Contact and Recovery Rate

00:01 Let's talk now about two very important indicators parameters called C and R.

00:06 These are of a contact rate and the recovery rate.

00:11 So, we talked about S as a function of t, S of t, because the number of people in the susceptible compartment changes over time.

00:21 But the differential equation describing the flow of individuals out of that compartment is given by minus c, multiplied by I, multiplied by S.

00:32 It doesn't matter what that means, except to say that there is a constant at play there, that helps define the rate of flow of people from the susceptible compartment into the infectious compartment.

00:45 And that constant is c.

00:50 The rate of flow of infectious people out of the infection compartment is given by a more complicated expression.

00:58 c multiplied by IS minus r multiplied by I and R is this other constant we're going to talk about, called the recovery rate.

01:07 From the people who recover include those newly infected minus those who have been removed.

01:14 That's why this expression is kind of important, because depending upon which expression is bigger, that tells us if the disease is waning, or waxing.

01:27 And lastly, of course, the rate of flow of people into the recovery compartment is given by this constant called r multiplied by number of people, I.

01:39 So, this constant c and r, their call, as I mentioned, the contact rate, or sometimes the transmission rate, and the recovery rate.

01:46 Sometimes that's given as h, which is the healing rate because it's the rate of people who get better, who heal.

01:53 And unlike the other aspects of this model, which are computed or imagined, c and r are often empirically measured.

02:04 They are characteristics of the infection as observed in the population.

02:09 The number of people that have given contact is contacting, and the number of people who are getting better in the hospital.

02:17 These are things that can be observed and therefore estimated.

02:22 As noted, c is how quickly people become infected.

02:25 It's the contact rate.

02:28 And r is how quickly people become healed or recovered.

02:33 These are important because they also open up the opportunity to talk about levers of public health intervention.

02:41 Because we can control how quickly people become infected.

02:45 We can control this contact rate.

02:48 c is much easier to control than r.

02:51 Because you can do things like introduce personal protective equipment for healthcare workers, masks for everybody else, that reduces the contact rate.

03:00 You're going to have people stand further apart if we're dealing here with an infection that is spread by droplets exhale through the mouth.

03:09 Or we can implement quarantine measures to keep infected people from interacting with uninfected people.

03:15 We can slow the rate of susceptible people becoming infectious people.

03:21 And we do that by changing c, contact rate.

03:26 r in the other hand, is quite difficult to change.

03:30 because it is the rate of people who get better.

03:33 If we knew how to make people better, that would solve so much of the problem.

03:38 But making people better is a function of clinical power, clinical expertise, medical technologies, whether or not we have treatments, and cures available. Right? Also as a function of healthcare system responsiveness and effectiveness.

03:55 So, a better healthcare system will have a higher rate of recovery.

04:00 More clinical expertise give you a higher rate of recovery.

04:05 A less overwhelmed system gives you a higher rate of recovery.

04:12 So again, the change of S over time is given by an expression related to the contact rate.

04:22 And the change of I over time is given to a more by more complicated expression.

04:27 And the change of r over time is given by an expression gated by their recovery rate.

04:32 Let's look at the middle one.

04:34 This complicated expression for the rate of flow of infected people.

04:40 c multiplied by IS minus r multiplied by I.

04:44 Clearly, if that expression is greater than zero, then the pandemic is worsening.

04:53 It means more people are entering that infectious compartment.

04:58 We would like that number to come down, not go up.

05:03 So, will not surprise you to know then that C and R are important for allowing us to compute their reproduction number.

05:11 The recall that the reproduction number is the average number of new cases produced by an existing case.

05:17 And reproduction number allows us to estimate the seriousness of an epidemic.

05:23 The basic reproduction number is actually given by c divided by r.

05:28 Contact rate, or how quickly people become infected via by the recovery rate, or how quickly they become healed? So again, here is our graph of a population of 500 people experiencing the SIR model.

05:46 And this green curve, the infection curve, and the one that we care most about is actually affected by the reproduction number.

05:58 So, the height and width of that curve will change as the value of the reproduction number changes.

06:03 And remember, the reproduction number is dependent on these constants C and R, which we can change through interventions like mask wearing and quarantine, and healthcare system improvements.

06:15 So, we can change the shape of this green curve by doing those interventions.