# American Institute of Mathematical Sciences

## The dynamics of a two host-two virus system in a chemostat environment

 1 Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan 2 Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, USA

* Corresponding author: Sze-Bi Hsu

Received  June 2020 Revised  August 2020 Published  October 2020

Fund Project: The first author is supported by the grant MOST 108-2115-M-007-007. The second author is supported by NSF Grant DMS 1411703

The coevolution or coexistence of multiple viruses with multiple hosts has been an important issue in viral ecology. This paper is to study the mathematical properties of the solutions of a chemostat model for two host species and two virus species. By virtue of the global dynamics of its submodels and the theories of uniform persistence and Hopf bifurcation, we derive sufficient conditions for the coexistence of two hosts with two viruses and coexistence of two hosts with one virus, as well as occurrence of Hopf bifurcation.

Citation: Sze-Bi Hsu, Yu Jin. The dynamics of a two host-two virus system in a chemostat environment. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020298
##### References:

show all references

##### References:
The projection of the phase diagram of model (1) onto the $N_1N_2$ plane. Left: $\beta = 11.5$; right: $\beta = 20$
The time series of model (1). Left: $\beta = 11.5$; right: $\beta = 20$
The conditions for existence and local stability of equilibria of (4). Here, an equilibrium exists means it is nonnegative for $E_1^{nnv}$-$E_4^{nnv}$ and positive for $E_5^{nnv}$
 Equilibrium Existence condition Stability condition $E_0^{nnv}=(0,0,0)$ $r_1 <\omega$, $r_2 <\omega$ $E_1^{nnv}=(\tilde{N}_1,0,0)$ $r_1 > \omega$ $r_1 >r_2$, $N_1^\ast >\tilde{N}_1$ $E_2^{nnv}=(0,\tilde{N}_2,0)$ $r_2 > \omega$ $r_1 \tilde{N}_2$ $E_3^{nnv}=(N_1^\ast,0,\tilde{V}^\ast)$ $N_1^\ast <\tilde{N}_1$ ($r_1 >\omega$ required) $\left(\frac{\phi_1}{\phi_2}-\frac{r_1}{r_2}\right)(N_1^\ast-\eta) >0$ $E_4^{nnv}=(0,N_2^\ast,V^\ast)$ $N_2^\ast <\tilde{N}_2$ ($r_2 >\omega$ required) $\left(\frac{\phi_1}{\phi_2}-\frac{r_1}{r_2}\right)(N_2^\ast-\eta) <0$ $\left(\frac{\phi_1}{\phi_2}-\frac{r_1}{r_2}\right)(r_1-r_2) >0$ $E_5^{nnv}=(N_1^c,N_2^c,V^c)$ $(N_1^\ast-\eta)\left(\frac{\phi_1}{\phi_2}-\frac{\beta_2}{\beta_1}\right) <0$ $\left(\frac{\phi_1}{\phi_2}-\frac{r_1}{r_2}\right)\left(\frac{\phi_1}{\phi_2}-\frac{\beta_2}{\beta_1}\right) >0$ $(N_2^\ast-\eta)\left(\frac{\phi_1}{\phi_2}-\frac{\beta_2}{\beta_1}\right) >0$
