Dynamics and density function of a stochastic COVID-19 epidemic model with Ornstein–Uhlenbeck process

Abstract

Two different approaches to incorporate environmental perturbations in stochastic systems are compared analytically and computationally. Then we present a stochastic model for COVID-19 that considers susceptible, exposed, infected, and recovered individuals, in which the contact rate between susceptible and infected individuals is governed by the Ornstein–Uhlenbeck process. We establish criteria for the existence of a stationary distribution of the system by constructing a suitable Lyapunov function. Next, we derive the analytical expression of the probability density function of the model near the quasi-equilibrium. Additionally, we establish sufficient conditions for the extinction of disease. Finally, we analyze the effect of the Ornstein–Uhlenbeck process on the dynamic behavior of the stochastic model in the numerical simulation section. Overall, our findings shed light on the underlying mechanisms of COVID-19 dynamics and the influence of environmental factors on the spread of the disease, which can inform policy decisions and public health interventions.

Appendix (Local asymptotic stability of endemic equilibrium in the deterministic model (1.1))

For the deterministic model (1.1), we rewrite it by letting $N(t)=S(t)+E(t)+I(t)+R(t)$ to obtain the following equivalent model:

$$\begin{aligned} \left\{ \begin{aligned} {\frac{\textrm{d}N}{\textrm{d}t}}&=\Pi -\mu N-\rho I,\\ {\frac{\textrm{d}E}{\textrm{d}t}}&{=}\beta (\sigma _1 I{+}\sigma _2 E)(N{-}E{-}I{-}R)-(\delta +\mu )E,\\ {\frac{\textrm{d}I}{\textrm{d}t}}&=\tau \delta E-(\epsilon +\rho +\mu )I,\\ {\frac{\textrm{d}R}{\textrm{d}t}}&=(1-\tau )\delta E+\epsilon I-(\mu +\eta )R. \end{aligned} \right. \nonumber \\ \end{aligned}$$

(A.1)

It is not difficult to obtain that there exists a unique endemic equilibrium for model (A.1) as follows:

$$\begin{aligned} \Theta ^*= & {} (N^*,E^*,I^*,R^*)\\= & {} \left( \frac{\Pi }{\mu } -\frac{\rho I^*}{\mu },\frac{\epsilon +\rho +\mu }{\tau \delta }I^*,I^*, \frac{ (1-\tau )( \rho +\mu ) +\epsilon }{(\mu +\eta )\tau }I^*\right) ,\\{} & {} \quad \quad I^*=\frac{\frac{\Pi }{\mu }\left( 1-\frac{1}{R_0}\right) }{1+\frac{\rho }{\mu }+\frac{\epsilon +\rho +\mu }{\tau \delta } +\frac{(1-\tau )(\rho +\mu )+\epsilon }{\tau (\mu +\eta )}}>0, \end{aligned}$$

such that the following equations hold

$$\begin{aligned} \left\{ \begin{aligned}&\Pi -\rho I^*=\mu N^*,\\&\beta (\sigma _1 I^*+\sigma _2 E^*)S^*=(\delta +\mu )E^*,\\&\tau \delta E^*=(\epsilon +\rho +\mu )I^*,\\&(1-\tau )\delta E^*+\epsilon I^*=(\mu +\eta )R^*. \end{aligned} \right. \end{aligned}$$

(A.2)

where $S^*=N^*-E^*-I^*-R^*$.

Lemma A.1

If $R_0 > 1$, then the endemic equilibrium $(N^*,E^*,I^*,R^*)$ of the system (A.1) is locally asymptotically stable..

Proof

Consider

$$\begin{aligned} (E-E^*)'= & {} \beta (\sigma _1 I^*+\sigma _2 E^*)\\{} & {} \quad [(N-N^*)-(E-E^*)-(I-I^*)\\{} & {} \quad -(R-R^*)]-\frac{\beta \sigma _1 S^* I^*}{E^*}\\{} & {} \quad (E-E^*)+\beta \sigma _1 S^* (I-I^*). \end{aligned}$$

In view of (A.2), we obtain

$$\begin{aligned} L\left( \frac{(E-E^*)^2}{2} \right)= & {} \beta (\sigma _1 I^*+\sigma _2 E^*)(E-E^*)\nonumber \\{} & {} [(N-N^*)-(E-E^*)\nonumber \\{} & {} -(I-I^*)-(R-R^*)]\nonumber \\{} & {} -\frac{\beta \sigma _1 S^* I^*}{E^*}((E-E^*)^2)\nonumber \\{} & {} +\beta \sigma _1 S^* (E-E^*)(I-I^*)\nonumber \\{} & {} =\beta (\sigma _1 I^*+\sigma _2 E^*)(E-E^*)\nonumber \\{} & {} [(N-N^*)-(E-E^*)\nonumber \\{} & {} -(I-I^*)-(R-R^*)]\nonumber \\{} & {} + \beta \sigma _1 S^*E^* I^* \left[ \left( \frac{E}{E^*}{-}1\right) \right. \nonumber \\{} & {} \left. \left( \frac{I}{I^*}{-}1\right) -\left( \frac{E}{E^*}-1\right) ^2\right] .\nonumber \\ \end{aligned}$$

