Solving the Paradox: A Dynamical Model of Disease Spread

[greswd]

I'm considering a dynamical model of the scenario of a disease spreading across the population, modelling the number of cases, and there's a mathematical puzzle which I'd like to solve.

It's kinda different from the S-I-R model, and its also a crude model.

First, we consider two numbers, N_T and N_A.
N_T is the total number of infected cases, and N_A is the number of active infectious cases, those individuals who are actively spreading the disease.

I'm assuming that after a certain fixed length of time, a case becomes no longer infectious, and this length of time is labelled t_S.

Making the reasonable assumption that the rate of infection, aka the rate of new cases, or the rate of change of the total number of cases, is directly proportional to the number of active cases, we get the differential equation: $$\frac{dN_{T}(t)}{dt}=k\cdot N_{A}(t)$$And for the rate of change of the number of active cases: $$\frac{dN_{A}(t)}{dt}=\frac{dN_{T}(t)}{dt}-\frac{dN_{T}(t-t_{S})}{dt}$$Assuming that k remains constant over time, its possible for this model to have a solution in which the rate of new cases is constant as well.
The function of N_T would be just a straight line. And so would the function of N_A, a straight and completely horizontal line.
Its possible to have this general linear, steady-state solution for any value of k and t_S.An infectious disease also has a basic reproduction number, R₀, which is the number of new cases an infectious person is likely to generate. It seems intuitive that when R₀ is greater than 1, the number of cases will experience exponential growth.In the model, each active infectious case remains active and infectious for a duration of t_S, and infects a total number of the basic reproduction number of R₀ people during that duration.
Combining this with the linear solution mentioned above, the value of k is: $$k=\frac{R_0}{t_S}$$
As you can see, R₀ has been fitted into a linear, steady-state solution. It works perfectly fine when R₀ is greater than 1.

But that appears to be paradoxical. If R₀ is greater than 1, shouldn't linear, steady-state growth be impossible, as growth should be exponential instead?

 

[Chestermiller]

What's wrong with the SIR model, and how does your model relate to the SIR model?

After about an hour of thinking, I derived essentially the SIR model without ever having seen or heard of it. I was shocked when I learned about the SIR model and found out that it was identical to mine.

 

[greswd]

What's wrong with the SIR model, and how does your model relate to the SIR model?

After about an hour of thinking, I derived essentially the SIR model without ever having seen or heard of it. I was shocked when I learned about the SIR model and found out that it was identical to mine.

My model doesn't factor the total susceptible population into the infection rate, and it makes use of the derivative $-\frac{dN_{T}(t-t_{S})}{dt}$

There's the puzzle relating to R₀ which I'd like to solve.

 

[Chestermiller]

I have no problem with your first equation, although, the number of new cases in the SIR model is also proportional to the number of people who are still uninfected. Your 2nd equation also makes some sense to me. combining it with your first equation, it should read $$\frac{dN_A}{dt}=kN_A(t)-kN_A(t-t_s)$$ So this is really a difference-differential equation. This is the same as the SiR model, except for the 2nd term, which is expressed as $\gamma N_A(t)$. It basically says that a person remains infected on average $1/\gamma$ days, so that rate at which infected people become recovered is $\gamma N_A(t)$. So, in this case, you would have:$$\frac{dN_A}{dt}=kN_A(t)-\gamma N_A(t)$$This would give an exponential rate of increase of infected, provided $k>\gamma$.

 

[Chestermiller]

Actually, I like your version of the model quite a bit, even more than the SIR model. I think it is a much more realistic representation of how the transition from I to R occurs. Instead of saying that, on average (mixing cup average), the amount of time one is infected is $1/\gamma$, it more realistically says that everyone is infected the same amount of time, $t_s$. In addition, it's just as easy to implement computationally. I'm going to do some calculations with it and see how it plays out.

In terms of the equivalent SIR model, your corresponding equations would read:
$$\frac{dN_S}{dt}=-kN_S(t)N_I(t)$$
$$\frac{dN_I}{dt}=+k[N_S(t)N_I(t)-N_S(t-t_s)N_I(t-t_s)]$$
$$\frac{dN_R}{dt}=+kN_S(t-t_s)N_I(t-t_s)$$

 

[Chestermiller]

I ran some calculations with this modified SiR model, and it worked out really nicely. I used ts = 12 days and a doubling time of 5 days.

