\displaystyle R_{e}={\frac {R_{t}S}{N}}<1}$, where ${\displaystyle S}$ is the number of susceptible population at the time and ${\displaystyle N}$ is the total population. When ${\displaystyle R_{e}<1}$, the spreading decays and daily infected cases go down.

The MSIR model

For many infections, including measles, 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 and additionally through colostrum). This is called passive immunity. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.

MSIR compartmental model

To indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:

${\displaystyle {\begin{aligned}{\frac {dM}{dt}}&=\Lambda -\delta M-\mu M\\[8pt]{\frac {dS}{dt}}&=\delta M-{\frac {\beta SI}{N}}-\mu S\\[8pt]{\frac {dI}{dt}}&={\frac {\beta SI}{N}}-\gamma I-\mu I\\[8pt]{\frac {dR}{dt}}&=\gamma I-\mu R\end{aligned}}}$

Carrier state

Some people who have had an infectious disease such as tuberculosis never completely recover and continue to carry 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, who infected 22 people with typhoid fever. 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).

SEIR compartmental model

Assuming that the latency period is a random variable with exponential distribution with parameter ${\displaystyle a}$ (i.e. the average latency period is ${\displaystyle a^{-1}}$), and also assuming the presence of vital dynamics with birth rate ${\displaystyle \Lambda }$ equal to death rate ${\displaystyle N\mu }$ (so that the total number ${\displaystyle N}$ is constant), we have the model:

${\displaystyle {\begin{aligned}{\frac {dS}{dt}}&=\mu N-\mu S-{\frac {\beta IS}{N}}\\[8pt]{\frac {dE}{dt}}&={\frac {\beta IS}{N}}-(\mu +a)E\\[8pt]{\frac {dI}{dt}}&=aE-(\gamma +\mu )I\\[8pt]{\frac {dR}{dt}}&=\gamma I-\mu R.\end{aligned}}}$

We have ${\displaystyle S+E+I+R=N,}$ but this is only constant because of the simplifying assumption that birth and death rates are equal; in general ${\displaystyle N}$ is a variable.

For this model, the basic reproduction number is:

${\displaystyle R_{0}={\frac {a}{\mu +a}}{\frac {\beta }{\mu +\gamma }}.}$

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

${\displaystyle \left(S(0),E(0),I(0),R(0)\right)\in \left\{(S,E,I,R)\in [0,N]^{4}:S\geq 0,E\geq 0,I\geq 0,R\geq 0,S+E+I+R=N\right\}}$

it holds that:

${\displaystyle R_{0}\leq 1\Rightarrow \lim _{t\to +\infty }\left(S(t),E(t),I(t),R(t)\right)=DFE=(N,0,0,0),}$

${\displaystyle R_{0}>1,I(0)>0\Rightarrow \lim _{t\to +\infty }\left(S(t),E(t),I(t),R(t)\right)=EE.}$

In case of periodically varying contact rate ${\displaystyle \beta (t)}$ the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients:

${\displaystyle {\begin{aligned}{\frac {dE_{1}}{dt}}&=\beta (t)I_{1}-(\gamma +a)E_{1}\\[8pt]{\frac {dI_{1}}{dt}}&=aE_{1}-(\gamma +\mu )I_{1}\end{aligned}}}$

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.

${\displaystyle {\color {blue}{{\mathcal {S}}\to {\mathcal {E}}\to {\mathcal {I}}\to {\mathcal {S}}}}}$

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:

${\displaystyle {\begin{aligned}{\frac {dS}{dt}}&=\Lambda -{\frac {\beta SI}{N}}-\mu S+\gamma I\\[6pt]{\frac {dE}{dt}}&={\frac {\beta SI}{N}}-(\epsilon +\mu )E\\[6pt]{\frac {dI}{dt}}&=\varepsilon E-(\gamma +\mu )I\end{aligned}}}$

The MSEIR model

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

${\displaystyle \color {blue}{{\mathcal {M}}\to {\mathcal {S}}\to {\mathcal {E}}\to {\mathcal {I}}\to {\mathcal {R}}}}$

${\displaystyle {\begin{aligned}{\frac {dM}{dt}}&=\Lambda -\delta M-\mu M\\[6pt]{\frac {dS}{dt}}&=\delta M-{\frac {\beta SI}{N}}-\mu S\\[6pt]{\frac {dE}{dt}}&={\frac {\beta SI}{N}}-(\varepsilon +\mu )E\\[6pt]{\frac {dI}{dt}}&=\varepsilon E-(\gamma +\mu )I\\[6pt]{\frac {dR}{dt}}&=\gamma I-\mu R\end{aligned}}}$

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.

