Researching | Dynamical analysis for hybrid virus infection system in switching environment

Journals >Chinese Physics B >Volume 29 >Issue 9 >Page > Article

Chinese Physics B
Vol. 29, Issue 9, (2020)

Dynamical analysis for hybrid virus infection system in switching environment

Dong-Xi Li^1、† and Ni Zhang²

Author Affiliations

¹College of Data Science, Taiyuan University of Technology, Taiyuan 030024, China

²College of Mathematics, Taiyuan University of Technology, Taiyuan 03004, China

show less

DOI: 10.1088/1674-1056/ab8c3f Cite this Article

Dong-Xi Li, Ni Zhang. Dynamical analysis for hybrid virus infection system in switching environment[J]. Chinese Physics B, 2020, 29(9): Copy Citation Text

show less

Abstract

We investigate the dynamical behavior of hybrid virus infection systems with nonlytic immune response in switching environment, which is modeled as a stochastic process of telegraph noise and represented as a multi-state Markov chains. Firstly, The existence of unique positive solution and boundedness of the new hybrid system is proved. Furthermore, the sufficient conditions for extinction and persistence of virus are established. Finally, stochastic simulations are performed to test and demonstrate the conclusions. As a consequence, our work suggests that stochastic switching environment plays a crucial role in the process of virus prevention and treatment.

Keywords

dynamical behavior extinction hybrid virus infection system switching environment

$$ \begin{eqnarray}\left\{\begin{array}{l}\dot{x}(t)=\lambda -\delta x(t)-\displaystyle \frac{\beta x(t)y(t)}{1+qz(t)},\\ \dot{y}(t)=\displaystyle \frac{\beta x(t)y(t)}{1+qz(t)}-ay(t)-py(t)z(t),\\ \dot{z}(t)=cy(t)-bz(t).\end{array}\right.\end{eqnarray}$$

(1)

View in Article

$$ \begin{eqnarray}\begin{array}{ll} & {\mathbb{P}}\{r(t+\Delta t)=j| r(t)=i\}\\ = & \left\{\begin{array}{ll}{\nu }_{ij}\Delta t+o(\Delta t), & {\rm{if}}\,\,i\ne j,\\ 1+{\nu }_{ii}\Delta t+o(\Delta t), & {\rm{if}}\,\,i=j,\end{array}\right.\end{array}\end{eqnarray}$$

(2)

View in Article

$$ \begin{eqnarray*}{\pi }_{1}=\displaystyle \frac{{\nu }_{21}}{{\nu }_{12}+{\nu }_{21}}\,\,{\rm{and}}\,\,{\pi }_{2}=\displaystyle \frac{{\nu }_{12}}{{\nu }_{12}+{\nu }_{21}}.\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray*}\hat{\nu }=\mathop{\max }\limits_{k\in {\mathbb{S}}}\nu (k),\,\,\,\,\check{\nu }=\mathop{\min }\limits_{k\in {\mathbb{S}}}\nu (k).\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray}\left\{\begin{array}{l}\dot{x}(t)={\lambda }_{r(t)}-{\delta }_{r(t)}x(t)-\displaystyle \frac{{\beta }_{r(t)}x(t)y(t)}{1+{q}_{r(t)}z(t)},\\ \dot{y}(t)=\displaystyle \frac{{\beta }_{r(t)}x(t)y(t)}{1+{q}_{r(t)}z(t)}-{a}_{r(t)}y(t)-{p}_{r(t)}y(t)z(t),\\ \dot{z}(t)={c}_{r(t)}y(t)-{b}_{r(t)}z(t),\end{array}\right.\end{eqnarray}$$

(3)

View in Article

$$ \begin{eqnarray}\begin{array}{lll}\dot{x}(t) & = & {\lambda }_{r(t)}-{\delta }_{r(t)}x(t)-\displaystyle \frac{{\beta }_{r(t)}x(t)y(t)}{1+{q}_{r(t)}z(t)}\\ & \lt & \hat{\lambda }-\check{\delta }x(t).\end{array}\end{eqnarray}$$

(4)

