# American Institute of Mathematical Sciences

## A viral transmission model for foxes-cottontails-hares interaction: Infection through predation

 1 Dipartimento di Matematica "Giuseppe Peano", Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy 2 Department of Life Sciences and Systems Biology, University of Turin, via Accademia Albertina 13, 10123 Torino, Italy

* Corresponding author: Ezio Venturino

Received  November 2020 Revised  April 2021 Early access  June 2021

Fund Project: The last author has been partially supported by the project "Metodi numerici nelle scienze applicate" of the Dipartimento di Matematica "Giuseppe Peano" of the Università di Torino
EV is a member of the research group GNCS

The Eastern cottontail Sylvilagus floridanus is a lagomorph native to North America, introduced in Italy since the 1960s. In Central and Northern Italy, the cottontail overlaps its range with the native European hare Lepus europaeus and affects the predator-prey dynamics of native hares and foxes. Field data indicate that the cottontail is susceptible to infection by the European brown hare syndrome (EBHS) virus. Although the real role of cottontails and native foxes in the spreading of EBHS viruses is yet uncertain, we present a cottontail-hare-fox model including possible effects of EBHS, imported by foxes, through environmental contamination. A rather complete map of the possible system equilibria and their mutual relationship and transition is established.

Citation: Simona Viale, Elisa Caudera, Sandro Bertolino, Ezio Venturino. A viral transmission model for foxes-cottontails-hares interaction: Infection through predation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021158
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##### References:
Equilibrium $E_4$, attained for the parameter values $r = \log(3)$, $s = \log(4.5)$, $u = \log(5)$, $r_H = 0.3$, $r_U = 0.9$, $r_I = 0.4$, $m = \frac 27$, $\mu = 0.1$, $n = 4/5$, $p = 2/11$, $a = 0.2$, $b = 0.1$, $c = 0.2$, $h = 0.4$, $e = 0.91$, $c_{VV} = \log(3)-\frac 27$, $c_{VU} = c_{VV}-0.1$, $c_{UU} = \log(3)-\frac 27$, $c_{UV} = c_{UU}+0.1$, $c_{SS} = \log(4.5)/100-4/500$, $c_{LL} = \log(5)/30-2/330$, $\zeta = 0.4$, $\theta = 0.7$, $\xi = 3$, $\eta = 2$ and initial conditions (29)
Equilibrium $E_7$, attained for the parameter values $r = \log(3)$, $s = \log(4.5)$, $u = \log(5)$, $r_H = 0.3$, $r_U = 0.9$, $r_I = 0.4$, $m = 3$, $\mu = 0.1$, $n = 4/5$, $p = 2/11$, $a = 5$, $b = 3$, $c = 4$, $h = 4$, $e = 0.91$, $c_{VV} = \log(3)-\frac 27$, $c_{VU} = c_{VV}-0.1$, $c_{UU} = \log(3)-\frac 27$, $c_{UV} = c_{UU}+0.1$, $c_{SS} = \log(4.5)/100-4/500$, $c_{LL} = \log(5)/30-2/330$, $\zeta = 0.8$, $\theta = 0.7$, $\xi = 0.3$, $\eta = 0.2$ and initial conditions (29). Interestingly, this equilibrium shows damped oscillations
Equilibrium $E_9$. Coexistence is attained for the parameter values $r = \log(3)$, $s = \log(4.5)$, $u = \log(5)$, $r_H = 0.3$, $r_U = 0.9$, $r_I = 0.4$, $m = \frac 27$, $\mu = 0.1$, $n = 4/5$, $p = 2/11$, $a = 0.2$, $b = 0.1$, $c = 0.2$, $h = 0.4$, $e = 0.91$, $c_{VV} = \log(3)-\frac 27$, $c_{VU} = c_{VV}-0.1$, $c_{UU} = \log(3)-\frac 27$, $c_{UV} = c_{UU}+0.1$, $c_{SS} = \log(4.5)/100-4/500$, $c_{LL} = \log(5)/30-2/330$, $\zeta = 0.4$, $\theta = 0.7$, $\xi = 0.3$, $\eta = 0.2$ and initial conditions (29)
