key: cord-0687356-3qyf970n
authors: Chakraborty, Avishek
title: A remark on “COVID-19: Perturbation dynamics resulting chaos to stable with seasonality transmission” [Chaos, Solitons and Fractals 145 (2021) 110772]
date: 2022-01-24
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2022.111831
sha: f527f0fa52ffb5ef5bf14828b1ddf58f62ea226b
doc_id: 687356
cord_uid: 3qyf970n

In [2], introducing an extension of the well-known susceptible-exposed-infected-recovered (SEIR) model with seasonality transmission of SARS-CoV-2, the author has derived and discussed various analytical and numerical results. Careful scrutiny of the said article brings about some genuine issues pertaining to the model formulation, analysis and numerical studies carried out in [2]. Given the present pandemic and the havoc it has been causing throughout the world, and the responsibility of giving out rightful information/results backed by scientific proofs, there is a pressing need to address issues of such kind right away.

In [2] the following model was proposed as system (1):

= −(β 1 I 1 + β 2 I 2 )S, dE dt = (β 1 I 1 + β 2 I 2 )S − aE, dI 1 dt = aE − (γ 1 + α)I 1 , dI 2 dt = αI 1 − (γ 2 + µ)I 2 , dR dt = γ 1 I 1 + γ 2 I 2 , dD dt = µI 2 .

(1)

Next, the author has incorporated a probability factor and seasonality in the transmission rates of system (1) of [2] , thus leading to: dS dt = −(β e E 1 + β 0 I 0 + β 1 I 1 + β 2 I 2 )Sσ(t),

Here p denotes probability and σ(t)

] is a periodic function. Next, the author has considered spatio-temporal model followed by theoretical and numerical analyses.

Careful reading of [2] gives rise to the following points/questions:

Observations drawn from system (1) of [2] are as follows:

1. In the first equation of system (1) of [2] there is no positive input to rate of change of S population. According to this equation S → 0 as t → ∞ (where β 1 , β 2 , I 1 , I 2 > 0). When S = 0, the model fails to make sense.

The author has shown the stability result of system (1) of [2] in Theorem 3.1. Suppose (for the purpose to calculate the equilibrium point) dS dt = 0, then either S = 0 or β 1 I 1 + β 2 I 2 = 0. Let us consider S = 0, then from the rest of the equations ( dE dt = 0, dI 1 dt = 0, dI 2 dt = 0, dR dt = 0, dD dt = 0) we get E = I 1 = I 2 = R = 0, but nothing can be said about the equilibrium densities of the variables R and D. If we calculate from bottom to top, from dD dt = 0 we get I 2 = 0, if I 2 = 0 then from dR dt = 0 we get I 1 = 0. If I 1 = 0 then from dI 1 dt = 0 we get E = 0. If I 1 = I 2 = 0 then from dS dt = 0 we get S is either positive constant or 0. But we cannot say anything about the equilibrium densities of the variables R and D. Then, how could the author arrive at the result of Theorem 3.1? Also, how could the author find the Hopf bifurcation result of endemic equilibrium point in Theorem 4.3 of [2] . According to the calculation of endemic equilibrium point (all the populations must exist), there is no existence of such an equilibrium point for system (1) of [2] .

3. In Theorem 5.1, the author has shown the global stability result of system (1) of [2] . The proof begins with the coexistence equilibrium point (E * , I * 1 , I * 2 , R * , D * ) of system (7) of [2] . But, calculating E * , I * 1 , I * 2 , R * , D * from equation (7) of [2] , it is obvious from the last equation dD dt = 0 we get I * 2 = 0. From dR dt = 0 we get I * 1 = 0 (already I * 2 = 0). Next, if dI 1 dt = 0 or dE dt = 0, we can get E * = 0. But, from system (7) of [2] , nothing can be said about R * and D * . Also, calculating the characteristic equation of system (7) of [2] ,

it is different from equation (9) of [2] . Concentrating on equation (10) of [2] , we observe two eigen values to be 0, which means the equilibrium (E * , I * 1 , I * 2 , R * , D * ) is always a neutral point (by local stability analysis). Thus, the equilibrium point is not locally asymptotically stable. How could the author calculate the global stability in Theorem 5.1?

Observations drawn from system (2) of [2] are as follows:

1. System (2) of [2] is non-autonomous. Thus, finding the equilibrium point is not possible given the time-dependency of the system parameter(s) [3, 6, 8] . But, in [2] the author has shown the stability result of system (2) in Theorem 3.4 using the eigen values of system (2) . Here, the author stated that the eigen values of system (2) can be calculated, which is impossible.

In the proof of Theorem 5.2, the author stated "the same process will be followed as Theorem 5.1". But, system (2) of [2] is non-autonomous and the methodologies used in proving the global stability of an autonomous system [1, 4, 5, 7] and a non-autonomous system [3, 6, 8] vary drastically.

Listed below are some critical observations regarding the numerical results obtained in [2]:

1. In Fig 2(a) of [2] , there is no graph for variable D. The time series of system (1) of [2] with the given parameter set will be as shown in Figure 1 of the present manuscript. Also, the Matlab code is given in Figure 2. 2. In Fig 2(b) of [2] , the author ploted the time series of system (2) of [2] which is nonautonomous with respect to the periodic term σ(t) = 1 + cos(2π(t − φ)) in the first two equations. In 3. System (2) is non-autonomous and thus we cannot find the equilibrium point (author mentioned it as final destination in the text). So, with periodic function σ(t) in system (2), how can the author plot Figures 5, 6 and 7? 4. In Figures 8, 9 and 10, the author has shown Hopf bifurcation diagram of system (1) 2D plane and 3D space of [2] . Let us first look at Figure 8 . In Hopf bifurcation any system shows unstable focus (stable periodic solution around equilibrium point) and stable focus (stable equilibrium point) at two different ranges of a parameter. In Figure 8(a) , the author has plotted max and min points of periodic solution when 0 < α < 0.4, but the same phenomena cannot be seen in Figure 8(b) . Most importantly, in Figure 8 (b) the author has failed to show a stable periodic solution, instead some spiral graphs are seen. Also, when α > 0.4, the equilibrium density of S and R are different in Figures 8(a) and 8(b) .

The range of max and min points of the periodic solution of E and I 1 of [2] in Figures 9(a,b) are different from Figure 9 (c) when 0 < α < 0.4. Also, when α > 0.4, E and I 1 are tending to 0 (Figures 9(a,b) ), but from Figure 9 (c) equilibrium density (E, I 1 ) is observed to be increasing even far away from 0 as the bifurcation parameter increases. 

In the current manuscript some genuine issues regarding the result analysis carried out in [2] have been identified. Such issues need to be dealt with straight away in order to prevent the researchers working on covid/other disease models or dynamical studies from being mislead. Figure 3 of the present manuscript (after Figure 4) .

The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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 Figure 6 : Continuation of Matlab code of Figure 3 of the present manuscript (after Figure 5 ).