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January  2021, 26(1): 415-441. doi: 10.3934/dcdsb.2020298

## 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, 2021, 26 (1) : 415-441. doi: 10.3934/dcdsb.2020298
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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 $E_0=(0,0,0,0)$ $r_1 <\omega$, $r_2 <\omega$ $E_1=(\tilde{N}_1,0,0,0)$ $r_1 > \omega$ $r_1 >r_2$, $N_{1,1}^\ast >\tilde{N}_1$, $N_{1,2}^\ast >\tilde{N}_1$ $E_2=(0,\tilde{N}_2,0,0)$ $r_2 > \omega$ $r_1 \tilde{N}_2$, $N_{2,2}^\ast >\tilde{N}_2$ $E_3=(N_{1,1}^\ast,0,\frac{r_1(\tilde{N}_1-N_{1,1}^\ast)}{K\phi_{11}},0)$ $N_{1,1}^\ast <\tilde{N}_1$ $\begin{array}{l} B\Phi_3 >0 \\ \Phi R_1\cdot (N_{1,1}^\ast-\eta_1) >0 \end{array}$ $E_4=(N_{1,2}^\ast,0,0,\frac{r_1(\tilde{N}_1-N_{1,2}^\ast)}{K\phi_{12}})$ $N_{1,2}^\ast <\tilde{N}_1$ $\begin{array}{l} B\Phi_3 <0 \\ \Phi R_2\cdot (N_{1,2}^\ast-\eta_2) >0 \end{array}$ $E_5=(0,N_{2,1}^\ast,\frac{r_2(\tilde{N}_2-N_{2,1}^\ast)}{K\phi_{21}},0)$ $N_{2,1}^\ast <\tilde{N}_2$ $\begin{array}{l} B\Phi_4 >0 \\ \Phi R_1\cdot (N_{2,1}^\ast-\eta_1) <0 \end{array}$ $E_6=(0,N_{2,2}^\ast,0,\frac{r_2(\tilde{N}_2-N_{2,2}^\ast)}{K\phi_{22}})$ $N_{2,2}^\ast <\tilde{N}_2$ $\begin{array}{l} B\Phi_4 <0 \\ \Phi R_2\cdot(N_{2,2}^\ast-\eta_2) <0 \end{array}$ $E_7=(N_1^c,N_2^c,V_1^c,0)$ $\begin{array}{l} (N_{2,1}^\ast-\eta_1)\cdot B\Phi_1 >0 \\ (N_{1,1}^\ast-\eta_1)\cdot B\Phi_1 <0 \\ (r_1-r_2)\Phi R_1 >0 \end{array}$ $\begin{array}{l} NN <0 \\ \Phi R_1\cdot B\Phi_1 >0 \end{array}$ $E_8=(\hat{N}_1^c,\hat{N}_2^c,0,\hat{V}_2^c)$ $\begin{array}{l} (N_{2,2}^\ast-\eta_2)\cdot B\Phi_2 >0 \\ (N_{1,2}^\ast-\eta_2)\cdot B\Phi_2 <0 \\ (r_1-r_2)\Phi R_2 >0 \end{array}$ $\begin{array}{l} NN_h <0 \\ \Phi R_2\cdot B\Phi_2 >0 \end{array}$ $E_9=(N_1^p,N_2^p,V_1^p,V_2^p)$ $\begin{array}{l}B\Phi\cdot B\Phi_3 <0 \\ B\Phi\cdot B\Phi_4 >0 \\ \Phi R_1\cdot B\Phi_1\cdot NN\cdot B\Phi \cdot \Phi\Phi >0 \\ \Phi R_2\cdot B\Phi_2\cdot NN_h\cdot B\Phi \cdot \Phi\Phi >0 \end{array}$ $\begin{array}{l} B\Phi \cdot \Phi\Phi >0 \\ (14) \end{array}$
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