In this section, we give some numerical results to show complex disease dynamic outcomes of SDE model (4) by using Milstein’s method [34], and the numerical scheme for model (4) under consideration is given by

$$\begin{aligned} I_{k+1} =&I_{k}+I_{k} \biggl[ \biggl( \beta_{1}-\frac{\beta_{2}I_{k}}{b+I _{k}} \biggr) ( 1-{I_{k}}/{N} ) -(\mu +\gamma) \biggr] \Delta t +\sigma I_{k} ( 1-{I_{k}}/{N} ) \eta_{k}\sqrt{\Delta t} \\ &{} +\frac{\sigma^{2}}{2} \bigl(I_{k}(1-{I_{k}}/{N}) \bigr)^{2}\bigl(\eta_{k} ^{2}-1\bigr)\Delta t, \end{aligned}$$

where \(\eta_{k}\) (\(k=1,2,\ldots,n\)) are independent Gaussian random variables \(\mathbf{N}(0, 1)\), Δ*t* is the time step size.

From (7), we note that \(\mathcal{R}_{0}^{s}=R_{0}-\frac{ \sigma^{2}}{2(\mu +\gamma)}< R_{0}\), if \(R_{0}<1\), then \(\mathcal{R} _{0}^{s}<1\). We can know that if \(R_{0}<1\), \(I(t)\) goes to extinction for the deterministic model (4) (see [21]); and from Theorem 2.1, \(I(t)\) almost surely tends to zero exponentially with probability one for the stochastic model (4). Therefore we only consider the case of \(R_{0}>1\).

Following [22, 25], the choice for the following parameters remains unaltered:

$$ \beta_{1}=0.15,\qquad \beta_{2}=0.1, \qquad \mu =0.05,\qquad \gamma =0.02,\qquad b=10,\qquad N=1000 $$

(19)

and the initial value is \(I_{0}=10\). In this case, \(R_{0}=2.143>1\), and the deterministic model (6) has an unstable disease-free equilibrium \(E_{0}=0\) and a unique globally stable endemic equilibrium \(E^{*}=34.4494\).

Next we will focus on the role of noise intensity *σ* on the resulting dynamics for SDE model (4).

### 4.1 Stochastic disease-free dynamics

First of all, we adopt \(\sigma =0.405\), in this case, \(\mathcal{R} _{0}^{s}=0.97125<1\). From Theorem 2.1(i), we know that the disease \(I(t)\) will go extinct with probability one.

In Fig. 1, we show the stochastic disease dynamics of the evolution of the single path of \(I(t)\) obtained from two different numerical simulations run with the same parameters. One can see that \(I(t)\) is strongly oscillatory at the beginning and finally dies out with probability one. Easy to know that, in these cases above, \(I(t)\) tends to the disease-free equilibrium \(E_{0}=0\) of the deterministic model (6) almost surely at last. It should be noted that, for model (6), \(R_{0}=2.143>1\), \(E_{0}=0\) is unstable. Hence we can conclude that, in this case, environmental noise can make unstable \(E_{0}\) to a stable one.

Furthermore, we repeat 10,000 simulations with the same parameters as in Fig. 1, we can calculate the average extinction time for \(I(t)\), and the results are shown in Fig. 2. For example, when \(\sigma =0.405\), the average extinction time for \(I(t)\); when \(\sigma =0.45\), it is 278.3943. We can conclude that the average extinction time decreases with the increase of noise intensity *σ*.

### 4.2 Stochastic endemic dynamics

In this subsection, we will focus on the stochastic endemic dynamics of (4) in the case of \(\mathcal{R}_{0}^{s}>1\). For this reason, we choose \(\sigma =0.01, 0.05, 0.10, 0.15\) implies that \(\mathcal{R}_{0} ^{s}=2.142\), 2.125, 2.071, 1.982, respectively. From Theorem 2.1(ii), we can conclude that the disease will persist almost surely. In Fig. 3, we show the single path of \(I(t)\) for model (4) and its corresponding deterministic model (6) with \(\sigma =0.01, 0.05, 0.10, 0.15, 0.25, 0.35\), respectively, and we can see that the solutions \(I(t)\) of SDE model (4) fluctuate around the endemic equilibrium \(E^{*}=34.4494\) of the deterministic model (6), respectively. In addition, we can find that the bigger noise intensity *σ*, the stronger oscillatory \(I(t)\).

For the sake of learning the effects of the intensity of noise *σ* on the stochastic disease dynamics of SDE model (4), we have repeated the simulation 10,000 times, keeping the same parameters as in Fig. 3 and never observing any extinction scenario up to \(t=500\). These results are respectively confirmed by the histograms and the probability density functions in Fig. 4, showing the stationary distributions of \(I(t)\) at \(t=500\) for model (4). And the numerical method for them can be found in [9].

From Fig. 4, one can see that the solution to SDE model (4) in the persistent case also suggests for lower *σ* (e.g., \(\sigma =0.01\) and 0.05), the amplitude of fluctuation is slightly and the oscillations to be more symmetrically distributed (cf. Figs. 4(a) and 4(b)), and the fluctuations are reflected at the stationary distributions. While for higher *σ* (e.g., \(\sigma =0.1\) and 0.15), the amplitude of fluctuation is remarkable and the distributions of the solutions are skewed (cf. Figs. 4(c) and 4(d)) and the fluctuations are also reflected at the stationary distributions.

Furthermore, in the case of \(\sigma =0.01\) and 0.05, the distribution of \(I(t)\) appears closer to a normal distribution (see Figs. 4(a) and 4(b)). Simple computations show that, when \(\sigma =0.01\), the distribution of \(I(t)\) closely obeys the normal distribution \(\mathbf{N}(34.4577, 1.6995^{2})\), and the values less than one standard deviation away from the mean account \((32.7582, 36.1572)\) for 67.94% of the set; while two standard deviations from the mean account \((31.1530, 37.7624)\) for 94.92%; and three standard deviations account \((30.0517, 38.8637)\) for 98.99% (see Fig. 5(a)). Simple calculations show that the skewness (i.e., the measure of the asymmetry of the probability distribution of a real-valued random variable about its mean) in this case is 0.1662598, which is a positive skew. And in the case of \(\sigma =0.05\), the distribution of \(I(t)\) closely obeys \(\mathbf{N}(33.7969, 8.8664^{2})\), and the values less than three standard deviations away from the mean account \((8.0924, 59.5015)\) for 98.75% (see Fig. 5(b)), the positive skewness is 1.045055.

In order to understand the effect of the intensity of noise *σ* on the skewness of the distribution of \(I(t)\), we show the four probability density functions with \(\sigma =0.01, 0.05, 0.1\), and 0.15 in Fig. 6. Easy to see that, as *σ* increases, the means of \(I(t)\) become smaller and smaller, and the positive skewness of the distributions of \(I(t)\) becomes bigger and bigger.