${\displaystyle {\color {blue}{{\mathcal {M}}\to {\mathcal {S}}\to {\mathcal {E}}\to {\mathcal {I}}\to {\mathcal {R}}\to {\mathcal {S}}}}}$

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 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

${\displaystyle F=\beta (t){\frac {I}{N}},\quad \beta (t+T)=\beta (t)}$

with period T equal to one year.

Thus, our model becomes

${\displaystyle {\begin{aligned}{\frac {dS}{dt}}&=\mu N-\mu S-\beta (t){\frac {I}{N}}S\\[8pt]{\frac {dI}{dt}}&=\beta (t){\frac {I}{N}}S-(\gamma +\mu )I\end{aligned}}}$

(the dynamics of recovered easily follows from ${\displaystyle R=N-S-I}$), 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:

${\displaystyle {\frac {1}{T}}\int _{0}^{T}{\frac {\beta (t)}{\mu +\gamma }}\,dt<1\Rightarrow \lim _{t\to +\infty }(S(t),I(t))=DFE=(N,0),}$

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

${\displaystyle {\begin{aligned}&\partial _{t}S=D_{S}\nabla ^{2}S-{\frac {\beta IS}{N}},\\[6pt]&\partial _{t}I=D_{I}\nabla ^{2}I+{\frac {\beta IS}{N}}-\gamma I,\\[6pt]&\partial _{t}R=D_{R}\nabla ^{2}R+\gamma I,\end{aligned}}}$^[32]

where ${\displaystyle D_{S}}$, ${\displaystyle D_{I}}$ and ${\displaystyle D_{R}}$ are diffusion constants. Thereby, one obtains a reaction-diffusion equation. (Note that, for dimensional reasons, the parameter ${\displaystyle \beta }$ 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.^[33] Extensions of this model have been used to incorporate, e.g., effects of nonpharmaceutical interventions such as social distancing.^[34]

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.^[32] 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^[35] 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.^[36]

SIR_SS 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 models of social stress complement the classical epidemics models.^[37]

An example of using the numerical SIR model to fit the COVID-19 data (from U.S. CDC) in the state of Ohio, U.S.A. ${\displaystyle R_{t}}$ fitted using the SIR and the SIRV model are both shown. Note that although the SIR model can model an individual wave, a more complex model like SIR_SS would better model multiple waves.^[21][22]

The simplest SIR-social stress (SIR_SS) 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 (S_ign), rationally resistant (S_res), and exhausted (S_exh) that do not react on the external stimuli (this is a sort of refractory period). In other words: S(t) = S_ign(t) + S_res(t) + S_exh(t). Symbolically, the social stress model can be presented by the "reaction scheme" (where I denotes the infected individuals):

• ${\displaystyle {\color {blue}{{\mathcal {S_{ign}}}+2{\mathcal {I}}\to {\mathcal {S_{res}}}+2{\mathcal {I}}}}}$ – mobilization reaction (the autocatalytic form here means that the transition rate is proportional to the square of the infected fraction I);
• ${\displaystyle {\color {blue}{{\mathcal {S_{res}}}\to {\mathcal {S_{exh}}}}}}$ – exhaustion process due to fatigue from anti-epidemic restrictions;
• ${\displaystyle {\color {blue}{{\mathcal {S_{exh}}}\to {\mathcal {S_{ign}}}}}}$ – slow relaxation to the initial state (end of the refractory period).

The main SIR epidemic reaction

• ${\displaystyle {\color {blue}{{\mathcal {S_{...}}}+{\mathcal {I}}\to {\mathcal {2I}}}}}$

has different reaction rate constants ${\displaystyle \beta }$ for S_ign, S_res, and S_exh. Presumably, for S_res, ${\displaystyle \beta }$ is lower than for S_ign and S_ign.