View in Article

$$ \begin{eqnarray*}x(t)\lt \displaystyle \frac{\hat{\lambda }}{\check{\delta }}+1,\,\,\,\,{\rm{for}}\,\,{\rm{all}}\,\,{\rm{large}}\,\,t,{\rm{say}}\,\,t\gt {t}_{0}.\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray*}\begin{array}{lll}\dot{x}(t)+\dot{y}(t) & = & {\lambda }_{r(t)}-{\delta }_{r(t)}x(t)-{a}_{r(t)}y(t)-{p}_{r(t)}y(t)z(t)\\ & \lt & \hat{\lambda }-\check{\delta }x(t).\end{array}\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray*}x(t)+y(t)\ge C+\displaystyle \frac{\hat{\lambda }}{\check{\delta }}+1,t\gt {t}_{0},\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray*}\begin{array}{cc}x(t)+y(t)\lt C+\displaystyle \frac{\hat{\lambda }}{\check{\delta }}+1, & {\rm{for\,\,all}}\,\,t\gt {t}_{1}.\end{array}\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray*}\begin{array}{cc}\dot{z}(t)\lt \hat{c}\left(C+\displaystyle \frac{\hat{\lambda }}{\check{\delta }}+1\right)-\check{b}z, & {\rm{for}}\,\,{\rm{large}}\,\,t\end{array}\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray}\begin{array}{lll}{E}_{1} & = & \{\bar{x},\bar{y},\bar{z}\}=\left\{\displaystyle \frac{c\lambda (1+q\bar{z})}{c\delta +(b\beta +c\delta q)\bar{z}},\displaystyle \frac{b\bar{z}}{c},-\displaystyle \frac{pc\delta +ab\beta +ac\delta q}{2(bp\beta +c\delta pq)}+\displaystyle \frac{\sqrt{{(pc\delta +ab\beta +ac\delta q)}^{2}-4p(b\beta +c\delta q)(ac\delta -c\lambda \beta )}}{2(bp\beta +c\delta pq)}\right\},\end{array}\end{eqnarray}$$

(5)

View in Article

$$ \begin{eqnarray*}{\alpha }_{r(t)}={\beta }_{r(t)}{\lambda }_{r(t)}-{\delta }_{r(t)}{a}_{r(t)}.\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray*}{R}_{0}^{* }=\displaystyle \frac{({\pi }_{1}{\beta }_{1}+{\pi }_{2}{\beta }_{2})({\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2})}{({\pi }_{1}{a}_{1}+{\pi }_{2}{a}_{2})({\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2})}\lt 1,\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray}\mathop{\mathrm{lim}\inf }\limits_{t\to \infty }x(t)\le \displaystyle \frac{{\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2}}{{\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}}\,\,{\rm{a}}.{\rm{s}}.\end{eqnarray}$$

(6)

View in Article

$$ \begin{eqnarray}\mathop{\mathrm{lim}\sup }\limits_{t\to \infty }x(t)\ge \displaystyle \frac{{\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2}}{{\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}}\,{\rm{a}}.{\rm{s}}.\end{eqnarray}$$

(7)

View in Article

$$ \begin{eqnarray}\mathop{\mathrm{lim}}\limits_{t\to \infty }y(t)=0\,{\rm{a}}.{\rm{s}}.\end{eqnarray}$$

(8)

View in Article

$$ \begin{eqnarray}\mathop{\mathrm{lim}}\limits_{t\to \infty }z(t)=0\,{\rm{a}}.{\rm{s}}.\end{eqnarray}$$

(9)

View in Article

$$ \begin{eqnarray}{\varOmega }_{1}=\left\{\omega \in \varOmega :\mathop{\mathrm{lim}\inf }\limits_{t\to \infty }x(t)\gt \displaystyle \frac{{\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2}}{{\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}}+\varepsilon \right\}.\end{eqnarray}$$

(10)

View in Article

$$ \begin{eqnarray}\begin{array}{ll} & \mathop{\mathrm{lim}}\limits_{t\to \infty }\displaystyle \frac{1}{t}\left({\lambda }_{r(s)}-{\delta }_{r(s)}\left[\displaystyle \frac{{\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2}}{{\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}}+\varepsilon \right]\right){\rm{d}}s\\ = & {\pi }_{1}\left({\lambda }_{1}-{\delta }_{1}\left[\displaystyle \frac{{\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2}}{{\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}}+\varepsilon \right]\right)\\ & +{\pi }_{2}\left({\lambda }_{2}-{\delta }_{2}\left[\displaystyle \frac{{\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2}}{{\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}}+\varepsilon \right]\right)\\ = & -({\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2})\varepsilon .\end{array}\\ \end{eqnarray}$$

(11)