(A.3)

Note that

$$\begin{aligned} \left( N-\frac{\rho }{\epsilon +\rho +\mu }I\right) '=\Pi -\mu N-\frac{\tau \delta \rho }{\epsilon +\rho +\mu }E. \end{aligned}$$

Thus we have

$$\begin{aligned}{} & {} L\left( \frac{1}{2}\left( (N-N^*)-\frac{\rho }{\epsilon +\rho +\mu }(I-I^*)\right) ^2 \right) \\{} & {} \quad =\left( -\mu (N-N^*)-\frac{\tau \delta \rho }{\epsilon +\rho +\mu }(E-E^*)\right) \\{} & {} \qquad \left( (N-N^*)-\frac{\rho }{\epsilon +\rho +\mu }(I-I^*)\right) \\{} & {} \quad =-\mu (N-N^*)^2-\frac{\tau \delta \rho }{\epsilon +\rho +\mu }(N-N^*)\\{} & {} \qquad (E-E^*)+ \frac{\rho \mu }{\epsilon +\rho +\mu }(N-N^*)(I-I^*) \\{} & {} \qquad +\frac{\tau \delta \rho ^2}{(\epsilon +\rho +\mu )^2} (E-E^*)(I-I^*). \end{aligned}$$

Since

$$\begin{aligned}{} & {} L\left( \frac{(N-N^*)^2}{2} \right) \\{} & {} \quad =-\mu (N-N^*)^2-\rho (N-N^*)(I-I^*). \end{aligned}$$

Denote

$$\begin{aligned} V_1= & {} \frac{1}{2}\left( (N-N^*)-\frac{\rho }{\epsilon +\rho +\mu }(I-I^*)\right) ^2\\{} & {} \quad +\frac{\mu }{2(\epsilon +\rho +\mu )} (N-N^*)^2. \end{aligned}$$

Then we have

$$\begin{aligned} LV_1= & {} -\mu \left( 1+\frac{\mu }{ \epsilon +\rho +\mu }\right) (N-N^*)^2\nonumber \\{} & {} \quad -\frac{\tau \delta \rho }{\epsilon +\rho +\mu }\nonumber \\{} & {} \quad (N-N^*)(E-E^*) +\frac{\tau \delta \rho ^2}{(\epsilon +\rho +\mu )^2} \nonumber \\{} & {} \quad (E-E^*)(I-I^*). \end{aligned}$$

(A.4)

Next define

$$\begin{aligned} V_2=\frac{(E-E^*)^2}{2}+\frac{\beta (\sigma _1 I^*+\sigma _2 E^*)(\epsilon +\rho +\mu )}{\tau \delta \rho }V_1. \end{aligned}$$

Combining (A.3) and (A.4), we have

$$\begin{aligned} L V_2= & {} -\beta (\sigma _1 I^*+\sigma _2 E^*)(E-E^*)^2 \nonumber \\{} & {} \quad - \beta (\sigma _1 I^*+\sigma _2 E^*) (E-E^*)\nonumber \\{} & {} \quad \left( \frac{\mu +\epsilon }{\epsilon +\rho +\mu }(I-I^*)+(R-R^*)\right) \nonumber \\{} & {} \quad -\frac{\mu \beta (\sigma _1 I^*+\sigma _2 E^*)(\epsilon +\rho +\mu )}{\tau \delta \rho } \nonumber \\{} & {} \quad \left( 1+\frac{\mu }{ \epsilon +\rho +\mu }\right) (N-N^*)^2\nonumber \\{} & {} \quad + \beta \sigma _1 S^*E^* I^* \left[ \left( \frac{E}{E^*}-1\right) \left( \frac{I}{I^*}-1\right) \right. \nonumber \\{} & {} \left. -\left( \frac{E}{E^*}-1\right) ^2\right] . \end{aligned}$$

(A.5)

Note that

$$\begin{aligned}{} & {} \left( \frac{\mu +\epsilon }{\epsilon +\rho +\mu } I+R\right) '=\left( \frac{\tau (\mu +\epsilon )}{\epsilon +\rho +\mu }+1-\tau \right) \\{} & {} \quad \delta E-\mu I-(\mu +\eta )R. \end{aligned}$$

