key: cord-1017238-6dkgysrz authors: Yu, Zhenhua; Arif, Robia; Fahmy, Mohamed Abdelsabour; Sohail, Ayesha title: Self Organizing Maps for the Parametric Analysis of COVID-19 SEIRS Delayed Model date: 2021-06-24 journal: Chaos Solitons Fractals DOI: 10.1016/j.chaos.2021.111202 sha: 538d56a9580f6ccd31e379c0b0a411911b45114b doc_id: 1017238 cord_uid: 6dkgysrz Since 2019, entire world is facing the accelerating threat of Corona Virus, with its third wave on its way, although accompanied with several vaccination strategies made by world health organization. The control on the transmission of the virus is highly desired, even though several key measures have already been made, including masks, sanitizing and disinfecting measures. The ongoing research, though devoted to this pandemic, has certain flaws, due to which no permanent solution has been discovered. Currently different data based studies have emerged but unfortunately, the pandemic fate is still unrevealed. During this research, we have focused on a compartmental model, where delay is taken into account from one compartment to another. The model depicts the dynamics of the disease relative to time and constant delays in time. A deep learning technique called “Self Organizing Map” is used to extract the parametric values from the data repository of COVID-19. The input we used for SOM are the attributes on which, the variables are dependent. Different grouping/clustering of patients were achieved with 2- dimensional visualization of the input data ([Formula: see text]). Extensive stability analysis and numerical results are presented in this manuscript which can help in designing control measures. The spread rate of SARS-2 infection is alarming currently due to its, mutated versions and the 30 continuous waves [1, 2] in different parts of the world, especially Europe and America. The new variants 31 of the disease are emerging with the passage of time, increasing the pressure on "designing accurate 32 control measures". 33 Coronavirus (SARS-CoV2) belongs to the beta-coronavirus genus. The transmission mechanism fo-34 cuses on the viral S protein (spike protein) that binds the virus to a reaction catalyzed by some enzymes 35 located on the surface of the host cell. 36 Different control measure studies, at different levels, have recently emerged in the literature [3, 4] . 37 The daily based incidence data of the disease positive case-counts, was discussed by Clouston et'al. [3] . 38 They examined the spread and severity of the infectious disease with the aid of statistical analysis, by 39 keeping in view different variables, such as age, gender and race. As a major outcome, the importance of 40 social distancing was concluded. 41 Computational frameworks have been reported in the literature, to address the compartmental trans- inference at the beginning to the the disease outbreak, especially before January 23 when Wuhan was 48 quarantined and locked down, and that there was a lack of reliable data, except for the confirmed case 49 data that could be used for model calibration. 50 It has been reported in the literature that delay played an important role in the dynamics of the 51 disease transmission [8] . [9] reported the delayed dynamics due to the quarantine strategy. Recently the 52 research conducted by [10] reported that the pandemic was delayed due to the control measures practiced 53 by different countries in different ways, including extensive testing, contact tracking, and quarantining; 54 thus the widespread measures, enforcing "social isolation" were successful but not ideal for long term 55 practice due to the socioeconomic pressure. analysis is provided in the manuscript to make it more authentic. Numerical results, leading to some 61 proposals to control the pandemics are reported at the end. The "SEIRS" model utilized during this research is given as: Description Recovered nodes. 