 

[Chestermiller]

A more fundamental form of the equations for this model is:
$$\frac{dC_S}{dt}=-k'C_S(t)C_I(t)$$
$$\frac{dC_I}{dt}=+k'[C_S(t)C_I(t)-C_S(t-t_s)C_I(t-t_s)]$$
$$\frac{dC_R}{dt}=+k'C_S(t-t_s)C_I(t-t_s)$$where the C's are the population densities of the three populations, A is the area over the sample, and k'=kA. Note that, unlike k, the "rate constant" k' is a fundamental physical parameter of our "chemically reacting" system (that depends on the social distancing life styles of our population).

Another even more interesting form is obtained if we normalize the population densities of the three populations with respect to the overall population density N/A, where N is the total of the three populations over the area:

$$\frac{dx_S}{dt}=-\frac{k'N}{A}x_S(t)x_I(t)=-k^*x_S(t)x_I(t)\tag{1a}$$
$$\frac{dx_I}{dt}=+\frac{k'N}{A}[x_S(t)x_I(t)-x_S(t-t_s)x_I(t-t_s)]=+k^*[x_S(t)x_I(t)-x_S(t-t_s)x_I(t-t_s)]\tag{1b}$$
$$\frac{dN_R}{dt}=+\frac{k'N}{A}x_S(t-t_s)x_I(t-t_s)=+k^*x_S(t-t_s)x_I(t-t_s)\tag{1c}$$
with $$k*=\frac{k'N}{A}\tag{2}$$and with $$N=N_S+N_I+N_R$$and $x_S=N_S/N$, etc.

Since, as mentioned previously, the basic rate parameter $k'$ depends only on the social distancing life style of our population, the rate constant k* in these equations depends on k' and on the population density. This means that, as the overall population density increases, the rate constant k* increases in proportion. From this we can conclude that a lower overall population density is basically equivalent to practicing more stringent social distancing. So dense populations which practice social distancing are equivalent to lower population densities in rural areas that do not practice social distancing. This suggests that it might be a viable approach to lower or even remove social distancing restrictions from less populated areas without serious consequences.

 

[greswd]

A more fundamental form of the equations for this model is:
$$\frac{dC_S}{dt}=-k'C_S(t)C_I(t)$$
$$\frac{dC_I}{dt}=+k'[C_S(t)C_I(t)-C_S(t-t_s)C_I(t-t_s)]$$
$$\frac{dC_R}{dt}=+k'C_S(t-t_s)C_I(t-t_s)$$where the C's are the population densities of the three populations, A is the area over the sample, and k'=kA. Note that, unlike k, the "rate constant" k' is a fundamental physical parameter of our "chemically reacting" system (that depends on the social distancing life styles of our population).

nice, and yeah, a pandemic is very much like a chemical reaction lol
both physical and organic chemistry

 

[greswd]

Actually, I like your version of the model quite a bit, even more than the SIR model. I think it is a much more realistic representation of how the transition from I to R occurs. Instead of saying that, on average (mixing cup average), the amount of time one is infected is $1/\gamma$, it more realistically says that everyone is infected the same amount of time, $t_s$.

thanks, do you know how they get the co-efficient of $\gamma$ based on the average time being $1/\gamma$

 

[Chestermiller]

It's an approximation which says that, on average, the time for being infects is $t_s$ rather than saying that the time for every single person being infected is $t_s$ (like in your model).

 

[greswd]

It's an approximation which says that, on average, the time for being infects is $t_s$ rather than saying that the time for every single person being infected is $t_s$ (like in your model).

oh i see, though do you know how it is derived? thanks, I'm very curious

Also, I think the basic model needs another condition: $N_A(t)=N_T(t)-N_T(t-t_S)$

 

[Chestermiller]

oh i see, though do you know how it is derived? thanks, I'm very curious

It treats the population A as a well-mixed tank.

Also, I think the basic model needs another condition: $N_A(t)=N_T(t)-N_T(t-t_S)$

I don't think that another condition is needed, and I don't think this condition is correct.

 

[Chestermiller]

I want to also mention that the model won't be even close to realistic unless you also include a balance equation on the yet uninfected population. Ultimately, depleting the uninfected population is the key mitigating factor.

 

[greswd]

It treats the population A as a well-mixed tank.

I don't think that another condition is needed, and I don't think this condition is correct.

cos I need to figure out the derivation, I’ve searched a lot for it, but sadly can’t find anything

 

[Chestermiller]

cos I need to figure out the derivation, I’ve searched a lot for it, but sadly can’t find anything

Are you saying that you haven't seen the formulation of the SIR model equations?

 

[greswd]

Are you saying that you haven't seen the formulation of the SIR model equations?

