key: cord-0836615-vp4q04mr authors: Abdel-Gawad, Hamdy I.; Abdel-Gawad, Ahmed H. title: Discrete and continuum models of COVID-19 virus, formal solutions, stability and comparison with real data date: 2021-05-14 journal: Math Comput Simul DOI: 10.1016/j.matcom.2021.05.016 sha: b84b1cd6c172c221de263b5d5c0e858c8a8ca9ab doc_id: 836615 cord_uid: vp4q04mr Very recently, various mathematical models, for the dynamics of COVID-19 with main contribution of suspected-exposed-infected-recovered people have been proposed. Some models that account for the deceased, quarantined or social distancing functions were also presented. However, in any local space the real data reveals that the effects of lock-down and traveling are significant in decreasing and increasing the impact of this virus respectively. Here, discrete and continuum models for the dynamics of this virus are suggested. The continuum dynamical model is studied in detail. The present model deals with exposed, infected, recovered and deceased individuals (EIRD), which accounts for the health isolation and travelers (HIT) effects. Up to now no exact solutions of the parametric-dependent, nonlinear dynamical system NLDS were found. In this work, our objective is to find the exact solutions of a NLDS. To this issue, a novel approach is presented where a NLDS is recast to a linear dynamical system LDS. This is done by implementing the unified method (UM), with auxiliary equations, which are taken coupled linear ODE’s (LDS). Numerical results of the exact solutions are evaluated, which can be applied to data in a local space (or anywhere) when the initial data for the IRD are known. Here, as an example, initial conditions for the components in the model equation of COVID-19, are taken from the real data in Egypt. The results of susceptible, infected, recovered and deceased people are computed. The comparison between the computed results and the real data shows an agreement up to a relative error [Formula: see text]. On the other hand it is remarked that locking-down plays a dominant role in decreasing the number of infected people. The equilibrium states are determined and it is found that they are stable. This reveals a relevant result that the COVID 19 can endemic in the case of a disturbance in the number of the exposed people. This result can be interpreted as in what follows. When a disturbance occurs in the exposed number increases, so the number of infected people increases. Or COVID-19 is endemic. This result is, globally, valid. Furthermore, initial states control is analyzed, where region of initial conditions for infected and exposed is determined. We developed a software tool to interact with the model and facilitate applying various data of different local spaces. Corona viruses belong to the Corona viridae family in the Nidovirales order. Corona represents crown-like spikes on the outer surface of the virus [1] . COVID- 19 virus received the attention of many research works in science and medicine [1, 2] . It is evident that this virus is transmitted between people through respiratory droplets and contact routes [3] [4] [5] [6] . It occurs when a person is in in close contact (within 1 m) with someone who has respiratory symptoms and is therefore at risk of having his/her mouth and nose exposed to potentially infective respiratory droplets. Estimation of the asymptomatic ratio, the percentage of carriers with no symptoms, will improve understanding of the virus transmission [7] . The application of a recent epidemiological model, suspected-exposed-infectedrecovered, namely SEIR with Social Distancing (SEIR-SD), is extended here through the denition of a social distancing function varying over time to assess the situation related to the spreading of the COVID-19 in Italy [8] . In this work, the most suitable values of its parameters are found. Due to Egypt's high population density, human to human social contact rate is very high. So, control of the pandemic COVID-19 is very urgent in the early stage and is a challenging problem. Several mathematical models were proposed to study the COVID-19 dynamics and also to identify the inuential parameters that leads to reduce the outbreak size [10] . A model was proposed in [11] that takes into account the combination of a global network mobility model with a local epidemiology model to simulate and predict the outbreak dynamics and outbreak control of COVID-19 across Europe. A stochastic transmission model was developed [12] by extending the Susceptible-Infected-Removed (SIR) that assesses the eectiveness of response strategies of avoiding crowded areas and predicts the spread of COVID-19 infections in Japan. The outbreak of COVID-19 disease in mainland China was characterized by a distinctive sub-exponential increase of conrmed cases during the early phase of the epidemic, contrasting with an initial exponential growth expected for an unconstrained outbreak. This eect 2 The model and formal exact solutions 2.1 Discrete and continuum models We propose a EIRD-HIT model by the dierence equations: By using (1), the continuum model is constructed as in what follows. We and similar equations for I n , R n ,and D n hold. Thus, we have where (a) α is the rate of exposed people to infection. (b) σ is the rate of isolated people. (c) β and γ are the rates of the interaction of recovered and deceased with the infected people respectively. (d) k 1 and k 2 are the pumping rates of exposed and infected travelers respectively. (e) N is the population number. Indeed, a model which is more realistic than what is presented in (3), is to take into consideration the fact that diseased or recovered people were antecedent infected. Thus a delay time has to be introduced in the last two equations in (3) in I(t). Thus the model in (3) becomes where τ 1 and τ 2 are the measures of time lag between infection and recovered/deceased. It is worthy to men that a nonlinear dynamical system is not in general integrable. It may be conditionally integrable. This holds in the present case and the condition for integrability will be depicted later on. The scheme of the model is shown in