Then we have

$$\begin{aligned}{} & {} L\left( \frac{1}{2}\left( \frac{\mu +\epsilon }{\epsilon +\rho +\mu } (I-I^*)+(R-R^*)\right) ^2 \right) \nonumber \\{} & {} =\left( \delta \left( \frac{\tau (\mu +\epsilon )}{\epsilon +\rho +\mu }+1-\tau \right) (E-E^*)\right. \nonumber \\{} & {} \quad \left. -\mu (I-I^*)-(\mu +\eta )(R-R^*)\right) \nonumber \\{} & {} \quad \left( \frac{\mu +\epsilon }{\epsilon +\rho +\mu } (I-I^*)+(R-R^*)\right) \nonumber \\{} & {} =\delta \left( \frac{\tau (\mu +\epsilon )}{\epsilon +\rho +\mu }+1-\tau \right) (E-E^*)\nonumber \\{} & {} \quad \left( \frac{\mu +\epsilon }{\epsilon +\rho +\mu } (I-I^*)+(R-R^*)\right) -\frac{\mu (\mu +\epsilon )}{\epsilon +\rho +\mu } \nonumber \\{} & {} \quad (I-I^*)^2-(\mu +\eta )(R-R^*)^2\nonumber \\{} & {} \quad -\left( \mu +\frac{(\mu +\epsilon )(\mu +\eta )}{\epsilon +\rho +\mu }\right) (I-I^*)(R-R^*).\nonumber \\ \end{aligned}$$

(A.6)

In addition, we get

$$\begin{aligned}{} & {} L\left( \frac{(I-I^*)^2}{2} \right) \nonumber \\{} & {} = (I-I^*)[\tau \delta (E-E^*)-(\epsilon +\rho +\mu )(I-I^*)]\nonumber \\{} & {} =\tau \delta E^* I^* \left[ \left( \frac{E}{E^*}-1\right) \left( \frac{I}{I^*}-1\right) -\left( \frac{I}{I^*}-1\right) ^2\right] .\nonumber \\ \end{aligned}$$

(A.7)

Define

$$\begin{aligned} V_3= & {} \frac{c_1}{2}\left( \frac{\mu +\epsilon }{\epsilon +\rho +\mu } (I-I^*)\right. \\{} & {} \quad \left. +(R-R^*)\right) ^2 +\frac{\beta \sigma _1 S^*}{2\tau \delta }(I-I^*)^2. \end{aligned}$$

with

$$\begin{aligned} c_1=\frac{\beta (\sigma _1 I^*+\sigma _2 E^*)}{ \delta \left( \frac{\tau (\mu +\epsilon )}{\epsilon +\rho +\mu }+1-\tau \right) }. \end{aligned}$$

Combining (A.5), (A.6) and (A.7), we have

$$\begin{aligned} L V_3= & {} -\beta (\sigma _1 I^*+\sigma _2 E^*)(E-E^*)^2\nonumber \\{} & {} \quad -\frac{\mu \beta (\sigma _1 I^*+\sigma _2 E^*)(\epsilon +\rho +\mu )}{\tau \delta \rho } \nonumber \\{} & {} \quad \left( 1+\frac{\mu }{ \epsilon +\rho +\mu }\right) (N-N^*)^2\nonumber \\{} & {} \quad {-}\frac{c_1\mu (\mu {+}\epsilon )}{\epsilon {+}\rho {+}\mu } (I{-}I^*)^2{-}c_1(\mu {+}\eta )(R-R^*)^2\nonumber \\{} & {} \quad {-}c_1\left( \mu {+}\frac{(\mu {+}\epsilon )(\mu {+}\eta )}{\epsilon {+}\rho {+}\mu }\right) (I{-}I^*)(R-R^*)\nonumber \\{} & {} \quad + \beta \sigma _1 S^*E^* I^* \left[ 2\left( \frac{E}{E^*}-1\right) \left( \frac{I}{I^*}-1\right) \right. \nonumber \\{} & {} \quad \left. -\left( \frac{E}{E^*}-1\right) ^2 -\left( \frac{I}{I^*}-1\right) ^2\right] \nonumber \\{} & {} \le -\beta (\sigma _1 I^*+\sigma _2 E^*)(E-E^*)^2\nonumber \\{} & {} \quad -\frac{\mu \beta (\sigma _1 I^*+\sigma _2 E^*)(\epsilon +\rho +\mu )}{\tau \delta \rho } \nonumber \\{} & {} \quad \left( 1+\frac{\mu }{ \epsilon +\rho +\mu }\right) (N-N^*)^2\nonumber \\{} & {} \quad {-}\frac{c_1\mu (\mu {+}\epsilon )}{\epsilon {+}\rho {+}\mu } (I{-}I^*)^2{-}c_1(\mu +\eta )(R-R^*)^2\nonumber \\{} & {} \quad {-}c_1\left( \mu {+}\frac{(\mu {+}\epsilon )(\mu {+}\eta )}{\epsilon {+}\rho {+}\mu }\right) (I-I^*)(R-R^*).\nonumber \\ \end{aligned}$$

