The classical models in epidemics are compartment models. In such models, the hosts are arranged into different populations such as susceptible populations, infected populations, and recovered populations.

The Kermack-McKendrick deterministic model describes the transitions between the different populations. The susceptible population \(S(t)\) decrease at a rate \(-\alpha I(t) S(t)\), where \(\beta\) is the average probability of getting infected by the infected population. The recovery rate of the infected population is \(\alpha\).

(45)¶\[\begin{split}\frac{dS(t)}{dt} &= -\beta I(t) S(t) \\
\frac{dI(t)}{dt} &= \beta S(t) I(t) - \alpha I(t),\end{split}\]

with the constraint

\[S(t) + I(t) + R(t) = N,\]

where \(N\) is the total population. [Martcheva2015]

The limiting behavior of the system is also intuitive. If the infected population is quarantined, i.e., \(\beta=0\), the infected population decays exponentially with a half-life \(1/\alpha\). Notice that the populations can only be integers. When \(S(t)<1\), the infected population moves to the exponential decay phase.

At a specific time \(t=t_t\), we might reach the threshold that

\[\beta S(t_t) - \alpha = 0.\]

Suppose the susceptible population is larger initially, we would have \(\beta S(t_0) - \alpha > 0\) initially. The infected population will grow. The threshold indicates a flipping moment when the infected population will start to decrease.

The threshold requires

\[\frac{\beta S(t_t)}{\alpha} = 1.\]

When \(\frac{\beta S(t_t)}{\alpha}>1\), the infected population is growing. It is convinient to define

\[\tau(t) = \frac{\beta S(t)}{\alpha},\]

with which equations (45) becomes

\[\begin{split}\frac{dS(t)}{dt} &= - \alpha \tau(t) I(t) \\
\frac{dI(t)}{dt} &= \alpha (\tau(t) - 1) I(t),\end{split}\]

If \(\tau(t)>1\), the infected population will grow. If \(\tau(t)<1\), the infected population will decrease.

In the **very beginning of a epidemic event**, the susceptible population fraction is almost constant, thus \(\tau(t)\) is almost constant, i.e., \(\tau(t) = \tau_0\), the growth rate of \(I(t)\) is

\[\frac{dI(t)}{dt} = \alpha (\tau_0 - 1) I(t).\]

The equations are solved

\[I(t) = I(t_0) e^{\alpha (\tau_0 - 1) t}.\]

The exponential growth rate is determined by

\[\alpha (\tau_0 - 1).\]

The days \(\Delta t\) required to double the infected population is a constant in the early stage. To find out \(\Delta t_d\),

\[\frac{I(t+\Delta t)}{I(t)} = e^{\alpha (\tau_0 - 1) \Delta t} \equiv 2,\]

which leads to

\[\Delta t = \frac{\ln 2}{\alpha(\tau_0 -1)} \sim \frac{0.7}{\alpha(\tau_0 -1)} .\]

The transition events of a susceptible person being infected (\(S\to I\)) and an infected person being recovered (\(I\to R\)).

The rates are determined by the equation (45),

\[Y_{S\to I}\left(\int_0^t \beta I(t') S(t') dt' \right)\]

and

\[Y_{I\to R}\left( \int_0^t \alpha I(t') dt' \right).\]

Whenever an event is triggered in the process \(Y_{S\to I}\), we will have one more infected person and one less susceptible person.

In fact, \(1/\alpha\) is the mean days an infected person remains in the infectd status. This is used to estimate the \(\alpha\) using data. The value of \(\beta\) is estimated using the relation [Martcheva2015]

\[\frac{\beta}{\alpha} = \frac{\ln (S(t_0)/S(t\to\infty))}{S(t_0) + I(t_0) - S(t\to\infty) }\]

Some epidemics such as influenza infect us repeatedly. One simple model for them is the SIS model shown in figure SIS model.,

The dynamics are determined by

\[\frac{dI(t)}{dt} = \beta I(t) S(t) - \alpha I(t),\]

with the constraint

\[S(t) + I(t) = N.\]

