David A. Tanzer, August 24, 2020
The standard compartmental models for SIR, SEIR etc. use the rate equations for the dynamics, and hence assume a continuous flow model.
This works well, particularly due to the fact that population sizes in epidemics are “large”, on the order of millions of individuals, and the law of large numbers kicks in, which tells us that we … Read more
David A. Tanzer, August 24, 2020
With the advent of the rate equation, we have shifted into a model of reaction networks consisting of “pumps” that move at continuous rates, transferring “mathematical fluid” between containers. Yet, as we have indicated, the real activity proceeds by discrete steps; an individual recovers, two molecules collide to form a compound.
How are these views to be reconciled?
For concreteness, … Read more
David A. Tanzer, August 17, 2020
Recall that for dynamics, we want to get at the way that the compartments S, I, R evolve over tune,
So far, we’ve defined the rates at which the reactions fire.
So now let’s summarize the rates for the SIR model:
David A. Tanzer, August 17, 2020
The same kinetics occurs in chemical reaction networks. To illustrate, suppose there were two types of molecules, A and B, and that one A and one B molecule could collide to form a C molecule. So the reaction would be written $$A + B \rightarrow C$$.
Now imagine that a large number of A, B and C molecules are … Read more
David A. Tanzer, August 17, 2020
In the previous article, we analyzed a simple system with a simple reaction:
$$\mathrm{Unpopped} \xrightarrow{\mathrm{Cooking}} \mathrm{Popped}$$
There we saw that the rate of the reaction is proportional both to a constant associated with the reaction, and to the count in compartment Unpopped. This will hold for all reactions that take a single input connection, including recovery … Read more
David A. Tanzer, August 17, 2020
Last time, we analyzed a system with a single reaction:
And derived its rate equation:
In this article, we will see the solution to the equation, which will tell us exactly how the average popping rate evolves over time.
Note this article strikes a comparatively more quantitative tone than the others in … Read more
David A. Tanzer, August 17, 2020
Now let’s work out the equation for how the popping evolves over time.
Let Rate(t) be the average rate at which popping takes place at time $t$. Once the popping has begun, we expect Rate(t) to be high, and then dwindle down towards zero as more and more of the kernels have popped.
Suppose that Rate(20) = 5, … Read more
David A. Tanzer, August 17, 2020
Let’s take a simple model, with just one reaction:
$$\mathrm{Sick} \xrightarrow{\mathit{Recovery}} \mathrm{Healthy}$$
Suppose that everyone starts out ill, and eventually recovers.
Suppose that, on any given day, there is a 10% chance that a sick person will recover. So, viewing the process of recovery as a whole, it will be like a popcorn process, where … Read more
David A. Tanzer, August 17, 2020
In the last series, we introduced the idea of a reaction network, which is a collection of processes that are moving individuals between the compartments. What was missing there, though, was any kind of description of how fast the reactions run. This has a significant impact on the overall behavior of the network.
Let’s make this point concrete with an example. … Read more
David A. Tanzer, July 27, 2020
Last time, we started a tour of extensions to the SIR model. Now we conclude the tour and this series by describing an extension SEIR which is used for epidemics like COVID-19.
SEIR gives a more accurate description of the facts on the ground. It stands for stands for Susceptible, Exposed, Infectious, Recovered.
In SEIR, the … Read more