I have, but I haven’t seen any derivation of why the rate co-efficient is the reciprocal of the recovery duration

 

[greswd]

I want to also mention that the model won't be even close to realistic unless you also include a balance equation on the yet uninfected population. Ultimately, depleting the uninfected population is the key mitigating factor.

depleting the uninfected population, a real life horror

 

[Chestermiller]

depleting the uninfected population, a real life horror

It may not be that bad if the current tallies of reported cases is only a small fraction of the actual number of infections. I've seen experts on TV make the estimate that only one out of every ten infected cases gets reported; the rest are mild and unreported. So we might already have made a sizable dent into the total population.

 

[Chestermiller]

I have, but I haven’t seen any derivation of why the rate co-efficient is the reciprocal of the recovery duration

This is just an approximation to the removal term in your derivation. Rather than requiring that each and every person entering the contagious population stays there for exactly 14 days, their approach continually mixes the entire contagious population and releases 1/14 of them each day at random. So in their approach, the average amount of time that a person is contagious is 14 days, while in your approach, the exact amount of time that every person is contagious is 14 days. I think they do this because it is simpler for them to implement mathematically. However, in my judgment, your approach, as a difference-differential equation, is much more realistic and accurate, and just as easy to implement mathematically. I have, in fact, been using your approach in the computer version (Excel spreadsheet) of the model that I have developed.

 

[greswd]

This is just an approximation to the removal term in your derivation. Rather than requiring that each and every person entering the contagious population stays there for exactly 14 days, their approach continually mixes the entire contagious population and releases 1/14 of them each day at random. So in their approach, the average amount of time that a person is contagious is 14 days, while in your approach, the exact amount of time that every person is contagious is 14 days. I think they do this because it is simpler for them to implement mathematically. However, in my judgment, your approach, as a difference-differential equation, is much more realistic and accurate, and just as easy to implement mathematically. I have, in fact, been using your approach in the computer version (Excel spreadsheet) of the model that I have developed.

Oh, that’s a pretty interesting way to model it

how different are the results of both models in excel?

 

[Chestermiller]

Oh, that’s a pretty interesting way to model it

how different are the results of both models in excel?

They are very similar, provided you specify the values of the rate parameters k a little differently in each case.

 

[greswd]

They are very similar, provided you specify the values of the rate parameters k a little differently in each case.

I think an excel simulation also automatically satisfies the condition in #11 right?

 

[Chestermiller]

I think an excel simulation also automatically satisfies the condition in #11 right?

Yes. I've been able to show that analytically.

 

[Chestermiller]

Model Results With- and Without Amelioration (Solution to Eqns. 1 in post #7)

FRACTION INFECTED PER DAY AS A FUNCTION OF NUMBER OF DAYS

Blue line - Without amelioration
Green line - With amelioration (hibernation and social distancing measures)

Initial conditions:
Fraction susceptible initially 0.999999
Fraction infected and contagious initially 0.000001
Fraction recovered initially 0.

Scenarios:
Without amelioration: k* = 0.17/day (representing a doubling time of about 5 days)
With amelioration: k* changed to 0.1/day at day 80 (corresponding to current estimates of Ro)
Time interval each person is contagious: 14 days

These results seem to be consistent with what they have been telling us on TV. An interesting feature is that the peak with amelioration comes only about 5 days after the peak without. I would have expected it to come much later. The ultimate fraction that is infected is almost 90 percent without amelioration, and only 55 percent with amelioration.

The results also suggest (not shown here) that we ought to wait about a month after the peak before starting to ease up on restrictions; otherwise there can be significant flare-up.

It's amazing that such a simple model can yield so much useful information.

 

[anorlunda]

Whas

Initial conditions:
Fraction susceptible initially 0.999999
Fraction infected and contagious initially 0.000001
Fraction recovered initially 0.

What are the final conditions for the two cases?

Edit: and Are those three the only states?

 

[Chestermiller]

Whas

What are the final conditions for the two cases?

As I mentioned, about 90% ultimately infected and about 55% ultimately infected.

Edit: and Are those three the only states?
[/QUOTE]
Yes, those are the only 3 populations the model follows. As you can see from the equations, it's a very simplified version of what happens.

I'm going to also look and an easing scenario where I go back half-way at day 125.

 

[Chestermiller]

EFFECT OF EASING RESTRICTIONS (HALF-WAY), 45 DAYS AFTER STRONG RESTRICTIONS WERE IMPOSED AT DAY 80

Probably it would be much better to wait another 15 days.