key: cord-142398-glq4mjau
authors: Dhar, Abhishek
title: A critique of the Covid-19 analysis for India by Singh and Adhikari
date: 2020-04-11
journal: nan
DOI: nan
sha: 
doc_id: 142398
cord_uid: glq4mjau

In a recent paper (arXiv:2003.12055), Singh and Adhikari present results of an analysis of a mathematical model of epidemiology based on which they argue that a $49$ day lockdown is required in India for containing the pandemic in India. We assert that as a model study with the stated assumptions, the analysis presented in the paper is perfectly valid, however, any serious comparison with real data and attempts at prediction from the model are highly problematic. The main point of the present paper is to convincingly establish this assertion while providing a warning that the results and analysis of such mathematical models should not be taken at face value and need to be used with great caution by policy makers.

A large number of recent papers have analyzed mathematical models, of varying degrees of complexity, in an attempt to understand and sometimes make predictions about the growth and spread of Covid-19 across the world. Such models often come with serious limitations arising from several facts such as: (i) compartmentalization of the actual degrees of freedom to make analysis simpler, (ii) too many parameters whose values are known with high degrees of uncertainties, (ii) inherent nonlinearities in the governing equations which make long time predictions difficult.

Nevertheless, there have been papers which make (or appear to make) definitive predictions and many of these end up generating a lot of media hype and get wide attention in the social media. One real risk is that this is actually noticed and used by policy makers. In the present note, we analyze the paper by Singh and Adhikari, which we believe is in this class.

In particular, the paper appears to make the claim that a 49 day lockdown is required to effectively control the growth of the Covid-19 pandemic in India. In this note, we explain why this claim should not be taken seriously.

At the outset we make it clear that the present comment does not question the technical validity of the results in [1] . However the present comment points out that there are many issues on details such as that of interpretation of variables while making comparison with real data and choice of model parameters -these make any attempts at predictions meaningless.

In Sec. (II) we first analyze a simpler version of the model studied in [1] and show that for a certain choice of model parameters, this already reproduces the main features seen in the more complicated model. However we show that, on making small changes of the parameters to somewhat more realistic values, the predictions change significantly. In Sec. (III) we point out a large number of other problems related to the study in [1] which imply that numerical predictions emerging from such studies are completely unreliable.

The basic model considered by the authors is one which compartmentalizes the population according to age. At any time each compartment has a number of susceptible (S) , infected (I) and recovered (R) individuals and these evolve with time. The division into age groups is to account for the fact that the levels of contact between different groups (and within groups) vary and so they have different probabilities of passing infections. To discuss the main aspects of the model, it is sufficient to ignore the compartmentalization into age groups and we present the simpler version here. This more basic model is one which divides the population of size N into individuals who are susceptible (S) , asymptomatic infectious (I a ), symptomatic infectious (I s ) or are recovered (R), with the constraint N = S + I a + I s + R. The number R also includes people who have died.

Dynamics: Let I = I a + I s be the total number of infected individuals. For a well mixed population, the probability that an S individual meets an I individual is ∝ (I/N ). We define the infection rate β, which controls the rate of spread and represents the probability of transmitting disease between a susceptible and an infectious individual. We assume that infected people typically either recover or die after T R days, so I → R happens at a rate γ = 1/T R . We also assume that a fraction α of the infected population is asymptomatic.

However, as in the analysis of [1] , we will set α = 0 which effectively means that we ignore asymptomatic carrier populations (if any).

We then have the following dynamics for the system

The time dependent function u(t) is the "lockdown" function that incorporates the effect of a lockdown on the rate of spreading of the infection. A reasonable form is one where u(t)

has the constant value (= 1) before the beginning of the lockdown, at time t on , and then it changes to 0 < u l < 1 over a characteristic time scale ∼ t w . Thus we can take a form

The number u l indicates the lowering of social contacts as a result of a lockdown. This was 3 set to the value 0 in [1] but we note that realistically it is expected that social contacts cannot be brought down to zero (for example, people still have to get groceries).

An important parameter is the so-called reproductive days. The evolution for parameter set (I) is very close to the results presented in [1] . Note also that the evolution for the three parameter sets are identical before the lockdown. Notably, the real data (for the period March 6 -April 7, 2020) is following a completely different trajectory (orange points). The inset shows the same data plotted on a semi-log scale.