View in Article

$$ \begin{eqnarray}\begin{array}{lll}dx(t) & = & \left[{\lambda }_{r(t)}-{\delta }_{r(t)}x(t)+\displaystyle \frac{{\beta }_{r(t)}x(t)y(t)}{{q}_{r(t)}z(t)+1}\right]{\rm{d}}t\\ & \le & ({\lambda }_{r(t)}-{\delta }_{r(t)}x(t)){\rm{d}}t.\end{array}\end{eqnarray}$$

(12)

View in Article

$$ \begin{eqnarray*}\begin{array}{lll}x(t) & \le & x(0)+\displaystyle {\int }_{0}^{T}({\lambda }_{r(s)}-{\delta }_{r(s)}x(s)){\rm{d}}s\\ & & +\displaystyle {\int }_{T}^{t}\left({\lambda }_{r(s)}-{\delta }_{r(s)}\left[\displaystyle \frac{{\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2}}{{\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}}+\varepsilon \right]\right){\rm{d}}s\end{array}\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray*}\mathop{\mathrm{lim}\sup }\limits_{t\to \infty }\displaystyle \frac{x(t)}{t}\le -({\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2})\varepsilon,\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray*}\mathop{\mathrm{lim}}\limits_{t\to \infty }x(t)=0.\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray*}\begin{array}{lll}\displaystyle \frac{x(t)-x(0)}{t} & = & \displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}({\lambda }_{r(s)}-{\delta }_{r(s)}x(s)-\displaystyle \frac{{\beta }_{r(s)}x(s)y(s)}{{q}_{r(s)}z(s)+1}){\rm{d}}s\\ & \le & \displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}({\lambda }_{r(s)}-{\delta }_{r(s)}x(s)){\rm{d}}s.\end{array}\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray}\displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}{\delta }_{r(s)}x(s){\rm{d}}s\le \displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}{\lambda }_{r(s)}{\rm{d}}s-\displaystyle \frac{x(t)-x(0)}{t}.\end{eqnarray}$$

(13)

View in Article

$$ \begin{eqnarray*}({\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2})\mathop{\mathrm{lim}}\limits_{t\to \infty }\displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}x(s){\rm{d}}s\le {\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2}.\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray*}\mathop{\mathrm{lim}}\limits_{t\to \infty }\displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}x(s){\rm{d}}s\le \displaystyle \frac{{\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2}}{{\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}}+\varepsilon .\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray*}{\rm{d}}\,\mathrm{ln}\,y(t)=\left[\displaystyle \frac{{\beta }_{r(t)}x(t)}{{q}_{r(t)}z(t)+1}-{a}_{r(t)}-{p}_{r(t)}z(t)\right]\,{\rm{d}}t.\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray}\begin{array}{ll} & \displaystyle \frac{\mathrm{ln}y(t)-\mathrm{ln}y(0)}{t}\\ = & \displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}\left(-{a}_{r(s)}+\displaystyle \frac{{\beta }_{r(s)}x(s)}{{q}_{r(s)}z(s)+1}-{p}_{r(s)}z(s)\right){\rm{d}}s\\ & \le \displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}({\beta }_{r(s)}x(s)-{a}_{r(s)})\,{\rm{d}}s.\end{array}\\ \end{eqnarray}$$

(14)

View in Article

$$ \begin{eqnarray*}\begin{array}{ll} & \mathop{\mathrm{lim}\sup }\limits_{t\to \infty }\displaystyle \frac{1}{t}\mathrm{ln}(y(t))\\ \le & [{\pi }_{1}{\beta }_{1}+{\pi }_{2}{\beta }_{2}]\displaystyle \frac{{\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2}}{{\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}}-({\pi }_{1}{a}_{1}+{\pi }_{2}{a}_{2}).\end{array}\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray*}({\pi }_{1}{\beta }_{1}+{\pi }_{2}{\beta }_{2})({\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2})\lt ({\pi }_{1}{a}_{1}+{\pi }_{2}{a}_{2})({\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}),\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray*}\mathop{\mathrm{lim}}\limits_{t\to \infty }y(t)=0.\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray}{\varOmega }_{3}=\left\{\omega \in \varOmega :\mathop{\mathrm{lim}\sup }\limits_{t\to \infty }x(t)\lt \displaystyle \frac{{\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2}}{{\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}}-\varepsilon \right\}.\end{eqnarray}$$

(15)

