Compartmental models in epidemiology

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 disease with a short infectious period, such as measles in the UK prior to the introduction of a vaccine 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, the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and recovered compartments. The disease cannot break out again until the number of susceptibles has built back up as a result of offspring being born into the susceptible compartment.

Blue=Susceptible, Green=Infected, and Red=Recovered

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.

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 βI, where β is the average number of contacts per person times the probability of disease transmission in a contact between a susceptible and an infectious subject.

Between I and R, the transition rate is γ (simply the rate of recovery or death). If the duration of the infection is denoted D, then γ = 1/D, since an individual experiences one recovery in D units of time.

It's assumed that the permanence of each single subject in the epidemic states is a random variable with exponential distribution. More complex and realistic distributions (such as Erlang distribution) can be equally used with few modifications.

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 set of ordinary differential equations:[2]

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

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

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

Where ${\displaystyle S}$ is the stock of susceptible population, ${\displaystyle I}$ is the stock of infected, and ${\displaystyle R}$ is the stock of recovered population.

This model was for the first time proposed by O. 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, and does not admit a generic analytic solution. Nevertheless, significant results can be derived analytically, and with Monte Carlo methods, such as the Gillespie algorithm.

Firstly note that from:

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

it follows that:

${\displaystyle S(t)+I(t)+R(t)={\textrm {Constant}}=N}$,

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.[3][4] 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 recovery is ${\displaystyle T_{r}=\gamma ^{-1}}$. From here it follows that, on average, the number of contacts by an infected individual with others before the infected has recovered 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). Thus, in the limit ${\displaystyle t\rightarrow +\infty }$, the proportion of recovered individuals obeys the following transcendental equation

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

This equation shows that at the end of an epidemic, unless ${\displaystyle S(0)=0}$, not all individuals of the population have recovered, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infected individuals rather than an absolute lack of susceptible subjects. The role of the basic reproduction number is extremely important. In fact, upon rewriting the equation for infectious individuals as follows:

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

it yields that if:

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

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

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

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 the basic reproduction number is extremely important.

Note that in the above model the function:

${\displaystyle F=\beta I,}$

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. 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 ${\displaystyle N}$):

${\displaystyle F=\beta {\frac {I}{N}}.}$

Capasso and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process.

In 2014, Harko T. et al.[5] derived an exact analytical solution to the SIR model. In the without vital dynamics setup, for ${\displaystyle {\mathcal {S}}(u)=S(t)}$, etc., it corresponds to the following time parametrization

${\displaystyle {\mathcal {S}}(u)=S(0)u}$

${\displaystyle {\mathcal {I}}(u)=N-R(0)+\rho \ln(u)-S(0)u}$

${\displaystyle {\mathcal {R}}(u)=R(0)-\rho \ln(u)}$

for ${\displaystyle t=\int _{u}^{1}{\frac {N}{\beta s(N-R(0)+\rho \ln(s)-S(0)s)}}ds}$, ${\displaystyle \rho ={\frac {\gamma N}{\beta }}}$, with initial conditions ${\displaystyle ({\mathcal {S}}(1),{\mathcal {I}}(1),{\mathcal {R}}(1))=(S(0),N-R(0)-S(0),R(0));}$ ${\displaystyle u_{T}<u<1}$, where ${\displaystyle u_{T}}$ satisfies ${\displaystyle {\mathcal {I}}(u_{T})=0}$. By the transcendental equation for ${\displaystyle R_{\infty }}$ above, it follows that ${\displaystyle u_{T}=e^{-(R_{\infty }-R(0))/\rho }(=S_{\infty }/S(0)}$, if ${\displaystyle S(0)\neq 0)}$ and ${\displaystyle I_{\infty }=0}$.

An equivalent analytical solution was found by Miller[6][7] yields

${\displaystyle S(t)=S(0)e^{-\xi (t)}}$

${\displaystyle I(t)=N-S(t)-R(t)}$

${\displaystyle R(t)=R(0)+N{\frac {\gamma }{\beta }}\xi (t)}$

${\displaystyle \xi (t)={\frac {\beta \int _{0}^{t}I(t^{*})\,dt^{*}}{N}}}$

Here ${\displaystyle \xi (t)}$ can be interpreted as the expected number of transmissions an individual has received by time ${\displaystyle t}$. The two solutions are related by ${\displaystyle e^{-\xi (t)}=u}$.

Effectively the same result can be found in the original work by Kermack and McKendrick.[8]

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

${\displaystyle {\frac {dS}{dt}}=\Lambda -\mu S-\beta IS}$

${\displaystyle {\frac {dI}{dt}}=\beta IS-\gamma I-\mu I}$

${\displaystyle {\frac {dR}{dt}}=\gamma I-\mu R}$

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 :

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

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

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

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

The point EE is called the Endemic Equilibrium. 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 or equal than one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.