65 We have the following system of equations: with the initial conditions (2) Here, ϕ i (ϑ) are the initial functions, where i= 1, 2, 3, 4. All the parameters are non negative. Each • algorithmic properties. The algorithm of SOM is actually based on the competitive learning, whereas, the algorithm of ANN is 80 based on the error correction learning. To preserve the topological properties of the given data, the SOM 81 algorithm is linked with the neighborhood function. There are two layers of the SOM-Kohonen Neural Networks. First layer is called the input layer, that 83 is fully connected with the competitive layer of the processing neurons whereas the next layer is terms as 84 the output layer. For a network, we use an n× 1 input vector in general practice. The components of the vector are: The n-components of this input vector, are connected with each neurons in the array (the output 88 layer of m × m processing neurons). The input layer is connected to the Kohonen layer by weight, w j = (w j1 , w j2 , , w jn ), where w ij is 90 the weight value associated with i th component of input vector to the j th neurons. Consequently, by the separating variables and then integrating (3) It shows that V 1 (t) will remain positive as the V 1 (0) is a positive and thus it is a property of positive invariant. To prove that V 2 (t) is positive we use (2) in the [0, τ ], the second equation is since the parametric values are non negative. By separating and integrating both side it gives Similarly, from 3rd and 4th equations of model (1) and by using (2) we have; dV 4 (t) dt ≥ −κV 4 (t) − ψV 4 (t). By separating V 3 (t) and V 4 (t) receptively and integrating both sides; it gives From preceding analysis it is conclude that the positivity of V 1 (t), V 2 (t),V 3 (t) and V 4 (t) depends on the 107 initial-history functions; for equation 2nd 3rd and 4th equations respectively we have By this approach, the above method can be generalized to any finite interval [0, t] and this prove that 113 Lemma 1 Given the 1st order differential inequality: the solution of this inequality will satisfy that 115 and hence Proof: Suppose we have 1st order differential equation; there is an integration factor e αt . By multiplying both sides of the equation (10) with the integrating factor e αt gives an exact differential equation Proof: By solving 1st equation of model (1) For that reason, the positivity of V 1 , V 2 and V 3 in the above equation implies that the maximum value 122 that V 1 can attain is n, since rate dV1 dt < 0 and which implies that Again by adding the 2nd and 3rd equations of the model (1) we obtain Where m = max{η, σ}. According to the lemma (10) we obtain Adding the 3rd and 4th equation by applying lemma (10) we obtain By combining (15), (16) and (17) it is established that its solution will ultimately be bounded in that 126 domain . The model (1) contain two equilibrium points. The infection free equilibrium point is The endemic equilibrium point exist if ρ > κ, otherwise nodes (exposed, infectious and recovered) will 130 become zero. where The coefficient A * is as follows: The characteristic equation of Jacobian matrix, at the equilibrium point E 0 is as follows nρ − e λτ1 (η + 2(κ + λ) + σ) = 0. Here we have two eigenvalues as λ 1 = −κ and λ 2 = −κ − ψ that are the negative. the model is stable if The stability can be proved by the following theorem. Proof: Consider the equilibrium point E 0 is stable for τ 1 = τ 2 = 0 its mean equation (23) as has all 142 roots to be negative That implies ((η − σ) + nρ) 2 + 4nρσ > nρ − (η + 2κ + σ). Now, suppose that τ i = 0 varies continuously 144 in a positive direction such that there is a τ * i which give one imaginary eigenvalue pair by putting λ = + − i , 145 > 0, Hence substituting λ = i in (23), after simplifying we obtain The equation (26) shows For the model (1) the basic reproduction number R 0 is as follows The sensitivity if the reproductive number is analyzed by taking the partial derivative with respect to the parameter. Reproductive number is increased with increments in n and infection rate while decreases with inclusion 153 or drop-out rate, the recovery rate and the outbreak rate. For the