COVID-19 - Model and ODE delay equations

I have a working model and parameter optimiser for the COVID-19 epidemy (Emmanuel-R8/COVID-19-Julia]([https://github.com/Emmanuel-R8/COVID-19-Julia). The model is inspired (but slighlty more detailed) by this impressive project Neherlab COVID-19. My version is explained there Forecasting COVID-19.

However, those model rely on delay equations which are often ignored for simplicity but clearly distort the actual value of parameters. I have not been able to find good examples of delay equations.

Any pointers?

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Hi,
If you are looking for general references, there are a number of books:

* J.Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations
* Diekman, O., van Gils, S. A., Verduyn Lunel, S. M., & Walther, H. Delay-Equations: Functional, Complex and Nonlinear Analysis
* Kolmanovskii, V.,Myshkis, A. Introduction to the theory and applications of functional-differential equations.
Odo Diekman, one of the authors of the second book is a leading expert in biological models and coauthored a book about this: Mathematical tools for understanding infectious disease dynamics.

The only book on numerical methods is Numerical Methods for Delay Differential Equations
by Alfredo Bellen and Marino Zennaro.
Hope that helps

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Thanks for the references., but I should have been clearer. I am looking for guidance to implement them in Julia.

Presumably you’ve seen https://docs.sciml.ai/dev/tutorials/dde_example/#Delay-Differential-Equations-1?

Oh, you can start with this tutorial from the official documentation.

I saw those links. But I have several time lags that I would like to optimise for. I couldn’t readily see how this fits into the examples provided.

Seems like the conclusion is that resources are meager (given that I was looking for the lazy option of copy/paste someone else’s hard work…)

Thanks for the help!

https://docs.sciml.ai/latest/analysis/parameter_estimation/

are both doing optimization of DDE parameters. You can basically copy paste from the second link.

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You may know about this already, but here is a common trick for converting delay differential equations to ODEs for the purposes of numerical approximation/simulation.

Suppose that some agents convert from state A to B after a delay of T. Introduce interim states I_1, I_2, \dots, I_n, with Poisson transitions rates \lambda_0 from A to I_1, \lambda_k from I_k to I_{k+1}, and finally \lambda_n from I_n to B.

Calibrate your parameters so that \sum 1/\lambda_i = T, so the expected time it takes to get from A to B matches. The higher n is, the smaller the variance of the expected time will be (this is easy to check and is a good exercise).

In practice, for a lot of processes, you can get away with n = 2, \dots, 5 and get a good approximation. For a lot of things transitions are not deterministic anyway, so this may be a better model to start with. Eg for COVID-19, you can assume transmission rates increasing in k for the I_k.

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I missed your answer. Apologies.

I didn’t know that trick. Seems like an easy way to optimise for the value of the delay.

I have not started on implementing the delay as I have been debugging the parameter optimisation.

How does this compare to classic Padé-approximation of the Laplace-transform of time delay? Essentially, y(t) = x(t-\tau) \Rightarrow^{\cal L} Y(s) = \exp(-s\tau) X(s), with Padé approximation \exp(-\tau s) = \exp(-\frac{\tau s}{2})/\exp(\frac{\tau s}{2}) where the numerator and denominator are approximation in power series,

\exp(-\frac{\tau s}{2}) \approx 1-\frac{\tau s}{2} + \frac{1}{2}\left(\frac{\tau s}{2}\right)^2 -\cdots \\ \exp(\frac{\tau s}{2}) \approx 1+\frac{\tau s}{2} + \frac{1}{2}\left(\frac{\tau s}{2}\right)^2 +\cdots

which is then transformed back to ODEs?
Thus, in the simplest case,

y(t) = x(t-\tau) \\ \Downarrow \\ \left(1+\frac{\tau s}{2}\right)Y(s) \approx \left(1-\frac{\tau s}{2}\right)X(s) \\ \Downarrow \\ y(t) + \frac{\tau}{2}\frac{dy}{dt} \approx x(t) - \frac{\tau}{2}\frac{dx}{dt} \\ \Downarrow \\ \frac{dy}{dt} \approx - \frac{2}{\tau}(y-x) - \frac{dx}{dt}

The “problem” here is \frac{dx}{dt}. If x is a signal, you need to differentiate it. If x is the state of another ODE, you just eliminate it by replacing it with the right hand side of that ODE.

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Interesting to compare those two. A compartment fed with poisson rates acts a bit like an damped integrator, do you typically get these additional derivative terms dx/dt (making x an integrator of that new coordinate as well)?

the problem of differentiating wrt to the delay D in DDE is not well posed mathematically as it affects the state space. By scaling time, you can assume D=1 and multiply your vector field by D. It becomes then possible to differentiate.

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Note: the right hand side derivatives can, of course, be avoided by writing:

\exp(-\tau s) = \frac{1}{\exp(\tau s)} \approx \frac{1}{1+\frac{\tau s}{2}+\frac{1}{2}\left(\frac{\tau s}{2}\right)^2 +\cdots}

but as I recall, that leads to a poorer approximation.

Yes, indeed, but if there are multiple delays present, then differentiation of the right-hand side with respect to the remaining (after scaling) delay parameters is still - in general - not possible.

(This actually had me wondering a bit while following this topic. It may be that (perhaps implicitly) the problem is circumvented by using that the history segments gain one degree of smoothness after each time step equal to the maximum delay.)

I think that a finer grid of interim transitions (again, 2–5 as I suggested above) may lead to a better approximation than the one you suggested while remaining very tractable (but this is just intuition, I did not do the math).

Also, I find it intuitive to think about people being in “stages” of infection, between which they transition stochastically. I think that a fixed lag \tau for the population is a convenience assumption; I don’t think the process is deterministic so I would not work very hard to maintain this assumption.

My feeling is that you can do it. The state space is something like C([-D, 0], R) where D is the maximum delay. If the rescaled delays stay below 1, it should be fine for this part.

Then the question of mild/strong solution comes in. Regularity of C0-semigroups wrt to parameters, I have never thought much about this.

I had not thought about it much either, until relatively recently. There is a comment about it in this article in Remark 21, but the issue was well-known before that, see the references in that remark and also the work by Krisztin and Walther on state-dependent DDEs. There the problem becomes even more serious.

It is not my intention to take this too far off-topic, but if you would like to, we can continue the discussion elsewhere.

Note: I don’t intend to suggest using first order approximation for the exponential functions; that’s just an example. Anyway, would your proposal be similar to setting (\tau\ge 0, \tau_i \ge 0):

\exp(\tau s) = \prod_{i=1}^n\exp(\tau_i s)

where \sum_{i=1}^n\tau_i = \tau (e.g., n=4 or n=5), using only first order approximations of \exp(\tau_i s), and then tune the \tau_i to get the best approximation?

[No, I have to calculate that more carefully].

could be interesting to you as well