22nd Jun, 2020

Polytechnic University of Catalonia (Universitat Politècnica de Catalunya)

Question

Asked 29th Mar, 2020

SIR models are simple epidemic models, but their generalizations are used in many instances for decision making in front of crisis like the present covid-19 epidemic. A population of N individuals, at time t, is partitioned into susceptible s(t), infected i(t), and recovered r(t). This last class includes recovered, immune and dead people. A simple SIR model (differential equation) can be written as

ds(t)/dt = - b s(t) i(t)

di(t)/dt = (b s(t) - a) i(t)

dr(t)/dt = a i(t)

where a, b are positive parameters of the model.

The question is whether this model can be considered consistent taking into account that s(t), i(t), r(t) are positive and add up to N (or any constant like N=1). Are the solutions and parameters dependent on N? Is positiveness of the solutions guaranteed? Are the derivatives meaningful?

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22nd Jun, 2020

Polytechnic University of Catalonia (Universitat Politècnica de Catalunya)

Dear Muhammad Ali,

Thank you for your interest.

However, my question is related to consistency of such models, not only simple SIR models but their generalizations. The problem has several implications, but the more relevant, from my point of view, is that the whole system of differential equations, including its parameters, dependd on the size of the assumed population. The interpretation of a model whose parameters depend on the size of the population is always difficult and its consistency may be doubious.

From my point of view, these kind of models should be "scale invariant", and its parameters should be the same whichever is the size of the population. Compositional data analysis offers ideas for the reformulation of SIR models, but further research is still required.

1 Recommendation

The SIR model is too simple to expect it will simulate the Covid19 epidemic (and most others too). In a response to someone's question about fractals and chaos occurring in the epidemic, I wrote about my model that has the population partitioned into 8 subpopulations. In answer to your questions concerning the SIR model, the parameter b will depend upon N if the modeling is done correctly since it has units of (per person per unit of time). The solution definitely depends upon N = s(0)+i(0)+r(0), which is preserved in this case (although one can include population growth by just having a nonzero birth rate for s, for example. With the equations as they are, the positive octant is invariant, i.e., if s, i, and r start with nonnegative values, they will have nonnegative values for all time.

The question about derivatives being meaningful is important and natural since we are considering a population of individuals, i.e., a discrete set and so if any of s, i, or r is changing, it is discontinuous. However, if we look at the problem probabilistically, then we can look at s, i, and r as population densities which take any values in the interval [0,1] in the limit as N \to \infty.

So, the short answer is: These equations are reasonable when looking at very large populations. When looking at small populations, they are not reasonable.

1 Recommendation

Dear Juan,

I agree with Dr Bates, the SIR model is too simplistic to model COVID-19 accurately, or for that matter, any other epidemic. However, it does capture the salient features over short time periods if the parameters a and b are known, or estimated. A major problem is that, in particular, the b parameter varies over time. This can be due, among other reasons, to interventions by governments such as enforcing 'social distancing', 'self isolation' or 'lock-down', all of which are designed to reduce the 'transmission rate' b. The parameter a is generally more stable unless there is a medical breakthrough, such as a new treatment or the discovery of an effective vaccine.

I am currently fitting the SIR model to reported COVID-19 data with reasonable accuracy. However, it is the projections into the future that are problematic due to uncertainty in b. One way of including for variability in b, is to modify this parameter over time, something like b = b0(p +t)/q+t), q<p, but one is still left needing a good estimate for p and q. Also, drastic intervention actions by governments can result in rapid changes in b, and the rate at which b changes is dependent upon the response of the public, which may vary country to country.

A further complication is that the true values for S, I and R may never be know with high accuracy as some people become infected and recover without the medical authorities knowing. Also, some deaths may be wrongly attributed. Reporting lags can also distort the statistics.

A recent paper:

Lin, et al. (2020), A conceptual model for the coronavirus disease 2019 (COVID-19) outbreak in Wuhan, China with individual reaction and governmental action, "International Journal of Infectious Diseases", vol 93, 211-216.

proposes a more detailed model that offers an interesting approach.

Kind regards,

Graham W Griffiths

1 Recommendation

I think there is a basic issue in the mentioned models, both the simple ones and those more complex, namely that SIR systems are compositional by definition. The categories, be it three or more, represent parts of a population, and the information of interest is relative, not absolute. The standard models do not take this fact into account and built on the assumption that we are dealing with real random variables, which sample space is the whole or a subset of the real line endowed with the usual Euclidean geometry. This does not work for compositional data. It is well known that it can lead to spurious results. I am not aware of epidemiologic models that build on the simplex, endowed with the Aitchison geometry, as sample space, but they would surely benefit from it.

1 Recommendation

Vera and Graham made good points. Spatial transport should be included but instead of just diffusion, the geometry of the distribution of the population needs to be included as well as connections between population centers. The diffusion coefficient (local travel) within each population patch will differ from patch to patch as will the rates of communication between pairs of patches. The model should be age-structured since response to infection depends upon age. My model does not do any of these but it does account for sub-populations that are infected/infectious without symptoms and length of time having been infected (3 bins) and also testing and two bins for those in quarantine. Mortality coefficients and infectiousness coefficients depend on sub-population. I have had to adjust parameters as time progresses and as the population starts to act appropriately. It has been fairly good at predicting numbers of confirmed cases and deaths. The future does not look very good unfortunately. I don't think this will be over in 6 months time and by then tens of millions will have been infected in the US.

2 Recommendations

1st Apr, 2020

Polytechnic University of Catalonia (Universitat Politècnica de Catalunya)

Thank you (Graham, Peter and Vera) for your responses. I think they will be useful for eventual readers and also for me. From them, I infer that (a) positiveness (or non negativeness) is guaranteed when the initial conditions are also non-negative and (b) SIR models and most of their generalizations (even spatial ones) depend on N although this dependence is negligible for large N. Even so, it seems that the dependence on N is pointing to a lack of consistence, as models are thought as applicable to different sizes of populations. Other points are pending. For instance, if the model is (almost) independent on N, how should the changes (or the derivatives) in s,i,r be accounted for? The scale of changes and derivatives depend on N as they are simple differences. Does compositional data analysis solve this question asking for scale invariant magnitudes in the model? Is it enough to force a, b or other parameters to be those estimated for a constant N (e.g. N=1, N=100,000)?

1 Recommendation

El modelo SIR, es muy criticado por muchos factores, en especial los que han mencionado,en este enlace hacen un gran analisis al problema que estamos viviendo

It is too simple model, however it can be generalized to nth order differential equation systems with fix initial conditions. The next step may be its use as optimization problem.

22nd Jun, 2020

Polytechnic University of Catalonia (Universitat Politècnica de Catalunya)

Dear Muhammad Ali,

Thank you for your interest.

However, my question is related to consistency of such models, not only simple SIR models but their generalizations. The problem has several implications, but the more relevant, from my point of view, is that the whole system of differential equations, including its parameters, dependd on the size of the assumed population. The interpretation of a model whose parameters depend on the size of the population is always difficult and its consistency may be doubious.

From my point of view, these kind of models should be "scale invariant", and its parameters should be the same whichever is the size of the population. Compositional data analysis offers ideas for the reformulation of SIR models, but further research is still required.

1 Recommendation

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