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), which results in the following differential equations:

${\displaystyle {\tfrac {dM}{dT}}=B-\delta M-\mu M}$

${\displaystyle {\tfrac {dS}{dT}}=\delta M-\beta SI-\mu S}$

${\displaystyle {\tfrac {dI}{dT}}=\beta SI-\gamma I-\mu I}$

${\displaystyle {\tfrac {dR}{dT}}=\gamma I-\mu R}$

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.

SIR compartmental model with carrier class, C

For many important infections there is a significant incubation period during which the individual has been infected but is not yet infectious themselves. During this period the individual is in compartment E (for exposed).

SEIR compartmental model

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

${\displaystyle {\frac {dS}{dt}}=\mu N-\mu S-\beta {\frac {I}{N}}S}$

${\displaystyle {\frac {dE}{dt}}=\beta {\frac {I}{N}}S-(\mu +a)E}$

${\displaystyle {\frac {dI}{dt}}=aE-(\gamma +\mu )I}$

${\displaystyle {\frac {dR}{dt}}=\gamma I-\mu R.}$

We have ${\displaystyle S+E+I+R=N}$, but this is only constant because of the (degenerate) 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 form 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:

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

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

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:

${\displaystyle {\frac {dE_{1}}{dt}}=\beta (t)I_{1}-(\gamma +a)E_{1}}$

${\displaystyle {\frac {dI_{1}}{dt}}=aE_{1}-(\gamma +\mu )I_{1}}$

is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).

The SEIS model takes into consideration the exposed or latent period of the disease, giving an additional compartment, E(t).

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

In this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible again, moving back into the S(t) compartment. The following differential equations describe this model:

${\displaystyle {\tfrac {dS}{dT}}=B-\beta SI-\mu S+\gamma I}$

${\displaystyle {\tfrac {dE}{dT}}=\beta SI-(\epsilon +\mu )E}$

${\displaystyle {\tfrac {dI}{dT}}=\varepsilon E-(\gamma +\mu )I}$

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

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

${\displaystyle {\tfrac {dM}{dT}}=B-\delta M-\mu M}$

${\displaystyle {\tfrac {dS}{dT}}=\delta M-\beta SI-\mu S}$

${\displaystyle {\tfrac {dE}{dT}}=\beta SI-(\varepsilon +\mu )E}$

${\displaystyle {\tfrac {dI}{dT}}=\varepsilon E-(\gamma +\mu )I}$

${\displaystyle {\tfrac {dR}{dT}}=\gamma I-\mu R}$

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.

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

It is well known that the probability of getting a disease is not constant in the time. 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

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

with period T equal to one year.

Thus, our model becomes

${\displaystyle {\frac {dS}{dt}}=\mu N-\mu S-\beta (t){\frac {I}{N}}S}$

${\displaystyle {\frac {dI}{dt}}=\beta (t){\frac {I}{N}}S-(\gamma +\mu )I}$

(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:

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

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.

In presence of a communicable diseases, one of main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate ${\displaystyle P\in (0,1)}$:

${\displaystyle {\frac {dS}{dt}}=\mu N(1-P)-\mu S-\beta {\frac {I}{N}}S}$

${\displaystyle {\frac {dI}{dt}}=\beta {\frac {I}{N}}S-(\mu +\gamma )I}$

${\displaystyle {\frac {dV}{dt}}=\mu NP-\mu V}$

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

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

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

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

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

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 have to vaccinate 94.4% of newborns in order to eradicate the disease.

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:

${\displaystyle P=P(I),P^{\prime }(I)>0.}$

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.

In case there also are vaccinations of non newborn at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows:

${\displaystyle {\frac {dS}{dt}}=\mu N(1-P)-\mu S-\rho S-\beta {\frac {I}{N}}S}$

${\displaystyle {\frac {dV}{dt}}=\mu NP+\rho S-\mu V}$

leading to the following eradication condition:

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

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:

${\displaystyle {\frac {dS}{dt}}=\mu N-\mu S-\beta {\frac {I}{N}}S,S(nT^{+})=(1-p)S(nT^{-})n=0,1,2,\dots }$

${\displaystyle {\frac {dV}{dt}}=-\mu V,V(nT^{+})=V(nT^{-})+pS(nT^{-})n=0,1,2,\dots }$

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:

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

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.[10]

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.[10] For more information on this type of model see the reference Population Dynamics of Infectious Diseases: Theory and Applications, by R. M. Anderson.[11]

Other occurrences which may need to be considered when modeling an epidemic include things such as the following:[10]

• Nonhomogeneous mixing

• Variable infectivity

• Distributions that are spatially non-uniform

• Diseases caused by macroparasites