(A.8)

Then from

$$\begin{aligned} \left( I-\frac{\tau }{1-\tau }R\right) '=\frac{\tau (\mu +\eta )}{1-\tau } R-\left( \rho +\mu +\frac{\epsilon }{1-\tau }\right) I. \end{aligned}$$

We have

$$\begin{aligned}{} & {} L\left( \frac{1}{2}\left( (I-I^*)-\frac{\tau }{1-\tau }(R-R^*)\right) ^2\right) \nonumber \\{} & {} =\left( \frac{\tau (\mu +\eta )}{1-\tau }(R-R^*)-\left( \rho +\mu +\frac{\epsilon }{1-\tau }\right) (I-I^*)\right) \nonumber \\{} & {} \quad \left( (I-I^*) -\frac{\tau }{1-\tau }(R-R^*)\right) \nonumber \\{} & {} =-\left( \rho +\mu +\frac{\epsilon }{1-\tau }\right) (I-I^*)^2- \frac{\tau ^2(\mu +\eta )}{(1-\tau )^2}(R-R^*)^2 \nonumber \\{} & {} \quad +\frac{ \tau }{1-\tau }\left( \rho +\eta +2\mu +\frac{\epsilon }{1-\tau }\right) (I-I^*)(R-R^*).\nonumber \\ \end{aligned}$$

(A.9)

Then define

$$\begin{aligned} V=V_2+\frac{c_2}{2}\left( (I-I^*)-\frac{\tau }{1-\tau }(R-R^*)\right) ^2, \end{aligned}$$

with

$$\begin{aligned} c_2=\frac{c_1\left( \mu +\frac{(\mu +\epsilon )(\mu +\eta )}{\epsilon +\rho +\mu }\right) (1-\tau ) }{ \tau \left( \rho +\eta +2\mu +\frac{\epsilon }{1-\tau }\right) }. \end{aligned}$$

Combining (A.8) and (A.9), we have

$$\begin{aligned}{} & {} LV \le -\beta (\sigma _1 I^*+\sigma _2 E^*)(E-E^*)^2 \nonumber \\{} & {} \quad -\frac{\mu \beta (\sigma _1 I^*+\sigma _2 E^*)(\epsilon +\rho +\mu )}{\tau \delta \rho } \nonumber \\{} & {} \quad \left( 1+\frac{\mu }{ \epsilon +\rho +\mu }\right) (N-N^*)^2\nonumber \\{} & {} \quad -\left( \frac{c_1\mu (\mu +\epsilon )}{\epsilon +\rho +\mu }+c_2\left( \rho +\mu +\frac{\epsilon }{1-\tau }\right) \right) (I-I^*)^2\nonumber \\{} & {} \quad -\left( c_1(\mu +\eta )+\frac{c_2\tau ^2(\mu +\eta )}{(1-\tau )^2}\right) (R-R^*)^2.\nonumber \\ \end{aligned}$$

(A.10)

In view of (A.10), one can obtain that there are positive constants $q_1$, $q_2$ and $q_3$ such that

$$\begin{aligned}{} & {} q_1[(E-E^*)^2+(I-I^*)^2+(R-R^*)^2\\{} & {} \quad +(N-N^*)^2]\le V(N,E,I,R)\le q_2[(E-E^*)^2\\{} & {} \quad +(I-I^*)^2+(R-R^*)^2+(N-N^*)^2], \end{aligned}$$

and

$$\begin{aligned} LV\le & {} -q_3[(E-E^*)^2+(I-I^*)^2\\{} & {} \quad +(R-R^*)^2+(N-N^*)^2]. \end{aligned}$$

Hence we have

$$\begin{aligned} \limsup _{t\rightarrow +\infty }\frac{\ln |V((S(t),E(t),I(t),R(t)) )|}{t}\le {-q_3,} \end{aligned}$$

which implies that the endemic equilibrium $(N^*,E^*,$$I^*,R^*)$ of system (A.1) is exponentially stable and hence the endemic equilibrium is locally asymptotically stable. This completes the proof. $\square$