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.^[37] These constants for this epidemic in all countries can be extracted by the fitting of the SIR_SS model to publicly available data ^[38]

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.^[39] The KdV-SIR equation is written as follows:

${\displaystyle {\frac {d^{2}I}{dt}}-\sigma _{o}^{2}I+{\frac {3}{2}}{\frac {\sigma _{o}^{2}}{I_{max}}}I^{2}=0}$.

Here,

${\displaystyle \sigma _{o}=\gamma (R_{o}-1)}$,

${\displaystyle R_{o}={\frac {\beta }{\gamma }}{\frac {S_{o}}{N}}}$,

and

${\displaystyle I_{max}={\frac {S_{o}}{2}}{\frac {(R_{o}-1)^{2}}{R_{o}^{2}}}}$.

${\displaystyle S_{o}}$ indicates the initial value of the state variable ${\displaystyle S}$. Parameters ${\displaystyle \sigma _{o}}$(σ-naught) and ${\displaystyle R_{o}}$ (R-naught) are the time-independent relative growth rate and basic reproduction number, respectively. ${\displaystyle I_{max}}$presents the maximum of the state variables ${\displaystyle I}$(for the number of infected persons). An analytical solution to the KdV-SIR equation is written as follows:

${\displaystyle I=I_{max}sech^{2}\left({\frac {\sigma _{o}}{2}}t\right)}$,

which represents a solitary wave solution.

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 ${\displaystyle P\in (0,1)}$:

${\displaystyle {\begin{aligned}{\frac {dS}{dt}}&=\nu N(1-P)-\mu S-\beta {\frac {I}{N}}S\\[8pt]{\frac {dI}{dt}}&=\beta {\frac {I}{N}}S-(\mu +\gamma )I\\[8pt]{\frac {dV}{dt}}&=\nu NP-\mu V\end{aligned}}}$

where ${\displaystyle V}$ is the class of vaccinated subjects. It is immediate to show that:

${\displaystyle \lim _{t\to +\infty }V(t)=NP,}$

thus we shall deal with the long term behavior of ${\displaystyle S}$ and ${\displaystyle I}$, for which it holds that:

${\displaystyle R_{0}(1-P)\leq 1\Rightarrow \lim _{t\to +\infty }\left(S(t),I(t)\right)=DFE=\left(N\left(1-P\right),0\right)}$

${\displaystyle R_{0}(1-P)>1,\quad I(0)>0\Rightarrow \lim _{t\to +\infty }\left(S(t),I(t)\right)=EE=\left({\frac {N}{R_{0}(1-P)}},N\left(R_{0}(1-P)-1\right)\right).}$

In other words, if

${\displaystyle P<P^{*}=1-{\frac {1}{R_{0}}}}$

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 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:

${\displaystyle P=P(I),\quad P'(I)>0.}$

In such a case the eradication condition becomes:

${\displaystyle P(0)\geq P^{*},}$

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:

${\displaystyle {\begin{aligned}{\frac {dS}{dt}}&=\mu N(1-P)-\mu S-\rho S-\beta {\frac {I}{N}}S\\[8pt]{\frac {dV}{dt}}&=\mu NP+\rho S-\mu V\end{aligned}}}$

leading to the following eradication condition:

${\displaystyle P\geq 1-\left(1+{\frac {\rho }{\mu }}\right){\frac {1}{R_{0}}}}$

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:

${\displaystyle {\begin{aligned}{\frac {dS}{dt}}&=\mu N-\mu S-\beta {\frac {I}{N}}S,\quad S(nT^{+})=(1-p)S(nT^{-}),&&n=0,1,2,\ldots \\[8pt]{\frac {dV}{dt}}&=-\mu V,\quad V(nT^{+})=V(nT^{-})+pS(nT^{-}),&&n=0,1,2,\ldots \end{aligned}}}$

It is easy to see that by setting I = 0 one obtains that the dynamics of the susceptible subjects is given by:

${\displaystyle S^{*}(t)=1-{\frac {p}{1-(1-p)E^{-\mu T}}}E^{-\mu MOD(t,T)}}$

and that the eradication condition is:

${\displaystyle R_{0}\int _{0}^{T}S^{*}(t)\,dt<1}$

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 referred to as 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.^[41]

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.^[41][42]

Others

Other occurrences which may need to be considered when modeling an epidemic include things such as the following:^[41]

• Non-homogeneous mixing
• Variable infectivity
• Distributions that are spatially non-uniform
• Diseases caused by macroparasites