 Equilibrium Existence condition Stability condition $E_0^{nnv}=(0,0,0)$ $r_1 <\omega$, $r_2 <\omega$ $E_1^{nnv}=(\tilde{N}_1,0,0)$ $r_1 > \omega$ $r_1 >r_2$, $N_1^\ast >\tilde{N}_1$ $E_2^{nnv}=(0,\tilde{N}_2,0)$ $r_2 > \omega$ $r_1 \tilde{N}_2$ $E_3^{nnv}=(N_1^\ast,0,\tilde{V}^\ast)$ $N_1^\ast <\tilde{N}_1$ ($r_1 >\omega$ required) $\left(\frac{\phi_1}{\phi_2}-\frac{r_1}{r_2}\right)(N_1^\ast-\eta) >0$ $E_4^{nnv}=(0,N_2^\ast,V^\ast)$ $N_2^\ast <\tilde{N}_2$ ($r_2 >\omega$ required) $\left(\frac{\phi_1}{\phi_2}-\frac{r_1}{r_2}\right)(N_2^\ast-\eta) <0$ $\left(\frac{\phi_1}{\phi_2}-\frac{r_1}{r_2}\right)(r_1-r_2) >0$ $E_5^{nnv}=(N_1^c,N_2^c,V^c)$ $(N_1^\ast-\eta)\left(\frac{\phi_1}{\phi_2}-\frac{\beta_2}{\beta_1}\right) <0$ $\left(\frac{\phi_1}{\phi_2}-\frac{r_1}{r_2}\right)\left(\frac{\phi_1}{\phi_2}-\frac{\beta_2}{\beta_1}\right) >0$ $(N_2^\ast-\eta)\left(\frac{\phi_1}{\phi_2}-\frac{\beta_2}{\beta_1}\right) >0$
Global or local dynamics of (4). $E_0^{nnv}$-$E_5^{nnv}$ are defined in (5). Conditions for $E_5^{nnv}$ to be positive or not may not be all listed. "-" represents that some compartments of the equilibrium are negative. "U" represents "unstable"; "GAS" represents "globally asymptotically stable", "S" represents "locally asymptotically stable"
 Condition $E_0^{nnv}$ $E_1^{nnv}$ $E_2^{nnv}$ $E_3^{nnv}$ $E_4^{nnv}$ $E_5^{nnv}$ (a) $r_1 <\omega$, $r_2 <\omega$ GAS - - - - - (b) $r_2 <\omega \tilde{N}_1$ U GAS - - - - (c) $r_2 <\omega \tilde{N}_2$ U - GAS - - - (e) $r_1 <\omega \omega, N_1^\ast <\tilde{N}_1, N_2^\ast <\tilde{N}_2 \\ (\phi_1r_2-\phi_2r_1)(N_1^\ast-\eta) <0 \\ (\phi_1r_2-\phi_2r_1)(N_2^\ast-\eta) <0 \end{array}$ U U U U GAS - (g) $\begin{array}{l} r_1,r_2 >\omega, , N_1^\ast <\tilde{N}_1, N_2^\ast <\tilde{N}_2 \\ (\phi_1r_2-\phi_2r_1)(N_1^\ast-\eta) >0 \\ (\phi_1r_2-\phi_2r_1)(N_2^\ast-\eta) >0 \end{array}$ U U U GAS U - (h) $r_1 >r_2 >\omega$, $N_1^\ast >\tilde{N}_1$, $N_2^\ast <\tilde{N}_2$ U GAS U - U - (i) $\omega \tilde{N}_2$ U U GAS U - - (j) $r_1 >r_2 >\omega$, $N_1^\ast <\tilde{N}_1$, $N_2^\ast >\tilde{N}_2$ U U U GAS - - (k) $\omega \tilde{N}_1$, $N_2^\ast <\tilde{N}_2$ U U U - GAS - (l) $r_1 >r_2 >\omega$, $N_1^\ast >\tilde{N}_1$, $N_2^\ast >\tilde{N}_2$ U GAS U - - - (m) $\omega \tilde{N}_1$, $N_2^\ast >\tilde{N}_2$ U U GAS - - - (n) $\begin{array}{l} \mbox{(a) }r_1 >r_2 >\omega, \phi_1 r_2 >\phi_2r_1, \\ \tilde{N}_1 >N_1^\ast >\eta >N_2^\ast, \tilde{N}_2 >N_2^\ast; \\ \mbox{or (b) } \omega r_2 >\omega, \phi_1 r_2 >\phi_2r_1, \\ N_1^\ast <\eta \tilde{N}_1 >N_1^\ast >\eta >N_2^\ast \end{array}$ U U U U U S (p) $r_1 >r_2 >\omega$, $\phi_1 r_2 >\phi_2r_1$, U S U - S U $N_1^\ast >\tilde{N}_1 >\tilde{N}_2 >\eta >N_2^\ast$ (q) $\begin{array}{l} \omega r_2 >\omega, \phi_1 r_2 >\phi_2r_1, \\ N_1^\ast <\eta <\tilde{N}_2 <\tilde{N}_1, \tilde{N}_2 \tilde{N}_1, \tilde{N}_2 >\tilde{N}_1 >\eta >N_2^\ast \end{array}$ U U U - U S
