In this section, the theorem below can be interpreted as follows; at least, the stochastic perturbations do not destabilize the system. Let (R0s,R0m,R0x) the basic reproduction number of drug-sensitive strain, MDR strain, and XDR strain. Let us define R∗ as follows:
$$\begin{array}{*{20}l} R_{\ast}&=\text{max}(R_{0s},R_{0m},R_{0x}), \ \ \ \text{where} \\ R_{0s}&=\frac{\beta_{s}(\varepsilon_{s}+(1-\lambda_{s})(d+t_{1s}))}{(\varepsilon_{s}+d+t_{1s})(t_{2s}+\delta_{s}+d)+\gamma_{s}(t_{1s}+d)},\notag\\ R_{0m}&=\frac{\beta_{m}(\varepsilon_{m}+(1-\lambda_{m})d)}{(\varepsilon_{m}+d)(t_{2m}+\delta_{m}+d)+d\gamma_{m}},\notag\\ R_{0x}&=\frac{\beta_{x}(\varepsilon_{x}+(1-\lambda_{x})d)}{(\varepsilon_{x}+d)(t_{2x}+\delta_{x}+d)+d\gamma_{x}}.\notag \end{array} $$
(10)
Theorem 1
If R∗<1, then disease-free equilibrium is a.s exponentially stable.
Proof
Assume that [33]:
$$\begin{array}{*{20}l} 0\leq\alpha_{ss}&\leq(1-\lambda_{s}), \end{array} $$
(11)
$$\begin{array}{*{20}l} 0\leq\alpha_{mm}&\leq(1-\lambda_{m}), \end{array} $$
(12)
$$\begin{array}{*{20}l} 0\leq\alpha_{xx}&\leq(1-\lambda_{x}). \end{array} $$
(13)
Using the fact that \(\frac {S(t)+\sigma _{s} R(t)}{N}<1\) and that \(I_{s}(t)\leq I_{s}^{\infty }\) at any t, it follows
$$\begin{array}{*{20}l} L_{s}^{\infty}\leq \frac{\lambda_{s}\beta_{s}+\gamma_{s}}{d+t_{1s}+\epsilon_{s}}I^{\infty}_{s}. \end{array} $$
(14)
Using assumption (11), and for simplicity, let us define a1:=(d+δs+t2s+γs), a2:=(d+εs+t1s)). The fact that\(\frac {S(t)+\sigma _{s} R(t)+L_{s}(t)}{N}<1\) and that \(L_{s}(t)\leq L_{s}^{\infty }\), together with Eq. (14), implies that
$$\begin{array}{*{20}l} 0&\leq [R_{0s}-1]\frac{1}{a_{2}(a_{2}a_{1}-\epsilon_{s}\gamma_{s})}I_{s}^{\infty}, \end{array} $$
(15)
since R∗=max(R0s,R0m,R0x), R∗<1 implies that R0s<1. Similarly, using assumptions (12) and (13), we can prove the following inequalities involving Im and Ix. Define a \(\mathbb {C}^{8}\)-function \(V:\mathbb {R}^{8}_{+}\rightarrow \mathbb {R_{+}}\) by V(S(t), Ls(t), Lm(t), Lx(t), Is(t), Im(t), Ix(t), R(t)) which is a non-negativity function. V(t)=(S(t)+Ls(t)+Lm(t)+Lx(t)+Is(t)+Im(t)+Ix(t)+R(t)). Thus, we can define Z= lnV by the It\(\hat {o}\) formula; we compute,