Parameter set (III): We keep γ = 1/14, β = 0.24, α = 0 but change both the time, for the lockdown to be effective, to t w = 4.0 days and take u l = 0.1 (since, as discussed above, contacts are never reduced to 0).

In Fig. (1) we show the results for these three parameter sets and a comparison with real data for the number of confirmed Covid-19 cases in India. [Note: For comparison with real data, we follow [1] and assume that I s (t) is equal to the number of reported confirmed cases. In the comments in the next section, we explain however that this is not a reasonable assumption.]

In Fig. (1) we observe that the results from parameter set (I) are already in close agreement to those presented in Fig. (4d) of [1] . The estimated lockdown period for the number of infected to reduce to a value less than 10 is 54 (which compares remarkably well with the number 49 in [1] , the small difference is expected in this type of analysis). For the three parameter sets we find that the time required for the infected population to reduce to the value I = 10 are respectively This is of course also true for the predictions of the more sophisticated model studied in [1] -the true data after the lockdown has no resemblance to their predicted evolution.

Warning: The improved estimates quoted in Eq. (6) should not be used as suggested periods of lockdown. They are given here simply to point out the limitations of the present model.

We end this section by emphasizing that while the analysis presented here ignores details such as the contact population's age break up, population distribution and connectivity matrices, we already recover the main qualitative (and to a large extent, even quantitative) aspects of the results in [1] . It is expected that the main message on sensitivity of model predictions to parameter choices would not change by increasing the complexity of the model.

In the previous section, through the analysis of a simpler model, we highlighted some of the reasons why models such as those studied in [1] cannot be taken seriously for their predictive powers. However there are many more serious problems in the analysis and claims of the paper. Some of these are:

1. The modeling ignores important factors such as latency periods, asymptomatic and presymptomatic transmission. Presymptomatic transmission could happen over the order of 5 − 10 days and is a period during which individuals show no symptoms (and are undetected) but are infecting others.

2. The modeling ignores the fact that the number of people who have been tested are relatively very small in India and the actual number of infected people could be much higher.

is compared with the total reported number of confirmed cases. However one expects that the reported confirmed cases (say C) are those that habve already been detected and therfore it does not make sense that they should be counted amongst the ones who are spreading the infection. It seems reasonable to us that I >> C. So the comparison with real data (C) through the identification I = C is flawed. In fact it seems to us that it makes more sense to instead compare R with the real data.

4. The choice of initial conditions in solving these equations is subtle and it is unclear how the authors fix these. Small changes in initial conditions can lead to drastically different predictions.

5. As we have demonstrated, small changes in parameter values would lead to drastically different predictions.

6. The choice of the lockdown parameters t w < 1 and u l = 0 are completely unrealistic.

We expect there should be a time lag for the lockdown to take effect and even under lockdown, social contacts are not reduced to zero.

7. The optimal lockdown period is computed based on the arbitrary criterion that the infected number drops to the value 10. Why not 100 which would give a completely different prediction for the required lockdown period ? We do not see why this completely arbitrary value should be used as a criterion.

8. The actual trajectory of infections (or rather number of confirmed cases) after the lockdown started in India has no resemblance whatsoever with what is predicted by the authors, even for short times. In fact, a data analysis of the effect of lockdowns in several countries on the confirmed number of cases, presented in [3] , shows that the evolution differs qualitatively in all the countries from the picture presented in [1] . In our opinion, this basically proves that the model has no predictive value and should not be used as guiding material for policy makers.

We have established in this brief note that the studies of mathematical models of disease spreading, such as the one in [1] , typically have very limited ability to make numerical predictions. Such models, if analyzed properly, could play an important role in our qualitative understanding of disease evolution and, with some effort, perhaps in making short term predictions. Unfortunately, completely absurd claims sometimes attract a lot of public attention and run the risk of leading to wrong policy decisions.

While there are many such examples in the recent literature, we have focused here on analyzing one specific model study [1] . This paper is technically correct in that they state their assumptions clearly and the stated results are obtained under these assumptions. However, 

Samriddhi Sankar Ray and Sanjib Sabhapandit for a careful reading of the manuscript and pointing out errors. I thank Rajesh Gopakumar

Age-structured impact of social distancing on the COVID-19 epidemic in India

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