View in Article

$$ \begin{eqnarray*}\begin{array}{ll} & \mathop{\mathrm{lim}}\limits_{t\to \infty }\displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}\left({\lambda }_{r(s)}-{\delta }_{r(s)}\left[\displaystyle \frac{{\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2}}{{\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}}-\varepsilon \right]\right)\,{\rm{d}}s\\ = & {\pi }_{1}\left({\lambda }_{1}-{\delta }_{1}\left[\displaystyle \frac{{\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2}}{{\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}}-\varepsilon \right]\right)\\ & +{\pi }_{2}\left({\lambda }_{2}-{\delta }_{2}\left[\displaystyle \frac{{\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2}}{{\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}}-\varepsilon \right]\right)\\ = & ({\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2})\varepsilon .\end{array}\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray*}\begin{array}{lll}x(t)-x(0) & = & \displaystyle {\int }_{0}^{t}({\lambda }_{r(s)}-{\delta }_{r(s)}x(s)){\rm{d}}s-\displaystyle {\int }_{0}^{t}\displaystyle \frac{{\beta }_{r(s)}x(s)y(s)}{{q}_{r(s)}z(s)+1}{\rm{d}}s\\ & \ge & \displaystyle {\int }_{0}^{T}({\lambda }_{r(s)}-{\delta }_{r(s)}x(s)){\rm{d}}s\\ & & +\displaystyle {\int }_{T}^{t}\left({\lambda }_{r(s)}-{\delta }_{r(s)}\left[\displaystyle \frac{{\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2}}{{\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}}\right]\right){\rm{d}}s\\ & & -\displaystyle {\int }_{0}^{t}\displaystyle \frac{{\beta }_{r(s)}x(s)y(s)}{{q}_{r(s)}z(s)+1}{\rm{d}}s\end{array}\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray*}\mathop{\mathrm{lim}\sup }\limits_{t\to \infty }\displaystyle \frac{x(t)}{t}\ge ({\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2})\varepsilon,\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray*}\mathop{\mathrm{lim}}\limits_{t\to \infty }x(t)=\infty .\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray*}{R}_{0}^{* }=\displaystyle \frac{({\pi }_{1}{\beta }_{1}+{\pi }_{2}{\beta }_{2})({\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2})}{({\pi }_{1}{a}_{1}+{\pi }_{2}{a}_{2})({\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2})}\gt 1,\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray}\begin{array}{lll}\displaystyle \frac{y(t)-y(0)}{t} & = & \displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}\left[\displaystyle \frac{{\beta }_{r(s)}x(s)}{{q}_{r(s)}z(s)+1}\right.\\ & & \left. -{a}_{r(s)}y(s)-{p}_{r(s)}y(s)z(s)\right]{\rm{d}}s.\end{array}\end{eqnarray}$$

(16)

View in Article

$$ \begin{eqnarray}\begin{array}{ll} & \displaystyle \frac{x(t)-x(0)}{t}+\displaystyle \frac{y(t)-y(0)}{t}\\ = & \displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}({\lambda }_{r(s)}-{\delta }_{r(s)}x(s)-{a}_{r(s)}y(s)-{p}_{r(s)}y(s)z(s)){\rm{d}}s.\end{array}\end{eqnarray}$$

(17)

View in Article

$$ \begin{eqnarray*}\begin{array}{ll} & \displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}{\lambda }_{r(s)}{\rm{d}}s\\ = & \displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}({\delta }_{r(s)}x(s)){\rm{d}}s-\displaystyle \frac{x(t)-x(0)}{t}-\displaystyle \frac{y(t)-y(0)}{t}\\ & -\displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}({a}_{r(s)}y(s)){\rm{d}}s-\displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}({p}_{r(s)}y(s)z(s)){\rm{d}}s.\end{array}\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray}\begin{array}{ll} & \mathop{\mathrm{lim}}\limits_{t\to \infty }\displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}x(s){\rm{d}}s\\ = & \displaystyle \frac{{\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2}}{{\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}}-\displaystyle \frac{{\pi }_{1}{a}_{1}+{\pi }_{2}{a}_{2}}{{\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}}\mathop{\mathrm{lim}}\limits_{t\to \infty }\displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}y(s){\rm{d}}s\\ & -\displaystyle \frac{1}{{\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}}\mathop{\mathrm{lim}}\limits_{t\to \infty }\displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}({p}_{r(s)}y(s)z(s)){\rm{d}}s\\ \le & \displaystyle \frac{{\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2}}{{\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}}-\displaystyle \frac{{\pi }_{1}{a}_{1}+{\pi }_{2}{a}_{2}}{{\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}}\mathop{\mathrm{lim}}\limits_{t\to \infty }\displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}y(s){\rm{d}}s.\end{array}\end{eqnarray}$$