gure 1. The unied method, used here, asserts that the exact solutions of NLDS are expressed in auxiliary functions which satisfy appropriate auxiliary equations. Here, these later equations are taken linear coupled ODEs, which may be considered as LDS. Now, in (4) when I(t) is known, then we get Thus, we are left withe rst and second equations in (4), which will be solved, hereafter..Here we nd the exact solutions by using the UM [17] [18] [19] , where the solutions are expressed in the rational form by where g j (t), j = 1, 2, 3 are the auxiliary functions which satisfy the auxiliary equations, By substituting from (6) and (7) into the rst and second equations in (3), and by setting the coecients of g k j , j = 1, 2, 3, k = 0, 1, 2, .., equal to zero, we get a set of algebraic equations which solve to together with the condition of integrability which is σ = β + γ, which leads to σ > β and σ > γ. Thus this condition has an impact on the dynamics of COVID-19, in the sense that the rate of isolated people should be greater than the rate of recovery and of diseased people. The solutions of the auxiliary equations (7) are 5 J o u r n a l P r e -p r o o f By substituting From (8) and (9) into (6), we get together with 6 J o u r n a l P r e -p r o o f We mention that when substituting from (11), for a 0 , a 3 and γ 3 , into (10), the results obtained are very lengthy to be produced here. To adapt the model for any country data: 1. Given the EIRD real data for a specic local space starting from the initial values and covering a certain period of time. 2. Experiment with these real data to nd the order of magnitude of the best-t for the β, γ, α, k 1 , and k 2 in equation (3). 3. Use the values of these parameters to predict the EIRD for a longer period of time. When the above steps were carried on the real data of Egypt, starting from March 1st to August 30th, 2020, we found that σ = O(β), γ αE(0)/N, N = 10 8 , where N is the population number of Egypt. The exact solutions in (10) and (11) In (i) k 2 = 5 and for dierent values of α. In (ii) α = 1 and for dierent values of k 2 . From Fig. 2 , we nd that the infected people at the end of September will be about 103683 when α = 1. After Fig. 2 (ii), we remark that the eects of the parameters k 1 and k 2 are insensitive. This may be argued to the small numerical (moderate) values taken, of these parameters. We think that by taking large numerical values, remarkable eects may hold. In gure 3, the exposed, deceased and recovered are shown. It is worth noticing that the parameters α, β, γ, ...,in (4) , are taken appropriately to match with the real data in local spaces. We implemented the nal equations (10 and 11) as a MATLAB software application to allow researchers to utilize this model for various real data of dierent countries. The tool can be reached through [20] . Figure 4 shows a screenshot for the tool. A comparison between the method used here and the known methods in the literature is done in the following: 1-In this paper, the unied method presented in [15] was used. After this nomenclature, this method unies all known methods such as, the Exp-function expansion [19, 20] , the tanh modied, and the extended versions; the F-expansion and the G'/G expansion method. 2-On the other hand, the extended unied method, proposed in [16] , may be sucient to replace the analysis of using the symmetries by inspecting the symmetries endowed by using Lie group in NLPDEs. 3-Using the generalized unied method, presented in [17] , is more powerful tool than using the Hirota method. The equilibrium states ES are determined by setting the RHS in (3) equal to zero. There exist two ESs, (i) When k 2 = 0, we have E e = k 1 /σ e = 0 and I e = 0. (ii) When k 2 = 0, we have E e = k1N αIe+N σ and I e = We consider the ES in (ii) and assume that E(t) = E e + ε 1 e λt and I(t) = I e + ε 2 e λt , by substituting in the rst and second equations in (3), we have M ε 1 ε 2 = 0, M = a 11 a 12 a 21 a 22 , a 21 = −I e α (I e α + N (β + γ)), a 22 = I e N α (β + γ + λ) + k 2 N 2 (k 1 α (β + γ) − N (β + γ) 2 (β + γ + λ)). By setting det(M ) = 0, we get the characteristic equation, which is solved in λ. The results are lengthy to be produced here. They are shown in gure 5 where we nd that the ES in the case (ii) is asymptotically stable. By the same way, it is found that the ES in the case (i) is also asymptotically stable. This result leads to that the systems returns to the equilibrium state under disturbance. Or the COVID-19 maybe endemic. The criteria of initial state control asserts that, we can control the number of exposed and infected people such that, initially, the rate of infected people is negative, dI(t) dt | t=0 < 0. By using equation (3) we get The equation (15) is shown in gure (6) . It is worthy to notice that we can control the parameters α, β, γ, and k i , i = 1, 2 to t the exact solutions with real data in any locality (country). In fact, in the previous works, only the order of magnitude estimate of each component in a COVID-19 model was done. A comparison between the method used here and the known methods in the literature is done in the following: 1-In this paper, the unied method presented in [15] was used. After this nomenclature, this method unies all known methods such as, the Expfunction expansion [19, 20] , the tanh, modied, and the extended versions, the F-expansion, the G'/G expansion method. 2-On the other hand, the extended unied method, proposed in [16] , may be sucient to replace the analysis of using the symmetries by inspecting the symmetries endowed by using Lie group in NLPDEs. 3-Using the generalized unied method, presented in [17] , is more powerful tool than using the Hirota method. In this work, discrete and continuum models account for the exposed, infected, shows that a disturbance in the number of exposed people, might lead to the virus becoming endemic. We mention that is model can be applied to arbitrary initial data. In view of the mathematical modeling, the number of infected people does not decrease to zero in the absence of antivirus eect. None. 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