stability of endemic equilibrium point we will suppose that R 0 > 1. The Jacobian matrix at endemic equilibrium point where as The characteristic equation for the equi point E * is where the constants are defined as follows a 2 = A(η + 3κ + σ + ψ) + η(3κ + σ + ψ) + ψ(3κ + σ) + 3κ(2κ + σ), a 3 = A η(2κ + σ + ψ) + 3κ 2 + 2κ(σ + ψ) + σψ + η 3κ 2 + 2κ(σ + ψ) + σψ + κ κ 2 (σ + 4) + 3κ(σ + ψ) + 2σψ , Where a = 0. To gain insight regarding the endemic equilibrium point E * we will discuss stability of endemic 164 equilibrium point and conditions of Hopf bifurcation of threshold parameters like τ 1 and τ 2 by considering 165 the following cases. 166 Case 1. When τ 1 = 0 and τ 2 = 0 then equation (28) will become Therefore, the endemic equilibrium point E * is asymptotically stable with following conditions if (R 1 ) Case 2. When τ 1 = 0 and τ 2 > 0, (28) will become By assuming that for some values of τ 2 > 0, there exist a real ξ such that λ = iξ by putting λ = iξ after − c 3 ξ cos(ξτ 2 ) + c 4 sin(ξτ 2 ), squaring and adding both equations in 32 175 υ 4 + e 1 υ 3 + e 2 υ 2 − e 3 υ + e 4 = 0, ξ 2 = υ. Where the constants are as follows By rule of signs of Descartes (32) has at least on real positive root if (D 2 ) c 1 > 0 and (a 4 + b 4 ) 2 < c 2 4 176 holds. Eliminating sin(ξ 0 τ 2 ) form the equations (31) we have where E = ξ 6 0 (c 2 − c 1 (a 1 + b 1 + c 1 )) + ξ 4 0 (c 2 (− (a 2 + b 2 )) + c 3 (a 1 + b 1 ) + c 1 (a 3 + b 3 + c 3 ) − c 4 ) 179 and j = 0, 1, 2, ... By differentiate (32) with respect to the τ 2 , transversality will be obtain as τ 2,j = τ 2 and ξ = ξ 0 that is where: If Re( dλ dτ2 ) −1 > 0 a Hopf bifurcation will occur for delay τ 2 we reached following theorems. Theorem 2.6. Suppose that (R 1 ) and (D 1 ) is hold with delay τ 1 = 0 and there exist a τ 2 > 0 such that 184 E * remain stable for τ 2 < τ * 2 and unstable for τ 2 > τ * 2 , where τ * 2 = min{τ 2 , j} (33). Furthermore, model (1) undergoes the hopf bifurcation at point E * when τ 2 = τ * 2 . Case 3. When τ 2 = 0 and τ 1 > 0 then equation (28) will be We suppose for some values of τ 1 the λ = iξ we have two equations 188 ξ 4 − a 2 ξ 2 + a 4 = ξ 3 (b 1 + c 1 ) sin(ξτ 1 ) + ξ 2 (b 2 + c 2 ) cos(ξτ 1 − ξ) (b 3 + c 3 ) sin(ξτ 1 ) Squaring and adding both equations Where the coefficients are: Similarly to previous case, we arrived at the following theorem. Theorem 2.7. Suppose that (R 1 ) and (D 2 ) is holds with delay τ 2 = 0 and there exists a τ 1 > 0 such 191 that E * is locally asymptotically stable for τ 1 < τ * 1 and unstable for τ 1 > τ * 1 , where τ * 1 = min{τ 1 , j}. Furthermore, model (1) undergoes the hopf bifurcation which occur at point E * when τ 1 = τ * 1 . Where B 1 = a 3 ξ 2 1 ξ 2 1 (b 1 + c 1 ) − b 3 − c 3 + a 1 ξ 4 1 ξ 2 1 (− (b 1 + c 1 )) + b 3 + c 3 and j = 0, 1, 2, .... Case 4. When τ 1 > 0 and τ 2 > 0 the equation (28) will be We suppose for some values of τ 1 and τ 2 there is a real number ξ we have two equations of real and 196 imaginary values at λ = iξ. By squaring and adding both equations in 41. Applying Rouches Theorem we have 198 υ 4 + g 1 υ 3 + g 2 υ 2 + g 3 υ + g 4 = 0, ξ 2 = υ. Where the coefficients are: By rule of signs of Descartes equation (41) has at least one positive real root if 199 (D 3 ) a 2 1 − 2a 2 > 2b 1 c 1 cos(ξτ ) 2 + b 2 1 + c 2 1 and a 2 4 − 2b 4 c 4 cos(ξτ ) 2 < b 2 4 + c 2 4 hold. By eliminating sin(ξ 0 τ 1 ) we have Where G 1 = ξ 0 (c 3 sin(ξ 0 τ 2 ) − ξ 0 (c 2 cos(ξ 0 τ 2 ) + b 2 + c 1 sin(ξ 0 τ 2 )ξ 0 )) + c 4 cos(ξ 0 τ 2 ) + b 4 , To study Hopf bifurcation we would fix τ 2 in the stable interval and take derivative with respect to τ 1 of 202 equation (40) while using substitutions of ξ = ξ 0 τ 1 = τ 1,0 Where C 1 = cos(ξ 0 τ 1,0 ), S 2 = sin(ξ 0 τ 2 ), We obtain 204 dRe(λ) dτ 1 = P 2 P 3 − P 1 P 4 P 2 2 − P 2 1 . Hopf bifurcation will occur for τ i = τ 1,0 if dR(λ) dτ1 > 0. Theorem 2.8. If E * is existent, such that (R 1 ) and (D 3 ) hold, with τ 3 = 0 and τ 1 ∈ (0, τ * 1 ), there is an 207 existent positive parameter τ 2 such that endemic equilibrium point is locally