(18)

View in Article

$$ \begin{eqnarray*}\displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}({\beta }_{r(s)}x(s)){\rm{d}}s\ge \displaystyle \frac{lny(t)-lny(0)}{t}+\displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}{a}_{r(s)}{\rm{d}}s.\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray}\begin{array}{ll} & \displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}x(s){\rm{d}}s\\ \ge & \begin{array}{c}\displaystyle \frac{1}{{\pi }_{1}{\beta }_{1}+{\pi }_{2}{\beta }_{2}}\displaystyle \frac{\mathrm{ln}y(t)-\mathrm{ln}y(0)}{t}+\displaystyle \frac{{\pi }_{1}{a}_{1}+{\pi }_{2}{a}_{2}}{{\pi }_{1}{\beta }_{1}+{\pi }_{2}{\beta }_{2}}.\\ \end{array}\end{array}\end{eqnarray}$$

(19)

View in Article

$$ \begin{eqnarray}\begin{array}{ll} & \begin{array}{c}\displaystyle \frac{{\pi }_{1}{a}_{1}+{\pi }_{2}{a}_{2}}{{\pi }_{1}{\beta }_{1}+{\pi }_{2}{\beta }_{2}}\\ \end{array}\\ \le & \displaystyle \frac{{\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2}}{{\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}}-\displaystyle \frac{{\pi }_{1}{a}_{1}+{\pi }_{2}{a}_{2}}{{\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}}\mathop{\mathrm{lim}}\limits_{t\to \infty }\displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}y(s){\rm{d}}s.\end{array}\end{eqnarray}$$

(20)

View in Article

$$ \begin{eqnarray*}\mathop{\mathrm{lim}\inf }\limits_{t\to \infty }\displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}y(s){\rm{d}}s\le \left(\displaystyle \frac{({\pi }_{1}{\beta }_{1}+{\pi }_{2}{\beta }_{2})({\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2})-({\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2})({\pi }_{1}{a}_{1}+{\pi }_{2}{a}_{2})}{({\pi }_{1}{\beta }_{1}+{\pi }_{2}{\beta }_{2})({\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2})}\right)\left(\displaystyle \frac{{\pi }_{1}{\beta }_{1}+{\pi }_{2}{\beta }_{2}}{{\pi }_{1}{a}_{1}+{\pi }_{2}{a}_{2}}\right).\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray*}({\pi }_{1}{\beta }_{1}+{\pi }_{2}{\beta }_{2})({\pi }_{1}{\lambda }_{1}+{\pi }_{2}{\lambda }_{2})\gt ({\pi }_{1}{a}_{1}+{\pi }_{2}{a}_{2})({\pi }_{1}{\delta }_{1}+{\pi }_{2}{\delta }_{2}),\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray*}\mathop{\mathrm{lim}\inf }\limits_{t\to \infty }\displaystyle \frac{1}{t}\displaystyle {\int }_{0}^{t}y(s){\rm{d}}s\gt 0\,{\rm{a}}.{\rm{s}}.\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray*}\begin{array}{l}{x}_{k+1}={x}_{k}+({\lambda }_{r(t)}-{\delta }_{r(t)}x(t)-\displaystyle \frac{{\beta }_{r(t)}x(t)y(t)}{1+{q}_{r(t)}z(t)})\Delta t,\\ {y}_{k+1}={y}_{k}+(\displaystyle \frac{{\beta }_{r(t)}x(t)y(t)}{1+{q}_{r(t)}z(t)}-{a}_{r(t)}y(t)-{p}_{r(t)}y(t)z(t))\Delta t,\\ {z}_{k+1}={z}_{k}+({c}_{r(t)}y(t)-{b}_{r(t)}z(t))\Delta t.\end{array}\end{eqnarray*}$$

()

