Comparison Between Deterministic and Stochastic Model for Interaction (COVID-19) With Host Cells in Humans

: In this paper, the deterministic and the stochastic models are proposed to study the interaction of the Coronavirus (COVID-19) with host cells inside the human body. In the deterministic model, the value of the basic reproduction number 𝑅 0 determines the persistence or extinction of the COVID-19. If 𝑅 0 < 1 , one infected cell will transmit the virus to less than one cell, as a result, the person carrying the Coronavirus will get rid of the disease .If 𝑅 0 > 1, the infected cell will be able to infect all cells that contain ACE receptors. The stochastic model proves that if 𝛼 1 & 𝛼 2 are sufficiently large then 𝛼 1 & 𝛼 2 maybe give us ultimate disease extinction although 𝑅 0 > 1 , and this facts also proved by computer simulation.


Introduction:
Coronavirus (COVID-19) is deadly and infectious which attacks and weakens cells containing the angiotensin-converting enzyme 2 (ACE-2) receptors, An example of these cells that the novel coronavirus disease attacks are the intestinal cells, myocardial cells, renal tubes, male reproductive cells, gallbladder, lungs, bronchi, and nasal mucosa. For this reason, the respiratory system is the first target organ, the heart scores second as an organ targeted by the Coronavirus. Until 13/2/2021 the total number of infected with COVID-19 around the world about (109156020) and almost (2406571) they have died since the first case by COVID-19 which appeared at the end of 2019 in Wuhan, China 1,2 . Despite massive development in technology and medical equipment, Scientists still can't find a complete cure for the COVID-19 virus. To control the epidemic it is important to understand the dynamical behavior of COVID-19 and how its interaction with host cells in humans 3−5 .
Mathematical models were developed to understand the dynamics of viral infections. The large amount of scientific research done on the model of the interaction of Coronavirus with host cells in humans has been largely restricted to ordinary differential equations (ODEs) 5−8 .In our article will present a stochastic differential equations model for interaction (COVID-19) with host cells in humans. Several reasons motivated us to use stochastic differential equations models instead of deterministic equations. Real-life is random, not deterministic, especially when modeling the phenomenon of the spread of the Coronavirus for example internal COVID-19 dynamics. This is because target cells that contain ACE-2 receptors that interacting with coronavirus particles In the same environmental conditions but give different outputs. This article presents the influence of introducing stochasticity on the deterministic mathematical model. The new method in mathematical modeling gave us more accurate results than deterministic differential equations models, because employment the stochastic differential equations model many times can build up a distribution of the predicted results, such as total numbers of infected cells with the coronavirus at time t, whereas the deterministic differential equations model will introduce to us just one expected value 9−12 . The paper is organized as follows. Firstly present the deterministic model and Stochastic model for interaction coronavirus with cells that contain ACE-2 receptors. Secondly, present the conditions required for persistence or extinction of COVID-19 and how the injured recovered with the virus. Then the main results present. finally, the conclusions and references are listed.

The Mathematical Model for Interaction (COVID-19) With Host Cells in Humans 1.Deterministic Model
Various mathematical models have been used to comprehend activity and movement coronavirus inside the human body. The simpler version includes three types: The dynamics of healthy target cells T(t), The infected cells dynamics I(t), and the dynamics of coronavirus particles C(t). They can be described by using the following set of (ODEs).
If there is a vaccine that prevents the coronavirus particles from attaching to healthy host cells. In addition to a vaccine that prevents the compilation of the virus particles correctly and this leads to new COVID-19 that are weakly, and unable to reproduce, so the set of the deterministic model (1,2 and 3) has the following form. To estimate the infection, let use the basic reproductive number denoted by ( 0 ) which is the total expected number of secondary infections produced by the infected cell. If 0 < 1 this means any infected cell will transmission of infection to less than one cell and this give as the virus is cleared out .otherwise if 0 > 1 so each infected cell produces on averages more than one new infected cells and in this case, the infection grows and the disease can invade the all cells, for model 4 -6 the basic reproduction number will be as follows:

Stochastic model
The clearance rate of COVID-19 particles may be affected via several important factors for example binding and entry into cells that contain receptors (ACE-2). since death rates from COVID-19 particles and target cells depend on many complex natural and biological phenomena, This made scientists believe that there is randomness in this death rate. This gives us important motivation to believe that's we can insertion the stochastic in the deaths rate of infected cells and COVID-19 particles. So the new system 4-6 in the stochastic model will be as follows 13−16 .
The parameters in the system 1 -9 are expressed in Table 1.  coronavirus, and heal the infected in the virus. i.e. when lim t→∞ I(t) = 0, lim t→∞ C(t) = 0, Theorem 1. When these conditions are satisfied . This give lim t→∞ I(t) = 0, & lim t→∞ C(t) = 0, in another meaning I(t) and C(t) will goes to their fixed point exponentially with Probability 1. Because the above matrix is not positivedeterminant with main eigenvalue is negative so: ≤ ( 2 ( ) + 2 ( )) = −| |( 2 ( ) + 2 ( )) . Therefore This gives us lim t→∞ I(t) = 0, and lim t→∞ C(t) = 0. ∎ So the conditions of Theorem1 will always be achieved when 1 2 & 2 2 are sufficiently large, then 1 2 & 2 2 give us ultimate disease extinction although 0 > 1 . We now concentrate on T(t) ,and will prove that how T(t) is expansively stable in distribution about the expected value ⁄ . To make this possible will present the stochastic process ( ) which may be defined by its primary condition (1)

Main results:
In this part of the article, we'll prove the analytical results obtained from theories (1 and 2)by using computer simulations. Note that by the theoretically results I(t) &C(t) are exponentially stable and lim t→∞ I(t) = 0, & lim t→∞ C(t) = 0, Theorem 1 are met, although R 0 > 1. Also can find the value of T(t) by φ(t) where φ(t) is the average return process . Note the computer simulation program was written using Matlab by the Euler method and the outputs were verified through run them extensively and repeatedly. When taking the Eq.9 dC(t) = ((1 − γ)SaI(t) − βC(t) − (1 − v)σT(t)C(t))dt + α 2 C(t)dW 2 (t), Also by substituting the parameter values in Eq.9 and solve the resulting equation by using Ito's formula, will find the stochastic solution of Eq.9 as: C(t) = 10 6 e −1.42t and deterministc solution C(t) = 10 6 e −0.7t , so the virus particles C(t) goes to zero exponentially in case t → ∞. The computer simulation in Fig. 2, by using the Euler Maruyama method (EM), support these results clearly. when taking the Eq.7 dT(t) = (G − nT(t) − (1 − )σT(t)C(t))dt + α 1 T(t)dW 1 (t), So by substitution, the parameter values in Eq. 7 and by using Ito's formula 14 ,will find the stochastic solution of Eq.7as follows,T(t) = 10 6 e 999998t and deterministic solution will be as ∶ T(t) = 10 6 e 999998.5t , So the healthy cells T(t) will not go to zero exponentially in case t → ∞. This means the person with the virus has recovered. The simulation programs in Fig. 3, support these results. : dI(t) = ((1 − )σT(t)C(t) − aI(t))dt + α 1 I(t)dW 1 (t), So by substitution the parameter values in Eq.8 and by using Ito's formula 14 , will find the stochastic solution as : I(t) = 10 4 e −0.4t and the deterministic solution as I(t) = 10 4 e 0.1t So the infected cells (I(t) ) go to zero exponentially in case t → ∞ in stochastic model but not in deterministic model.The simulation programs in Fig. 4, by using MATLAB, support these results. − (1 − )σT(t)C(t))dt + 2 C(t)dW 2 (t), Also by substitution, the parameter values in Eq.9 and by using Ito's formula 14 , will find the stochastic solution of Eq.9 Is C(t) = 10 4 e −0.03006t and deterministic solution equal to C(t) = 10 4 e 0.68994t , so the virus particles C(t) goes to zero exponentially in case t → ∞ in stochastic model but not in deterministic model. The computer simulation programs Fig.5, support these results. when taking the Eq.7 dT(t) = (G − nT(t) − (1 − )σT(t)C(t))dt + α 1 T(t)dW 1

(t)
So by substitution, the parameter values in Eq.7 and by using Ito's formula, will find the stochastic solution of Eq.7 as T(t) = 10 4 e 999999.3t , and the deterministic solution is T(t) = 10 4 e 999999.8t .

Conclusions:
This paper introduced environmental stochasticity into the deterministic model also explored the properties of COVID-19 and how the disease spreads inside the human body, through its interaction with cells that contain receptors for the ACE-2. As well in this paper, we construct the basic reproduction number R 0 and conditions required for extinction or persistence COVID-19. In general in the deterministic model if R 0 < 1, the disease will die out the injured person will be cured .If R 0 > 1 the disease will persist. In stochastic model prove that anyone infected with the Coronavirus can recover if the stochastic variance α 1 2 and α 2 2 are big sufficient this gives us lim t→∞ I(t) = 0, lim t→∞ C(t) = 0 and lim t→∞ T(t) = G n although R 0 > 1.