$$\begin{array}{*{20}l} LV&=\frac{1}{V}dS+\frac{1}{V}dL_{s}+\frac{1}{V}dL_{m}+\frac{1}{V}dL_{x}+\frac{1}{V}dI_{s}+\frac{1}{V}dI_{m}+\frac{1}{V}dI_{x}+\frac{1}{V}dR\notag\\ &-\frac{1}{2}\left[\frac{1}{V^{2}}dS^{2}\,+\,\frac{1}{V^{2}}dL_{s}^{2}\,+\,\frac{1}{V^{2}}dL_{m}^{2}\,+\,\frac{1}{V^{2}}dL_{x}^{2}\,+\,\frac{1}{V^{2}}dI_{s}^{2}\,+\,\frac{1}{V^{2}}dI_{m}^{2}\,+\,\frac{1}{V^{2}}dI_{x}^{2}+\frac{1}{V^{2}}dR^{2}\right]\notag\\ &=\left(b-dS-\beta_{s}\frac{SI_{s}}{N}-\beta_{m}\frac{SI_{m}}{N}-\beta_{x}\frac{SI_{x}}{N}\right)V^{-1}dt \end{array} $$
$$\begin{array}{*{20}l} &+\left(\lambda_{s}\beta_{s}\frac{SI_{s}}{N}+\sigma_{s}\lambda_{s}\beta_{s}\frac{RI_{s}}{N}-\alpha_{ss}\beta_{s}\frac{L_{s}I_{s}}{N}-\alpha_{sm}\beta_{m}\frac{L_{s}I_{m}}{N}-\alpha_{sx}\beta_{x}\frac{L_{s}I_{x}}{N}+\gamma_{s}I_{s}\right)\notag\\& -(d+\epsilon_{s}+t_{1s})L_{s})V^{-1}dt \notag\\&+\left(\lambda_{m}\beta_{m}\frac{SI_{m}}{N}+\sigma_{m}\lambda_{m}\beta_{m}\frac{RI_{m}}{N}+\alpha_{sm}\beta_{m}\lambda_{m}\frac{L_{s}I_{m}}{N}-\alpha_{mm}\beta_{m}\frac{L_{m}I_{m}}{N}-\alpha_{mx}\beta_{x}\frac{L_{m}I_{x}}{N}\right)\notag\\ &+\gamma_{m}I_{m}-(d+\varepsilon_{m})L_{m}+(1-P_{1})t_{1s}L_{s}+(1-P_{2})t_{2s}I_{s})V^{-1}dt\notag\\&+\left(\lambda_{x}\beta_{x}\frac{SI_{x}}{N}+\sigma_{x}\lambda_{x}\beta_{x}\frac{RI_{x}}{N}+\alpha_{sx}\beta_{x}\lambda_{x}\frac{L_{s}I_{x}}{N}+\alpha_{mx}\beta_{x}\lambda_{x}\frac{L_{m}I_{x}}{N}\right.\notag\\&-\left.{\vphantom{\frac{L_{s}I_{x}}{N}}}\alpha_{xx}\beta_{x}\frac{L_{x}I_{x}}{N}-(d+\varepsilon_{x})L_{x}+\gamma_{x}I_{x}+(1-P_{3})t_{2m}I_{m}\right)V^{-1}dt\notag \end{array} $$
$$\begin{array}{*{20}l} &+\left(\alpha_{ss}\beta_{s}\frac{L_{s}I_{s}}{N}+(1-\lambda_{s})\beta_{s}\left(\frac{SI_{s}}{N}+\sigma_{s}\frac{RI_{s}}{N}\right)+\varepsilon_{s}L_{s}\right.\notag\\&\left.