*David A. Tanzer, December 25, 2020*

## Leaf shape: an optimization in nature

On this Christmas day, let us take a moment to think of the tree, and wonder how it actually grows. Slowly move your attention to the leaves, and ponder how they grow. Next, forget about this type of leaf, for a conifer, with parallel veins — as our true subject is the growth of leaves with branching veins. … Read more

*David A. Tanzer, August 24, 2020*

# Continuous and discrete flows in epidemic models

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*

# Continuous vs. Discrete Flow

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*

# The Dynamics of SIR

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:

- Infection: Susceptible + Infected —> 2 Infected (with rate constant 1)
- Recovery: Infected —> Recovered (with rate constant 2)

…

Read more
*David A. Tanzer, August 17, 2020*

# Digression: reaction rates in chemistry

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*

# Analysis of reaction rates – second approach

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*

# Digression: solving the popcorn rate equation

Last time, we analyzed a system with a single reaction:

And derived its rate equation:

- Unpopped'(t) = – RateCoefficient * Unpopped(t)

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*

# Analysis of reaction rates: first approach

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*

# What is meant by the rate of a reaction?

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*

# The rates matter

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