# American Institute of Mathematical Sciences

January  2021, 26(1): 515-539. doi: 10.3934/dcdsb.2020261

## Rich dynamics of a simple delay host-pathogen model of cell-to-cell infection for plant virus

 1 School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA 2 Department of Mathematics and Computer Science, Lawrence Technological University, Southfield, MI 48075, USA 3 Agronomy Department, University of Florida, Gainesville, FL 32611, USA 4 Department of Ecology, Evolution, and Behavior, University of Minnesota, St. Paul, MN 55108, USA

* Corresponding author: Yang Kuang

Received  March 2020 Revised  July 2020 Published  August 2020

Fund Project: TP and YK are supported by NSF grants DMS-1615879 and DEB-1930728. YK is also partially supported by the NIGMS of the National Institutes of Health (NIH) under award number R01GM131405

Viral dynamics within plant hosts can be important for understanding plant disease prevalence and impacts. However, few mathematical modeling efforts aim to characterize within-plant viral dynamics. In this paper, we derive a simple system of delay differential equations that describes the spread of infection throughout the plant by barley and cereal yellow dwarf viruses via the cell-to-cell mechanism. By incorporating ratio-dependent incidence function and logistic growth of the healthy cells, the model can capture a wide range of biologically relevant phenomena via the disease-free, endemic, mutual extinction steady states, and a stable periodic orbit. We show that when the basic reproduction number is less than $1$ ($R_0 < 1$), the disease-free steady state is asymptotically stable. When $R_0>1$, the dynamics either converge to the endemic equilibrium or enter a periodic orbit. Using a ratio-dependent transformation, we show that if the infection rate is very high relative to the growth rate of healthy cells, then the system collapses to the mutual extinction steady state. Numerical and bifurcation simulations are provided to demonstrate our theoretical results. Finally, we carry out parameter estimation using experimental data to characterize the effects of varying nutrients on the dynamics of the system. Our parameter estimates suggest that varying the nutrient supply of nitrogen and phosphorous can alter the dynamics of the infection in plants, specifically reducing the rate of viral production and the rate of infection in certain cases.