{\vphantom{\frac{L_{s}I_{x}}{N}}}-(d+\delta_{s}+t_{2s}+\gamma_{s})I_{s}\right)V^{-1}dt\notag\\&+(1-\lambda_{m})\beta_{m}\left(\frac{SI_{m}}{N}+\sigma_{m}\frac{RI_{m}}{N}+\alpha_{sm}\frac{L_{s}I_{m}}{N}\right) +\alpha_{mm}\beta_{m}\frac{L_{m}I_{m}}{N}+\varepsilon_{m}L_{m} \notag\\&-(d+\delta_{m}+t_{2m}+\gamma_{m})I_{m})V^{-1}dt\notag\\ &+\left(\alpha_{xx}\beta_{x}\frac{L_{x}I_{x}}{N}+(1-\lambda_{x})\beta_{x}\left(\frac{SI_{x}}{N}+\sigma_{x}\frac{RI_{x}}{N}+\alpha_{sx}\frac{L_{s}I_{x}}{N}+\alpha_{mx}\frac{L_{m}I_{x}}{N}\right)+\epsilon_{x}L_{x}\right. \notag\\ &-\left.{\vphantom{\frac{L_{x}I_{x}}{N}}}(d+\delta_{x}+t_{2x}+\gamma_{x}+\epsilon_{4}u_{4}(t))I_{x}\right)V^{-1}dt\notag\\ &+\left(P_{1}t_{1s}L_{s}+P_{2}t_{2s}I_{s}+P_{3}t_{2m}I_{m}+t_{2x}I_{x}+\epsilon_{4}u_{4}(t)I_{x}-\sigma_{s}\beta_{s}\frac{RI_{s}}{N} \right.\notag\\ &\left.-\sigma_{m}\beta_{m}\frac{RI_{m}}{N}-\sigma_{x}\beta_{x}\frac{RI_{x}}{N} -dR\right)V^{-1}dt-\frac{1}{2}\left[\frac{\xi^{2}_{1} S}{V^{2}}+\frac{\xi^{2}_{2} L_{s}}{V^{2}}+\frac{\xi^{2}_{3} L_{m}}{V^{2}}+\frac{\xi^{2}_{4} L_{x}}{V^{2}}\right.\notag\\ &\left.+\frac{\xi^{2}_{5} I_{s}}{V^{2}}+\frac{\xi^{2}_{6} I_{m}}{V^{2}}+\frac{\xi^{2}_{7} I_{x}}{V^{2}}+\frac{\xi^{2}_{8} R}{V^{2}}\right].\notag \end{array} $$
Then,
$$\begin{array}{*{20}l} dZ&=LVdt+[\xi_{1}SdW_{1}+\xi_{2}L_{s}dW_{2}+\xi_{3}L_{m}dW_{3}+\xi_{4}L_{x}dW_{4}+\xi_{5}I_{s}dW_{5}\notag\\&+\xi_{6}I_{m}dW_{6}+\xi_{7}I_{x}dW_{7}+\xi_{8}RdW_{8}]V^{-1}.\notag \end{array} $$
Therefore,
$$\begin{array}{*{20}l} Z=Z_{0}+\int_{0}^{t}LVdt+\sum_{i=1}^{8}G_{i}(t), \end{array} $$
(16)
where each Gi(t) is a martingale defined as: \(G_{1}(t)= \int _{0}^{t}\frac {\xi _{1}SdW_{1}}{V}\), \(G_{2}(t)= \int _{0}^{t}\frac {\xi _{2}L_{s}dW_{2}}{V}\), \(G_{3}(t)= \int _{0}^{t}\frac {\xi _{3}L_{m}dW_{2}}{V}\), \(G_{4}(t)= \int _{0}^{t}\frac {\xi _{4}L_{x}dW_{4}}{V}\), \(G_{5}(t)= \int _{0}^{t}\frac {\xi _{5}I_{s}dW_{5}}{V}\), \(G_{6}(t)= \int _{0}^{t}\frac {\xi _{6}I_{m}dW_{6}}{V}\), \(G_{7}(t)= \int _{0}^{t}\frac {\xi _{7}I_{x}dW_{7}}{V}\), \(G_{8}(t)= \int _{0}^{t}\frac {\xi _{8}RdW_{8}}{V}.\) Then,
$${\lim}_{t\rightarrow \infty}\frac{Z}{t}={\lim}_{t\rightarrow \infty}\frac{Z_{0}}{t}+{\lim}_{t\rightarrow \infty}\frac{1}{t}\int_{0}^{t}LVdt$$
$$+{\lim}_{t\rightarrow \infty}\frac{1}{t}\sum_{i=1}^{8}G_{i}(t).$$