stable for τ 2 < τ 2 and unstable 208 for τ 2 > τ 2 , where τ 2 = min{τ 2 , j} as in (42). Moreover, model (1) will undergoes hopf bifurcation at 209 point E * when τ 2 = τ 2 . Note: Similarly, For τ 2 ∈ (0, τ * 2 ), there is exists threshold parameter τ 1 such that endemic equi point is 211 locally asymptotically a stable for τ 1 < τ 1 and unstable if τ 1 > τ 1 . Moreover, hopf bifurcation occur for Where α 1 = ξ 2 a 3 cos(ξ 2 τ 1 ) − b 1 ξ 2 2 + b 3 − a 1 ξ 3 2 cos(ξ 2 τ 1 ) + a 4 + ξ 4 2 sin(ξ 2 τ 1 ) − a 2 ξ 2 2 sin(ξ 2 τ 1 ), α 2 = b 4 − ξ 2 (a 3 s + b 2 ξ 2 ) + a 4 + ξ 4 2 cos(ξ 2 τ 1 ) − a 2 ξ 2 2 cos(ξ 2 τ 1 ) + a 1 ξ 3 2 sin(ξ 2 τ 1 ). Dynamics without Delay 215 Lower "Exposed to Infected" Rate 216 Figure 2 presents the dynamics for σ = 0.3, when the transmission from the exposed to the infected 217 compartment is less. From figure 2, we can see that the impact of different transmission rates on the pandemic are notewor-219 thy. For example, for the higher values of ψ, the rates at which the recovered individuals again become 220 susceptible, is higher. This leads to higher infection rates (blue line). On the other hand, the lower value of ψ corresponds to better recovery rate, with less probability of 222 getting infected again, and thus leads to lower infection rates. Next, from figure 3, we can see that the best dynamics for the better control strategies of the pandemic 224 are β ≤ 0.09 and ψ ≤ 0.035. Higher "Exposed to Infected" Rate 226 Figure 3 presents the dynamics for σ = 0.6, when the transmission from the exposed to the infected 227 compartment, is higher. 228 We thus emphasize on the fact that it is not only important to control the interaction of infected 229 people with the susceptible, but it is really necessary to discover a drug, which will reduce the probability In this manuscript, important agent based (each individual was treated as an agent), strategy is adopted 241 and the machine learning tool of self organizing maps is used to explore the parameters of the resulting 242 mathematical model. In this research, we conclude that it is not only important to emphasize on the isolation of susceptible 244 individuals in a population, it is also necessary to control the interaction among the infected, recovered 245 and asymptomatic individuals. We have verified this hypothesis with the aid of numerical simulations. 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Geroscience Nonlinear dynamics and chaos in a fractional-order financial system A new study 280 on the mathematical modelling of human liver with caputo-fabrizio fractional derivative Modified function projective synchronization 283 of chaotic system Computational biology of propagation in excitable media models of 285 cardiac tissue Ai-modelling of molecular 287 identification and feminization of wolbachia infected aedes aegypti A fractional differential equation 290 model for the covid-19 transmission by using the caputo-fabrizio derivative A mathematical theoretical study 293 of a particular system of caputo-fabrizio fractional differential equations for the rubella disease model Advances in Difference Equations Analysis of the model of hiv-1 296 infection of cd4+ cd4ˆ{+} t-cell with a new approach of fractional derivative An introduction to compartmental modeling for the 299 budding infectious disease modeler The authors declare that there is no conflict of interest. The authors contributed equally to the manuscript. The authors declare that they have no known competng fnancial interests or personal relatonships ☒ that could have appeared to influence the work reported in this paper.The authors declare the following fnancial interests/personal relatonships which may be considered ☐ as potental competng interests:Credit Author Statement seirs:ZY did supervision, worked on the idea and conceived the manuscript. RA did the dynamical analysis, MAF did the literature review and simulations, AS supervised and did simulations. All the four authors equally contributed in the preparation of the manuscript.