Citation: 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, 2021, 26 (1) : 515-539. doi: 10.3934/dcdsb.2020261
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Increasing $\tau$ changes the stability of the positive equilibrium, which gives rise to a stable orbit. For this simulation, we use $r = 0.3,K = 10^3,\beta = 0.1,\delta = 0.0001$ and $\tau$ varies from $1$ to $80$. We plot $\tau$ over a viable region. For smaller value of $\tau$, either the condition for the theorem is not satisfied or the positive steady does not exist. The switching between a stable positive steady state and a stable orbit takes place around $\tau \approx 50.5$
Corresponding examples for Figure 1. (a) $\tau = 50$, the oscillation is damping toward the positive steady state. (b) $\tau = 51$, the oscillation is stable
For this simulation, we start with the following values $r = 0.3,K = 10^3,\beta = 0.1,\delta = 0.0001$ and $\tau = 51$. (a) increasing the infection rate $\beta$ can have a destabilizing effect on the endemic equilibrium; however, as $\beta$ increases, $S$ and $I$ approach closely to 0. (b) Decreasing the death rate $\delta$ can be destabilizing as well. (c) The growth rate $r$ can be both stabilizing or destabilizing as it varies. As $r$ decreases, it can result in mutual extinction. (d) shows additional details of Figure 1 and 2. Note that varying the carrying capacity $K$ only changes the size but not the stability
Parameter fitting result for the cell-to-cell transmission model. The description of each experiment is given in subsection 1.2 and additional details can be found in Kendig et al. [26]
Comparison of data fitting between mass action model and ratio-dependent model. The description of each experiment is given in subsection 1.2 and additional details can be found in Kendig et al. [26]
Estimated parameter for cell-to-cell model. Note that $\delta$ is fixed to be $1/13$ day$^{-1}$. The value of $R_0$ is calculated based on the estimated parameters. The description of each experiment is given in subsection 1.2 and additional details can be found in Kendig et al. [26]
 Parameter Fitted (CTRL) Fitted (+N) Fitted (+P) Fitted (+NP) Units $r$ 0.9000 0.9000 0.9000 0.8860 day$^{-1}$ $K$ 515024 719563 400294 400000 cells $\beta$ 0.5387 0.4355 0.8925 0.6710 cells virion$^{-1}$ day$^{-1}$ $b$ 65 94 62 80 virions cell$^{-1}$ day$^{-1}$ $\tau$ 8.27 12.00 12.00 12.00 days $R_0$ 1.62 2.07 2.30 2.23 unitless
 Parameter Fitted (CTRL) Fitted (+N) Fitted (+P) Fitted (+NP) Units $r$ 0.9000 0.9000 0.9000 0.8860 day$^{-1}$ $K$ 515024 719563 400294 400000 cells $\beta$ 0.5387 0.4355 0.8925 0.6710 cells virion$^{-1}$ day$^{-1}$ $b$ 65 94 62 80 virions cell$^{-1}$ day$^{-1}$ $\tau$ 8.27 12.00 12.00 12.00 days $R_0$ 1.62 2.07 2.30 2.23 unitless
Fitting errors for the cell-to-cell transmission model. The description of each experiment is given in subsection 1.2 and additional details can be found in Kendig et al. [26]
 Experiment control +N +P +NP RMSE 4.27e+6 5.97e+6 3.66e+6 8.96e+6 MAPE 8.03e-1 6.35e-1 4.10e-1 7.14e-1
 Experiment control +N +P +NP RMSE 4.27e+6 5.97e+6 3.66e+6 8.96e+6 MAPE 8.03e-1 6.35e-1 4.10e-1 7.14e-1
Stability results and open questions in terms of $\beta, \delta, r$ and $\tau$ $\left(\text{note: } R_0 = \frac{\beta-\delta}{\beta e^{-\delta\tau}}\right)$
 Conditions Results or question 1. $\beta<\delta$ $(K,0)$ is globally asymptotically stable 2. $\beta>\delta$ and $\frac{\beta-\delta}{\beta e^{-\delta\tau}}<1$ $(K,0)$ is locally asymptotically stable 3. $\beta>\delta$ and $\frac{\beta-\delta}{\beta-\delta-r}>\frac{\beta-\delta}{\beta e^{-\delta\tau}}>1$ Open question 1: is $E^*$ stable? when does a periodic orbit occurs? 4. $\beta>\delta+r$ and $\frac{\beta-\delta}{\beta-\delta-r}<\frac{\beta-\delta}{\beta e^{-\delta\tau}}$ (0, 0) is globally asymptotically stable
 Conditions Results or question 1. $\beta<\delta$ $(K,0)$ is globally asymptotically stable 2. $\beta>\delta$ and $\frac{\beta-\delta}{\beta e^{-\delta\tau}}<1$ $(K,0)$ is locally asymptotically stable 3. $\beta>\delta$ and $\frac{\beta-\delta}{\beta-\delta-r}>\frac{\beta-\delta}{\beta e^{-\delta\tau}}>1$ Open question 1: is $E^*$ stable? when does a periodic orbit occurs? 4. $\beta>\delta+r$ and $\frac{\beta-\delta}{\beta-\delta-r}<\frac{\beta-\delta}{\beta e^{-\delta\tau}}$ (0, 0) is globally asymptotically stable
Estimated parameter for mass action model. The description of each experiment is given in subsection 1.2 and additional details can be found in Kendig et al. [26]
 Parameter Fitted (CTRL) Fitted (+N) Fitted (+P) Fitted (+NP) Units $r$ 0.0993 0.0100 0.8579 0.1549 day$^{-1}$ $K$ 4.0000e+5 6.0164e+5 4.0038e+5 1.0987e+6 cells $\beta$ 2.0273e-6 2.8651e-7 1.9817e-6 1.9188e-6 cells virion$^{-1}$ day$^{-1}$ $d$ 0.7129 0.1001 0.1001 0.1001 day$^{-1}$ $b$ 118.2189 199.9803 60.4637 56.2613 virions cell$^{-1}$ day$^{-1}$ $\tau$ 9.6880 21.0000 4.9741 7.4480 days
 Parameter Fitted (CTRL) Fitted (+N) Fitted (+P) Fitted (+NP) Units $r$ 0.0993 0.0100 0.8579 0.1549 day$^{-1}$ $K$ 4.0000e+5 6.0164e+5 4.0038e+5 1.0987e+6 cells $\beta$ 2.0273e-6 2.8651e-7 1.9817e-6 1.9188e-6 cells virion$^{-1}$ day$^{-1}$ $d$ 0.7129 0.1001 0.1001 0.1001 day$^{-1}$ $b$ 118.2189 199.9803 60.4637 56.2613 virions cell$^{-1}$ day$^{-1}$ $\tau$ 9.6880 21.0000 4.9741 7.4480 days
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