SIS model 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. We have the model: : \begin{align} \frac{dS}{dt} & = - \frac{\beta S I}{N} + \gamma I \\[6pt] \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, so that \forall I(0) > 0: : \begin{align} & \frac{\beta}{\gamma} \le 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{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). 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. 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 \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.
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}, \\[6pt] & \frac{dI}{dt} = \frac{\beta I S}{N} - \gamma I - \mu I, \\[6pt] & \frac{dR}{dt} = \gamma I, \\[6pt] & \frac{dD}{dt} = \mu I, \end{align} where \beta, \gamma, \mu are the rates of infection, recovery, and mortality, respectively.
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, \\[6pt] & \frac{dI}{dt} = \frac{\beta(t) I S}{N} - \gamma(t) I, \\[6pt] & \frac{dR}{dt} = \gamma(t) I, \\[6pt] & \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. This model uses the following system of differential equations for the population fractions S, I, R, V, D: : \begin{align} & \frac{dS}{dt} = - a(t) S I - v(t) S, \\[6pt] & \frac{dI}{dt} = a(t) S I - \mu(t) I - \psi(t) I, \\[6pt] & \frac{dR}{dt} = \mu(t) I, \\[6pt] & \frac{dV}{dt} = v(t) S,\\[6pt] & \frac{dD}{dt} = \psi(t) I \end{align} where a(t), v(t), \mu(t), \psi(t) are the infection, vaccination, recovery, and fatality rates, respectively. For the semi-time initial conditions S(0)=1-\eta, I(0)=\eta, R(0)=V(0)=D(0)=0 and constant ratios k=\mu(t)/a(t), b=v(t)/a(t), and q=\psi(t)/a(t) the model had been solved approximately, and exactly for some special cases, irrespective of the functional form of a(t). The kinetic equations become: : \begin{align} & \frac{dS}{dt} = - a(t) S I - v(t) S + b(t) [\mu(t) I + v(t) S ], \\[6pt] & \frac{dI}{dt} = a(t) S I - \mu(t) I, \\[6pt] & \frac{dR}{dt} = [1- b(t)] \mu(t) I , \\[6pt] & \frac{dV}{dt} = [1- b(t)] v(t) S ,\\[6pt] \end{align} where infection rate a(t) can be write as \beta(t) /N , recovery rate \mu(t) can be simplified to a constant \gamma , v(t) is the vaccination rate, b(t) is the break through ratio or fraction of immuned people susceptible to reinfection ( \begin{align} \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{align}
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.
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} \\[8pt] \frac{dE}{dt} & = \frac{\beta I S}{N} - (\mu + a ) E \\[8pt] \frac{dI}{dt} & = a E - (\gamma +\mu ) I \\[8pt] \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 [0,N]^4 : 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 \\[8pt] \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).
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 \\[6pt] \frac{dE}{dt} & = \frac{\beta SI}{N} - (\epsilon + \mu)E \\[6pt] \frac{dI}{dt} & = \varepsilon E - (\gamma + \mu)I \end{align}
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 \\[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{align}
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}}}
More complex general models When developing more detailed models for in-depth analysis, models are mostly generated for specific outbreak scenarios of specific diseases, including compartments for targeted research questions like hospitalization compartments or detection dynamics. Even though those models are often tailored for specific situations, there are complex models, still usable for a broad variety of different diseases. One of those attempts to create a general model includes twelve compartments, extending the well-known SEIR model by a second stage of infection, detection compartments, and two doses of vaccination. Additionally smear infections are incorporated via an external Pathogen P and a simplistic vector population is included by S_V and I_V. Moreover population dynamics like birth and death processes can be included. Such complex models enable a deeper understanding of infection dynamics and the introduction of different pharmaceutical and non-pharmaceutical interventions.
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 : 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 \\[8pt] \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 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}, \\[6pt] & \partial_t I = D_I \nabla^2 I + \frac{\beta I S}{N}- \gamma I, \\[6pt] & \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. 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. in the traveling wave coordinate. 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. == Heterogeneous (structured, Bayesian) model==