Regarding the quadratic variations of the stochastic integral Gi(t), we have \( \int _{0}^{t}\frac {(\xi _{1}S)^{2}}{V^{2}}ds\leq \xi _{1}^{2}t\), \( \int _{0}^{t}\frac {(\xi _{2}L_{s})^{2}}{V^{2}}ds\leq \xi _{2}^{2}t\), \( \int _{0}^{t}\frac {(\xi _{3}L_{m})^{2}}{V^{2}}ds\leq \xi _{3}^{2}t\), \( \int _{0}^{t}\frac {(\xi _{4}L_{x})^{2}}{V^{2}}ds\leq \xi _{4}^{2}t\), \( \int _{0}^{t}\frac {(\xi _{5}I_{s})^{2}}{V^{2}}ds\leq \xi _{5}^{2}t\), \( \int _{0}^{t}\frac {(\xi _{6}I_{m})^{2}}{V^{2}}ds\leq \xi _{6}^{2}t\), \( \int _{0}^{t}\frac {(\xi _{7}I_{x})^{2}}{V^{2}}ds\leq \xi _{7}^{2}t\), \( \int _{0}^{t}\frac {(\xi _{8}R)^{2}}{V^{2}}ds\leq \xi _{8}^{2}t.\) By the strong law of large numbers for martingales [36], we therefore have
$${\lim}_{t\rightarrow\infty}\sup\frac{1}{t}\sum_{i=1}^{8}G_{i}(t)=0\ \ \ (a.s).$$
It finally follows from (16) by dividing t on both sides and then letting t→∞ that
$${\lim}_{t\rightarrow\infty}\sup\frac{\ln Z(t)}{t}\leq{\lim}_{t\rightarrow\infty}\sup\frac{1}{t}\int_{0}^{t}LVdt\ \ \ (a.s).$$
We note that
$$\begin{array}{*{20}l} LV&\leq\frac{1}{V}dS+\frac{1}{V}dL_{s}+\frac{1}{V}dL_{m}+\frac{1}{V}dL_{x}+\frac{1}{V}dI_{s}+\frac{1}{V}dI_{m}+\frac{1}{V}dI_{x}+\frac{1}{V}dR\notag\\ &-\frac{1}{2}\left[\frac{1}{V^{2}}dS^{2}\,+\,\frac{1}{V^{2}}dL_{s}^{2}\,+\,\frac{1}{V^{2}}dL_{m}^{2}\,+\,\frac{1}{V^{2}}dL_{x}^{2}\,+\,\frac{1}{V^{2}}dI_{s}^{2}\,+\,\frac{1}{V^{2}}dI_{m}^{2}\,+\,\frac{1}{V^{2}}dI_{x}^{2}\,+\,\frac{1}{V^{2}}dR^{2}\right]\notag\\ &=\left(b-dS-\beta_{s}\frac{SI_{s}}{N}-\beta_{m}\frac{SI_{m}}{N}-\beta_{x}\frac{SI_{x}}{N}\right)V^{-1}dt\notag\\&+\left(\lambda_{s}\beta_{s}\frac{SI_{s}}{N}+\sigma_{s}\lambda_{s}\beta_{s}\frac{RI_{s}}{N}-\alpha_{ss}\beta_{s}\frac{L_{s}I_{s}}{N}-\alpha_{sm}\beta_{m}\frac{L_{s}I_{m}}{N}-\alpha_{sx}\beta_{x}\frac{L_{s}I_{x}}{N}+\gamma_{s}I_{s}\right)\notag\\& -(d+\epsilon_{s}+t_{1s})L_{s})V^{-1}dt \notag\\&+\left(\lambda_{m}\beta_{m}\frac{SI_{m}}{N}+\sigma_{m}\lambda_{m}\beta_{m}\frac{RI_{m}}{N}+\alpha_{sm}\beta_{m}\lambda_{m}\frac{L_{s}I_{m}}{N}-\alpha_{mm}\beta_{m}\frac{L_{m}I_{m}}{N}-\alpha_{mx}\beta_{x}\frac{L_{m}I_{x}}{N}\right)\notag\\& +\gamma_{m}I_{m}-(d+\varepsilon_{m})L_{m}+(1-P_{1})t_{1s}L_{s}+(1-P_{2})t_{2s}I_{s})V^{-1}dt\notag\\&+\left(\lambda_{x}\beta_{x}\frac{SI_{x}}{N}+\sigma_{x}\lambda_{x}\beta_{x}\frac{RI_{x}}{N}+\alpha_{sx}\beta_{x}\lambda_{x}\frac{L_{s}I_{x}}{N}+\alpha_{mx}\beta_{x}\lambda_{x}\frac{L_{m}I_{x}}{N}\right.