# Custom propensity functionΒΆ

In traditional compartmental modeling, the rate at which a reaction occurs obeys the law of mass action. In other words, it is directly proportional to the population of the reacting compartment(s). For example, an EMODL SEIR model with waning immunity will contain reaction terms as the following:

```
(reaction exposure (S I) (E I) (* Ki S I))
(reaction infection (E) (I) (* Kl E))
(reaction recovery (I) (R) (* Kr I))
(reaction waning (R) (S) (* Kw R))
```

However, mass action dynamics can be too restricting in modeling a complex epidemiological system with mechanisms such as seasonal forcing, pulse vaccination, and discrete aging. CMS provides a set of Mathematical operators and functions to aid formulating custom propensity functions. Below is a part of the Garki model [1] that involves seasonal forcing:

```
; seasonal parameter
(func C (* 0.2 (+ 1.01 (sin (* (/ time 365) 2 pi)))))
; infection rate
(func h (* g (- 1 (exp (/ (* (- C) Y1) totalpop)))))
(reaction recoveryY3 (Y3) (X3) (/ (* Y3 h) (- (exp (/ h r2)) 1)))
```

[1] | http://garkiproject.nd.edu/static/documents/garkiproject.pdf (Chapter 10, page 263) |