 Condition $E_0^{nnv}$ $E_1^{nnv}$ $E_2^{nnv}$ $E_3^{nnv}$ $E_4^{nnv}$ $E_5^{nnv}$ (a) $r_1 <\omega$, $r_2 <\omega$ GAS - - - - - (b) $r_2 <\omega \tilde{N}_1$ U GAS - - - - (c) $r_2 <\omega \tilde{N}_2$ U - GAS - - - (e) $r_1 <\omega \omega, N_1^\ast <\tilde{N}_1, N_2^\ast <\tilde{N}_2 \\ (\phi_1r_2-\phi_2r_1)(N_1^\ast-\eta) <0 \\ (\phi_1r_2-\phi_2r_1)(N_2^\ast-\eta) <0 \end{array}$ U U U U GAS - (g) $\begin{array}{l} r_1,r_2 >\omega, , N_1^\ast <\tilde{N}_1, N_2^\ast <\tilde{N}_2 \\ (\phi_1r_2-\phi_2r_1)(N_1^\ast-\eta) >0 \\ (\phi_1r_2-\phi_2r_1)(N_2^\ast-\eta) >0 \end{array}$ U U U GAS U - (h) $r_1 >r_2 >\omega$, $N_1^\ast >\tilde{N}_1$, $N_2^\ast <\tilde{N}_2$ U GAS U - U - (i) $\omega \tilde{N}_2$ U U GAS U - - (j) $r_1 >r_2 >\omega$, $N_1^\ast <\tilde{N}_1$, $N_2^\ast >\tilde{N}_2$ U U U GAS - - (k) $\omega \tilde{N}_1$, $N_2^\ast <\tilde{N}_2$ U U U - GAS - (l) $r_1 >r_2 >\omega$, $N_1^\ast >\tilde{N}_1$, $N_2^\ast >\tilde{N}_2$ U GAS U - - - (m) $\omega \tilde{N}_1$, $N_2^\ast >\tilde{N}_2$ U U GAS - - - (n) $\begin{array}{l} \mbox{(a) }r_1 >r_2 >\omega, \phi_1 r_2 >\phi_2r_1, \\ \tilde{N}_1 >N_1^\ast >\eta >N_2^\ast, \tilde{N}_2 >N_2^\ast; \\ \mbox{or (b) } \omega r_2 >\omega, \phi_1 r_2 >\phi_2r_1, \\ N_1^\ast <\eta \tilde{N}_1 >N_1^\ast >\eta >N_2^\ast \end{array}$ U U U U U S (p) $r_1 >r_2 >\omega$, $\phi_1 r_2 >\phi_2r_1$, U S U - S U $N_1^\ast >\tilde{N}_1 >\tilde{N}_2 >\eta >N_2^\ast$ (q) $\begin{array}{l} \omega r_2 >\omega, \phi_1 r_2 >\phi_2r_1, \\ N_1^\ast <\eta <\tilde{N}_2 <\tilde{N}_1, \tilde{N}_2 \tilde{N}_1, \tilde{N}_2 >\tilde{N}_1 >\eta >N_2^\ast \end{array}$ U U U - U S
The conditions for existence and stability of equilibria of model (1). Here, an equilibrium exists means it is nonnegative for $E_1$-$E_8$ and positive for $E_9$. The notations are defined in (9) and (10)
 Equilibrium Existence condition Stability condition $E_0=(0,0,0,0)$ $r_1 <\omega$, $r_2 <\omega$ $E_1=(\tilde{N}_1,0,0,0)$ $r_1 > \omega$ $r_1 >r_2$, $N_{1,1}^\ast >\tilde{N}_1$, $N_{1,2}^\ast >\tilde{N}_1$ $E_2=(0,\tilde{N}_2,0,0)$ $r_2 > \omega$ $r_1 \tilde{N}_2$, $N_{2,2}^\ast >\tilde{N}_2$ $E_3=(N_{1,1}^\ast,0,\frac{r_1(\tilde{N}_1-N_{1,1}^\ast)}{K\phi_{11}},0)$ $N_{1,1}^\ast <\tilde{N}_1$ $\begin{array}{l} B\Phi_3 >0 \\ \Phi R_1\cdot (N_{1,1}^\ast-\eta_1) >0 \end{array}$ $E_4=(N_{1,2}^\ast,0,0,\frac{r_1(\tilde{N}_1-N_{1,2}^\ast)}{K\phi_{12}})$ $N_{1,2}^\ast <\tilde{N}_1$ $\begin{array}{l} B\Phi_3 <0 \\ \Phi R_2\cdot (N_{1,2}^\ast-\eta_2) >0 \end{array}$ $E_5=(0,N_{2,1}^\ast,\frac{r_2(\tilde{N}_2-N_{2,1}^\ast)}{K\phi_{21}},0)$ $N_{2,1}^\ast <\tilde{N}_2$ $\begin{array}{l} B\Phi_4 >0 \\ \Phi R_1\cdot (N_{2,1}^\ast-\eta_1) <0 \end{array}$ $E_6=(0,N_{2,2}^\ast,0,\frac{r_2(\tilde{N}_2-N_{2,2}^\ast)}{K\phi_{22}})$ $N_{2,2}^\ast <\tilde{N}_2$ $\begin{array}{l} B\Phi_4 <0 \\ \Phi R_2\cdot(N_{2,2}^\ast-\eta_2) <0 \end{array}$ $E_7=(N_1^c,N_2^c,V_1^c,0)$ $\begin{array}{l} (N_{2,1}^\ast-\eta_1)\cdot B\Phi_1 >0 \\ (N_{1,1}^\ast-\eta_1)\cdot B\Phi_1 <0 \\ (r_1-r_2)\Phi R_1 >0 \end{array}$ $\begin{array}{l} NN <0 \\ \Phi R_1\cdot B\Phi_1 >0 \end{array}$ $E_8=(\hat{N}_1^c,\hat{N}_2^c,0,\hat{V}_2^c)$ $\begin{array}{l} (N_{2,2}^\ast-\eta_2)\cdot B\Phi_2 >0 \\ (N_{1,2}^\ast-\eta_2)\cdot B\Phi_2 <0 \\ (r_1-r_2)\Phi R_2 >0 \end{array}$ $\begin{array}{l} NN_h <0 \\ \Phi R_2\cdot B\Phi_2 >0 \end{array}$ $E_9=(N_1^p,N_2^p,V_1^p,V_2^p)$ $\begin{array}{l}B\Phi\cdot B\Phi_3 <0 \\ B\Phi\cdot B\Phi_4 >0 \\ \Phi R_1\cdot B\Phi_1\cdot NN\cdot B\Phi \cdot \Phi\Phi >0 \\ \Phi R_2\cdot B\Phi_2\cdot NN_h\cdot B\Phi \cdot \Phi\Phi >0 \end{array}$ $\begin{array}{l} B\Phi \cdot \Phi\Phi >0 \\ (14) \end{array}$
 Equilibrium Existence condition Stability condition $E_0=(0,0,0,0)$ $r_1 <\omega$, $r_2 <\omega$ $E_1=(\tilde{N}_1,0,0,0)$ $r_1 > \omega$ $r_1 >r_2$, $N_{1,1}^\ast >\tilde{N}_1$, $N_{1,2}^\ast >\tilde{N}_1$ $E_2=(0,\tilde{N}_2,0,0)$ $r_2 > \omega$ $r_1 \tilde{N}_2$, $N_{2,2}^\ast >\tilde{N}_2$ $E_3=(N_{1,1}^\ast,0,\frac{r_1(\tilde{N}_1-N_{1,1}^\ast)}{K\phi_{11}},0)$ $N_{1,1}^\ast <\tilde{N}_1$ $\begin{array}{l} B\Phi_3 >0 \\ \Phi R_1\cdot (N_{1,1}^\ast-\eta_1) >0 \end{array}$ $E_4=(N_{1,2}^\ast,0,0,\frac{r_1(\tilde{N}_1-N_{1,2}^\ast)}{K\phi_{12}})$ $N_{1,2}^\ast <\tilde{N}_1$ $\begin{array}{l} B\Phi_3 <0 \\ \Phi R_2\cdot (N_{1,2}^\ast-\eta_2) >0 \end{array}$ $E_5=(0,N_{2,1}^\ast,\frac{r_2(\tilde{N}_2-N_{2,1}^\ast)}{K\phi_{21}},0)$ $N_{2,1}^\ast <\tilde{N}_2$ $\begin{array}{l} B\Phi_4 >0 \\ \Phi R_1\cdot (N_{2,1}^\ast-\eta_1) <0 \end{array}$ $E_6=(0,N_{2,2}^\ast,0,\frac{r_2(\tilde{N}_2-N_{2,2}^\ast)}{K\phi_{22}})$ $N_{2,2}^\ast <\tilde{N}_2$ $\begin{array}{l} B\Phi_4 <0 \\ \Phi R_2\cdot(N_{2,2}^\ast-\eta_2) <0 \end{array}$ $E_7=(N_1^c,N_2^c,V_1^c,0)$ $\begin{array}{l} (N_{2,1}^\ast-\eta_1)\cdot B\Phi_1 >0 \\ (N_{1,1}^\ast-\eta_1)\cdot B\Phi_1 <0 \\ (r_1-r_2)\Phi R_1 >0 \end{array}$ $\begin{array}{l} NN <0 \\ \Phi R_1\cdot B\Phi_1 >0 \end{array}$ $E_8=(\hat{N}_1^c,\hat{N}_2^c,0,\hat{V}_2^c)$ $\begin{array}{l} (N_{2,2}^\ast-\eta_2)\cdot B\Phi_2 >0 \\ (N_{1,2}^\ast-\eta_2)\cdot B\Phi_2 <0 \\ (r_1-r_2)\Phi R_2 >0 \end{array}$ $\begin{array}{l} NN_h <0 \\ \Phi R_2\cdot B\Phi_2 >0 \end{array}$ $E_9=(N_1^p,N_2^p,V_1^p,V_2^p)$ $\begin{array}{l}B\Phi\cdot B\Phi_3 <0 \\ B\Phi\cdot B\Phi_4 >0 \\ \Phi R_1\cdot B\Phi_1\cdot NN\cdot B\Phi \cdot \Phi\Phi >0 \\ \Phi R_2\cdot B\Phi_2\cdot NN_h\cdot B\Phi \cdot \Phi\Phi >0 \end{array}$ $\begin{array}{l} B\Phi \cdot \Phi\Phi >0 \\ (14) \end{array}$
 [1] Tao Feng, Zhipeng Qiu, Xinzhu Meng. Dynamics of a stochastic hepatitis C virus system with host immunity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6367-6385. doi: 10.3934/dcdsb.2019143 [2] Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046 [3] Cameron J. Browne, Sergei S. Pilyugin. Global analysis of age-structured within-host virus model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 1999-2017. doi: 10.3934/dcdsb.2013.18.1999 [4] Shikun Wang. Dynamics of a chemostat system with two patches. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6261-6278. doi: 10.3934/dcdsb.2019138 [5] Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020344 [6] Hui li, Manjun Ma. Global dynamics of a virus infection model with repulsive effect. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4783-4797. doi: 10.3934/dcdsb.2019030 [7] Zhikun She, Xin Jiang. Threshold dynamics of a general delayed within-host viral infection model with humoral immunity and two modes of virus transmission. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020259 [8] Tin Phan, Bruce Pell, Amy E. Kendig, Elizabeth T. Borer, Yang Kuang. Rich dynamics of a simple delay host-pathogen model of cell-to-cell infection for plant virus. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020261 [9] Hossein Pourbashash, Sergei S. Pilyugin, Patrick De Leenheer, Connell McCluskey. Global analysis of within host virus models with cell-to-cell viral transmission. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3341-3357. doi: 10.3934/dcdsb.2014.19.3341 [10] Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735 [11] Stephen Pankavich, Nathan Neri, Deborah Shutt. Bistable dynamics and Hopf bifurcation in a refined model of early stage HIV infection. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2867-2893. doi: 10.3934/dcdsb.2020044 [12] Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325 [13] Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051 [14] Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507 [15] Tewfik Sari, Frederic Mazenc. Global dynamics of the chemostat with different removal rates and variable yields. Mathematical Biosciences & Engineering, 2011, 8 (3) : 827-840. doi: 10.3934/mbe.2011.8.827 [16] Yan-Xia Dang, Zhi-Peng Qiu, Xue-Zhi Li, Maia Martcheva. Global dynamics of a vector-host epidemic model with age of infection. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1159-1186. doi: 10.3934/mbe.2017060 [17] Cuicui Jiang, Kaifa Wang, Lijuan Song. Global dynamics of a delay virus model with recruitment and saturation effects of immune responses. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1233-1246. doi: 10.3934/mbe.2017063 [18] Cuicui Jiang, Wendi Wang. Complete classification of global dynamics of a virus model with immune responses. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1087-1103. doi: 10.3934/dcdsb.2014.19.1087 [19] Feng-Bin Wang, Junping Shi, Xingfu Zou. Dynamics of a host-pathogen system on a bounded spatial domain. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2535-2560. doi: 10.3934/cpaa.2015.14.2535 [20] Xiulan Lai, Xingfu Zou. A reaction diffusion system modeling virus dynamics and CTLs response with chemotaxis. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2567-2585. doi: 10.3934/dcdsb.2016061

2019 Impact Factor: 1.27

## Tools

Article outline

Figures and Tables