\notag\\&-\left.{\vphantom{\frac{L_{s}I_{x}}{N}}}\alpha_{xx}\beta_{x}\frac{L_{x}I_{x}}{N}-(d+\varepsilon_{x})L_{x}+\gamma_{x}I_{x}+(1-P_{3})t_{2m}I_{m}\right)V^{-1}dt\notag\\&+\left(\alpha_{ss}\beta_{s}\frac{L_{s}I_{s}}{N}+(1-\lambda_{s})\beta_{s}\left(\frac{SI_{s}}{N}+\sigma_{s}\frac{RI_{s}}{N}\right)+\varepsilon_{s}L_{s}-(d+\delta_{s}+t_{2s}+\gamma_{s})I_{s}\right)V^{-1}dt\notag\\&+\left((1-\lambda_{m})\beta_{m}\left(\frac{SI_{m}}{N}+\sigma_{m}\frac{RI_{m}}{N}+\alpha_{sm}\frac{L_{s}I_{m}}{N}\right) +\alpha_{mm}\beta_{m}\frac{L_{m}I_{m}}{N}+\varepsilon_{m}L_{m}\right. \notag\\&-\left.{\vphantom{\frac{RI_{m}}{N}}}(d+\delta_{m}+t_{2m}+\gamma_{m})I_{m}\right)V^{-1}dt\notag\\&+\left(\alpha_{xx}\beta_{x}\frac{L_{x}I_{x}}{N}+(1-\lambda_{x})\beta_{x}\left(\frac{SI_{x}}{N}+\sigma_{x}\frac{RI_{x}}{N}+\alpha_{sx}\frac{L_{s}I_{x}}{N}+\alpha_{mx}\frac{L_{m}I_{x}}{N}\right)+\epsilon_{x}L_{x}\right. \notag\\&-\left.{\vphantom{\frac{RI_{m}}{N}}}(d+\delta_{x}+t_{2x}+\gamma_{x}+\epsilon_{4}u_{4}(t))I_{x}\right)V^{-1}dt\notag\\&+\left(P_{1}t_{1s}L_{s}+P_{2}t_{2s}I_{s}+P_{3}t_{2m}I_{m}+t_{2x}I_{x}+\epsilon_{4}u_{4}(t)I_{x}-\sigma_{s}\beta_{s}\frac{RI_{s}}{N}\right. \notag\\&\left.{\vphantom{\frac{RI_{m}}{N}}}-\sigma_{m}\beta_{m}\frac{RI_{m}}{N}-\sigma_{x}\beta_{x}\frac{RI_{x}}{N} -dR\right)V^{-1}dt-\frac{1}{2}\left[\frac{\xi^{2}_{1} S}{V^{2}}+\frac{\xi^{2}_{2} L_{s}}{V^{2}}+\frac{\xi^{2}_{3} L_{m}}{V^{2}}+\frac{\xi^{2}_{4} L_{x}}{V^{2}}\right.\notag\\&\left.+\frac{\xi^{2}_{5} I_{s}}{V^{2}}+\frac{\xi^{2}_{6} I_{m}}{V^{2}}+\frac{\xi^{2}_{7} I_{x}}{V^{2}}+\frac{\xi^{2}_{8} R}{V^{2}}\right].\notag \end{array} $$
and in fact
$${\lim}_{t\rightarrow\infty}\sup\frac{1}{t}\int_{0}^{t}LVdt<0, $$
therefore,
$${\lim}_{t\rightarrow\infty}\sup\frac{\ln Z(t)}{t}<0. $$
This finally proves the (a.s.) exponential stability. □