View in Article

$$ \begin{eqnarray*}\begin{array}{lll}{E}^{* } & = & \{{x}^{* },{y}^{* },{z}^{* }\}=\left\{\displaystyle \frac{{\pi }_{1}{c}_{1}{\lambda }_{1}(1+{q}_{1}{\bar{z}}_{1})+{\pi }_{2}{c}_{2}{\lambda }_{2}(1+{q}_{2}{\bar{z}}_{2})}{{\pi }_{1}{c}_{1}{\delta }_{1}+({b}_{1}{\beta }_{1}+{c}_{1}{\delta }_{1}{q}_{1}){\bar{z}}_{1}+{\pi }_{2}{c}_{2}{\delta }_{2}+({b}_{2}{\beta }_{2}+{c}_{2}{\delta }_{2}{q}_{2}){\bar{z}}_{2}},\displaystyle \frac{{\pi }_{1}{b}_{1}{\bar{z}}_{1}+{\pi }_{2}{b}_{2}{\bar{z}}_{2}}{{\pi }_{1}{c}_{1}+{\pi }_{2}{c}_{2}},\right.\\ & & \left. \displaystyle \frac{{\pi }_{1}(-({p}_{1}{c}_{1}{\delta }_{1}+{a}_{1}{b}_{1}{\beta }_{1}+{a}_{1}{c}_{1}{\delta }_{1}{q}_{1}))}{2{\pi }_{1}({b}_{1}{p}_{1}{\beta }_{1}+{c}_{1}{\delta }_{1}{p}_{1}{q}_{1})+2{\pi }_{2}({b}_{2}{p}_{2}{\beta }_{2}+{c}_{2}{\delta }_{2}{p}_{2}{q}_{2})}\right.\\ & & \left. +\displaystyle \frac{{\pi }_{1}(\sqrt{({p}_{1}{c}_{1}{\delta }_{1}+{a}_{1}{b}_{1}{\beta }_{1}+{a}_{1}{c}_{1}{\delta }_{1}{q}_{)}{}^{2}-4{p}_{1}({b}_{1}{\beta }_{1}+{c}_{1}{\delta }_{1}{q}_{1})({a}_{1}{c}_{1}{\delta }_{1}-{c}_{1}{\lambda }_{1}{\beta }_{1})})}{2{\pi }_{1}({b}_{1}{p}_{1}{\beta }_{1}+{c}_{1}{\delta }_{1}{p}_{1}{q}_{1})+2{\pi }_{2}({b}_{2}{p}_{2}{\beta }_{2}+{c}_{2}{\delta }_{2}{p}_{2}{q}_{2})}\right.\\ & & \left. +\displaystyle \frac{{\pi }_{2}(-({p}_{2}{c}_{2}{\delta }_{2}+{a}_{2}{b}_{2}{\beta }_{2}+{a}_{2}{c}_{2}{\delta }_{2}{q}_{2}))}{2{\pi }_{1}({b}_{1}{p}_{1}{\beta }_{1}+{c}_{1}{\delta }_{1}{p}_{1}{q}_{1})+2{\pi }_{2}({b}_{2}{p}_{2}{\beta }_{2}+{c}_{2}{\delta }_{2}{p}_{2}{q}_{2})}\right.\\ & & \left. +\displaystyle \frac{{\pi }_{2}(\sqrt{({p}_{2}{c}_{2}{\delta }_{2}+{a}_{2}{b}_{2}{\beta }_{2}+{a}_{2}{c}_{2}{\delta }_{2}{q}_{)}{}^{2}-4{p}_{2}({b}_{2}{\beta }_{2}+{c}_{2}{\delta }_{2}{q}_{2})({a}_{2}{c}_{2}{\delta }_{2}-{c}_{2}{\lambda }_{2}{\beta }_{2})})}{2{\pi }_{1}({b}_{1}{p}_{1}{\beta }_{1}+{c}_{1}{\delta }_{1}{p}_{1}{q}_{1})+2{\pi }_{2}({b}_{2}{p}_{2}{\beta }_{2}+{c}_{2}{\delta }_{2}{p}_{2}{q}_{2})}\right\}.\end{array}\end{eqnarray*}$$

()

View in Article

Figures&Tables (4)

References (36)

Copy Citation Text

Dong-Xi Li, Ni Zhang. Dynamical analysis for hybrid virus infection system in switching environment[J]. Chinese Physics B, 2020, 29(9):

Download Citation

Save the article for my favorites

Paper Information

Recommended Topics

Nuclear physics

Particles and fields

Particle and nuclear astrophysics and cosmology

Set citation alerts for the article

Please enter your email address