The dynamics of the basic SIS model is determined by one single first-order differential equation

\[\begin{split}\frac{dI(t)}{dt} &= \beta I(t) (N - I(t)) - \alpha I(t) \\
&= (\beta N - \alpha )I(t) - \beta I(t) I(t) \\
&= (\beta N - \alpha )I(t) \left( 1 - \frac{I(t)}{(\beta N - \alpha)/\beta} \right) \\
&\equiv r \left(1 - \frac{I}{r/\beta}\right) I(t),\end{split}\]

where we defined the growth rate

\[r \equiv \beta N - \alpha = \alpha(\frac{\beta}{\alpha} N - 1) \equiv \alpha (\mathscr R_0 - 1).\]

The parameter \(\mathscr R_0\) is the basic reproduction number,

\[\mathscr R_0 = \frac{\beta}{\alpha} N .\]

If \(\mathscr R_0 > 1\), we get a positive growth grate for \(I(t)\). Otherwise, the infected population will decrease.

Basic Reproduction Rate

A quote from the Martcheva [Martcheva2015]

Epidemiologically, the reproduction number gives the number of secondary cases one infectious individual will produce in a population consisting only of susceptible individuals.

Some diseases are transmitted from one host to another with some intermediate living carriers such as arthropod. An intermediate living carrier is called a vector. Vectors do not get sick because of the pathogenic microorganism but they will carry the pathogenic microorganism throughout their lives.

To model the vector-borne diseases, two populations are added to the model, the infected population of vectors \(I_v(t)\) and the susceptible population of vectors \(S_v(t)\). Apart from being infected by the infected hosts, the birth rate \(\Lambda_v\) and the death rate \(\mu\) of the vectors are also related to the two populations. Thus the two populations are coupled to the different populations of the hosts,

\[\begin{split}\frac{S_v(t)}{dt} &= \Lambda_v - p a S_v(t) I(t) -\mu S_v(t) \\
\frac{I_v(t)}{dt} &= p a S_v(t) I(t) - \mu I_v,\end{split}\]

where \(a\) is the rate of a vector biting a host, \(p\) is the rate of a vector being infected when biting an infected host. The product \(pa\) is the rate of a vector being infected. [Martcheva2015]

Because most vector-borne diseases are repeatative, we combine the dynamics of the vectors with the SIS model with the constraint \(S(t) + I(t) = N\),

\[\frac{I(t)}{dt} = qa S(t) I_v(t) -\alpha I(t),\]

where \(q\) is the rate of being transmitted from the vector to the host, \(\alpha\) is the recovery rate. The recovered hosts become susceptible.

A general compartment model is a model may include other stages of the disease. The differential equations are easily translated from the flowcharts.

In the disease progression, four stages are relevant. [Martcheva2015]

Exposed stage E or latent stage L: infected but not infectious;

Asymptomatic stage A: the asymptomatic stage describes the asymptomatic infection or subclinical infection where the host is infected by no symptoms are shown;

Carrier stage C: infected but not sick;

Passive immunity stage M: antibodies are transferred between hosts.

Compartments related to epidemic control can also be integrated into the models.

Quarantine Q

Treatment T

Vaccination V

Each compartment is also subject to variants based on the demographics of the populations, heterogeneities of the pathogens and hosts. [Martcheva2015]

Consider the SIR model (45) and add birth rate \(\Lambda\) and death rate \(\mu\) to the model,

\[\begin{split}\frac{dS(t)}{dt} &= \Lambda - \beta I(t) S(t) - \mu S(t)\\
\frac{dI(t)}{dt} & = \beta I(t) S(t) - \alpha I(t) - \mu I(t) \\
\frac{dN(t)}{dt} &= \alpha I(t) - \mu N(t),\end{split}\]

with

\[R(t) + I(t) + S(t) = N(t).\]

The reproduction number is

\[\mathscr R_0 = \frac{\Lambda \beta}{\mu(\alpha + \mu)}.\]

Since the equation for \(N(t)\) is independent of \(S(t)\) and \(I(t)\), the stability is determined by the first two equations. Denote the equilibrium point as \(S_0, I_0\), we linearize the equations using Linear Stability Analysis and find that the characteristic equation is related to \(\mathscr R_0\), i.e., the growth rate of the linearized system \(\lambda\) is a function of \(\mathscr R_0\) thus the stability of the system is also related to the reproduction number. Though the compartment model can be complicated as more stages are added, linear stability analysis is a very effective tool to analyze the stability.

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