Mathematical study for the phase-based transmissibility of a novel COVID-19 Coronavirus

In this paper, a mathematical dynamical system modeling a SEIRW model of infectious disease transmission for a transmissibility of a novel COVID-19 Coronavirus is studied. A qualitative analysis such as the local and global stability of equilibrium points is carried out. It is proved that if R ≤ 1, then the disease-free equilibrium is globally asymptotically stable and if R > 1, then the disease-persistence equilibrium is globally asymptotically stable


Introduction
On December 31, 2019, the World Health Organization (WHO) alerted several cases of pneumonia of unknown origin in the city of Wuhan (Hubei Province of China).But this virus was unlike any known virus.On January 7, 2020, the Chinese authorities confirmed that it was indeed a new virus from the coronavirus family.At first it was temporarily baptized "2019-nCoV" then definitively COVID-19 or SARS-CoV-2.Since then, human cases have been reported from almost all countries and COVID-19 has been classified as a pandemic by the World Health Organization (WHO).As the epidemic of pneumonia due to the new coronavirus 2019-nCoV spreads around the world, Chinese researchers had in recent months, on several occasions, alerted the international scientific community to the risk of seeing soon emerge a human infection by a coronavirus from bats in China.The seafood market in Wuhan, central China's Hubei province, may be the source of the new coronavirus (2019-nCoV) epidemic, according to Chinese researchers [6,17].This epidemic obliges us to propose models allowing to estimate the transmissibility and dynamic of the transmission of the virus.There were several researches focusing on mathematical modelling [6,7,8].However, the transmission route form the seafood market to people were not considered in the published models.Since public concerns were focusing on the transmission from Huanan Seafood Wholesale Market (reservoir) to people, we proposed a simplified model as a Reservoir-People (RP) transmission network model.This paper is organized as follows.In section 2, a mathematical dynamical system involving "SEIRW" epidemic model is considered.The basic reproduction number (R) was calculated using the next generation matrix method to assess the transmissibility of the SARS-CoV-2.A profound qualitative analysis is given.The analysis of the local and global stability of equilibrium points is carried out in sections 3 and 4, respectively.It is proved that if the reproduction number R > 1, then the disease-persistence (endemic) equilibrium is globally asymptotically stable.However, if R ≤ 1, then the disease-free equilibrium is globally asymptotically stable.Finally, in section 5, some numerical tests are done in order to validate the obtained results.
2 Mathematical model and some properties W denotes the SARS-CoV-2 in reservoir (the seafood market).The individuals were divided into four compartments: susceptible individuals (S), exposed individuals (E), infected individuals (I) including symptomatic and asymptomatic infected individuals, and removed individuals (R) including recovered and death individuals.The rate of individuals traveling out from the city was defined as m.n/m denotes the total number of individuals traveling into the city.The incubation period and latent period of human infection are assumed to be equal and defined as 1/ω.The infectious period of compartment I was defined as 1/γ.The compartment S will be infected through sufficient contact with compartment I and compartment W , and the transmission rates were defined as b 1 and b 2 , respectively.

Susceptible
Note that symptomatic and asymptomatic infected people are merged into one compartment I as infected people.The closed non-negative cone in R 5 , is positively invariant by the system (1).More precisely, Proposition 1.
Proof.The solution is positive due to the fact that : Since S = 0 then Ṡ = n > 0, if E = 0 then Ė = b 1 SI + b 2 SW > 0, once I = 0 then İ = ωE > 0, if R = 0 then Ṙ = γI > 0, and if W = 0 then Ẇ = εI > 0. The boundedness of solutions of system (1) can be proved by adding all equations of system (1), and then one obtains, for T = S + E + I + R + W − n m , the following equation for the total individuals : Then the boundedness of the solution of system (1) holds since all compartments of T are positive.
One can easily deduce from equality (2) that the set Ω 1 is positively invariant attractor for system (1).
Define Q = ( n m , 0, 0, 0, 0) as the disease free equilibrium point and Q * = (S * , E * , I * , R * ,W * ) as the endemic equilibrium point of system (1) where S * , E * , I * , R * > 0 and W * > 0 satisfying Diekmann, et al. [4] proposed a method to calculate the basic reproduction number R for complex compartmental models by using the next generation matrix method.This method elaborated later by van den Driessche and Watmough [5].
T be the number of individuals in each compartment, where the first M < N compartments contain infected individuals.Consider these equations written in the form is the rate of appearance of new infections in compartment i, and V i (x) is the rate of other transitions between compartment i and other infected compartments.It is assumed that F i and V i ∈ C 2 , and non-negative and V is a non-singular M-matrix [4,5].Then FV −1 is the next generation matrix and where ρ is the spectral radius.In our case The determinent of V is given by det(V ) = ε(ω + m)(γ + m) > 0 and therefore Then the basic reproduction number for model ( 1) is given by .

Local Analysis
Theorem 1.
• If R < 1, then the disease-free equilibrium Q is locally asymptotically stable.
• If R > 1, then the disease-free equilibrium Q is unstable.
Proof.The matrix J is evaluated at Q = ( n m , 0, 0, 0, 0) is given by : The characteristic equation is where Therefore Then if R < 1, then a 1 > 0, a 3 > 0 and a 1 a 2 − a 3 > 0, and if R > 1, then a 1 > 0 and a 3 < 0 and thus using Routh-Hurwitz criterion, all eigenvalues have negative real parts if R < 1.This completes the proof.
Proof.The matrix J * is evaluated at E * = (S * , E * , I * , R * ,W * ) is given by : The characteristic equation is given by P * (λ ) = −(λ + m)P 4 (λ ) with where ), Therefore, after calculations we get 3 > 0 and by the Routh Hurwitz criteria we end the proof.
Proof.It is proved in Proposition 1 that Ω 1 is a positive invariant attractor set of all solution of system (1).Now, since Ṡ(t) < 0 for then lim infW (t) ≤ I(t).This completes the proof.
Theorem 3. If R ≤ 1, then the disease-free equilibrium Q is globally asymptotically stable.
The global stability of the disease-persistence (endemic) equilibrium Q * is given in the following theorem.
Proof.Consider the following Lyapunov function: The equilibrium Q * is the only internal stationary point of system (1).The function V 2 (t) admits its minimum value and V 2 (t) → +∞ at the boundary of the positive quadrant.Consequently, Q * is the global minimum point, and the function is bounded from below.The derivative, of V 2 (t), along solutions of system (1) is given by Using the fact that We recall also the following inequality : Since arithmetical mean of nonnegative real numbers is greater than the geometrical one, we have the following inequalities Therefore V2 ≤ 0 .Thank's to the stability Lyapunov theorem, one deduces that Q * = (S * , E * , I * , R * ,W * ) is stable.

Numerical Simulations
We validate numerical simulations for system (1).Four cases were considered; two of them (Figure 2) confirming the global stability of the disease-free equilibrium Q when R ≤ 1.The other two tests (Figure 3) confirm the global stability of the disease-persistence equilibrium Q * when R > 1.We remark that the solution of system (1) converge asymptotically to Q.Only Susceptible compartment persist, the other compartments vanish.In this case, the solution of system (1) converge asymptotically to Q * and all compartments persist.

Conclusion
A mathematical 5D dynamical system modelling an SEIRW model of transmissibility of the SARS-CoV-2 is studied.A profound qualitative analysis is given.The analysis of the local and global stability of equilibrium points is carried out.It is proved that if R > 1 then the disease-persistence equilibrium is globally asymptotically stable.However, if R ≤ 1, then the disease-free equilibrium is globally asymptotically stable.

Figures Figure 1 Flowchart
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