20  Serocatalytic models

20.1 Force of infection

Recall from the SIR model Section 1.1, that the rate of transition from \(S\) to \(I\), \(\lambda = \beta I\), is called the force of infection (FOI). The FOI represents the risk of a susceptible individual becoming infected over a given period of time.

20.2 Serocatalytic models

Serocatalytic models provide a simple framework: the population is divided into two groups seronegative (\(N\)) and seropositive (\(P\)). Assuming no seroreversion, individuals transition from \(N\) to \(P\) at a rate \(\lambda\), which is equivalent to the FOI.

The term “catalytic model” comes from its analogy to a chemical reaction, where reagent \(N\) is transformed into reagent \(P\) at a rate \(\lambda\).

The FOI can either be constant \(\lambda\) or vary over time \(\lambda(t)\).

20.2.1 Constant FOI

Consider a cohort where we track seroprevalence over time (i.e., we observe \(P(t)\)). The system can be described by the following differential equations:

\[\frac{dN(t)}{dt} = -\lambda N(t)\]

\[\frac{dP(t)}{dt} = \lambda N(t)\]

In which \(N(t) + P(t) = 1\) or \(N(t) = 1 - P(t)\). We can rewrite the latter equation as:

\[\frac{dP(t)}{dt} = \lambda(1 - P(t))\]

We can solve \(P\) analytically as:

\[P(t) = 1 - e^{-\lambda t} \tag{20.1}\]

\[\begin{align} P(t) &= 1 - e^{-\lambda t} \Leftrightarrow e^{-\lambda t} = (1 - P(t)) \\ P(t) &= 1 - e^{-\lambda t} \\ \Leftrightarrow \frac{dP(t)}{dt} &= -(-\lambda)e^{-\lambda t} \\ &= \lambda e^{-\lambda t} \\ &= \lambda (1 - P(t)) \end{align}\]

20.2.2 Constant FOI (endemic model)

In a population where a disease is at endemic equilibrium without vaccination, seronegative (\(N\)) and seropositive (\(P\)) remain constant over time (because it is equilibrium). However, these proportions vary with age. By replacing the time term in Equation 20.1 with age, we can express the seropositive \(P(a)\) as a function of age:

\[P(a) = 1 - e^{-\lambda a}\]

20.2.3 Time-varying FOI

FOI is a function of time \(\lambda(t)\). Then Equation 20.1 become:

\[\frac{dP(t)}{dt} = \lambda(t)(1 - P(t))\]

We can solve \(P\) analytically as:

\[P = 1 - (1 - P_0) e^{-\int_0^t \lambda(u) du}\]

\[\begin{align} \frac{dP}{dt} &= \lambda(t)(1 - P) \\ \Leftrightarrow \frac{dP}{1 - P} &= \lambda(t)dt \end{align}\]

Now integrate both sides:

\[\begin{align} \int \frac{dP}{1 - P} &= \int \lambda(t)dt \\ \Leftrightarrow -\log(1 - P) &= \int \lambda(t)dt + C \\ \Leftrightarrow \log(1 - P) &= -\int \lambda(t)dt - C \end{align}\]

where \(C\) is the constant of integration. Exponentiating both sides:

\[1 - P = e^{-\int \lambda(t)dt - C}\]

Simplifying further:

\[1 - P = A e^{-\int \lambda(t) dt}\]

where \(A = e^{-C}\) is a constant determined by initial conditions. Solving for \(P\):

\[P = 1 - A e^{-\int \lambda(t) dt}\]

If the initial condition is \(P(0) = P_0\), substitute \(t = 0\) into the solution:

\[P_0 = 1 - A e^0 = 1 - A\]

Thus, \(A = 1 - P_0\), and the final solution is:

\[P = 1 - (1 - P_0) e^{-\int_0^t \lambda(u) du}\]

Usually we assume \(P_0 = 0\) (the population is fully susceptible or seronegative at the beginning \(t = 0\)), so:

\[P(t) = 1 - e^{-\int_0^t \lambda(u) du}\]

20.2.4 Time-varying FOI across different ages

When the sample includes individuals of different ages, the seroprevalence \(P(a, t)\) depends on both the individual’s age \(a\) and the calendar time \(t\).

\[P(a, t) = 1 - e^{-\int_{t - a}^t \lambda(u) du}\]

where \(\int_{t - a}^t \lambda(u)\) represents the cumulative FOI experienced by an individual from birth \(t - a\) up to the current time \(t\).

20.2.5 Seroreversion

If individuals lose immunity over time (waning immunity) with a constant rate \(\rho\).