Picture of the whole bifurcation structure of the model (17) arising from the analysis of Section 4. In each node, the equilibrium name is reported, together with its nonvanishing populations. The red lines indicate the bifurcations that have been found only numerically. Bifurcation thresholds: $r_{0, 1} = m$, $\mu_{1, 2} = \zeta r_U-c_{UV}V_1-m$, $s_{1, 4} = n+a V_1$, $u_{1, 6} = p+b V_1$, $s_{2, 4} = n+a V_2+c U_2$, $u_{2, 7} = p+bV_2+hU_2+\eta \frac{U_2}{V_2+U_2}$, $\mu_{0, 2} = \zeta r_U-m$, $s_{0, 3} = n$, $u_{3, 8} = p+\xi$, $u_{0, 5} = p$, $r_{5, 6} = m-r_H be L_5$, $s_{5, 8} = n$, $\mu_{5, 7} = \zeta r_U-m+\zeta r_I he L_5$, $\mu_{6, 7} = \zeta r_U-c_{UV} V_6-m+\zeta r_I he L_6$, $s_{6, 9} = n+a V_6$, $u_{4, 9} = p+bV_4+hU_4+\frac{\xi S_4+\eta U_4}{V_4+U_4+S_4}$, $s_{7, 9} = n+aV_7+cU_7$. The asymmetry in the graph could be interpreted biologically as follows. Consider for instance the absence of the arc $E_8-E_4$ while arc $E_2-E_7$ is present. Both apparently connect points with 2 and 3 populations, but in reality $E_2$ contains only foxes, susceptible $V$ and infected $U$, while $E_8$ has two different system populations, cottontails $S$ and hares $L$. A similar claim can be made for the pairs $E_0-E_8$ and $E_0-E_2$
Transcritical bifurcation relating equilibria $E_2$ and $E_0$ of the system (17) in terms of the bifurcation parameter $\mu \in [1, 4]$, the foxes virus-related mortality
Transcritical bifurcation relating equilibria $E_3$ and $E_4$ of the system (17) in terms of the healthy foxes reproduction rate $r\in [1, 3.5]$ as bifurcation parameter. The parameters are given by (28) with the exception of those in (31)
Transcritical bifurcation connecting equilibria $E_8$ and $E_9$ of the system (17) for the bifurcation parameter given by the healthy foxes reproduction rate $r \in [3, 6]$. The parameters are given by (28) with the exception of those in (32)
The model parameters and their meaning
 Parameter Interpretation $r$ healthy foxes reproduction rate on other resources $r_U$ infected foxes reproduction rate on other resources $r_H$ healthy foxes reproduction rate on cottontails and hares $r_I$ infected foxes reproduction rate on cottontails and hares $m$ foxes natural mortality rate $c_{UU}$, $c_{VV}$, $c_{UV}$, $c_{VU}$ foxes intraspecific competition coefficients $\zeta \le 1$ foxes vertical virus transmission $\theta$ foxes transmission rate by feeding on infected cottontail $a$ healthy foxes predation rate on cottontails $b$ healthy foxes predation rate on hares $c$ infected foxes predation rate on cottontails $h$ infected foxes predation rate on hares $e$ foxes conversion coefficient of captured prey $\mu$ foxes mortality rate due to virus, possibly $\mu=0$ $s$ cottontails reproduction rate $c_{SS}$ cottontails intraspecific competition $n$ cottontails natural mortality rate $u$ hares reproduction rate $c_{LL}$ hares intraspecific competition $p$ hares natural mortality rate $\xi$ hares infection, i.e. direct mortality rate, by cottontails environment pollution $\eta$ hares infection, i.e. direct mortality rate, by infected foxes environment pollution
 Parameter Interpretation $r$ healthy foxes reproduction rate on other resources $r_U$ infected foxes reproduction rate on other resources $r_H$ healthy foxes reproduction rate on cottontails and hares $r_I$ infected foxes reproduction rate on cottontails and hares $m$ foxes natural mortality rate $c_{UU}$, $c_{VV}$, $c_{UV}$, $c_{VU}$ foxes intraspecific competition coefficients $\zeta \le 1$ foxes vertical virus transmission $\theta$ foxes transmission rate by feeding on infected cottontail $a$ healthy foxes predation rate on cottontails $b$ healthy foxes predation rate on hares $c$ infected foxes predation rate on cottontails $h$ infected foxes predation rate on hares $e$ foxes conversion coefficient of captured prey $\mu$ foxes mortality rate due to virus, possibly $\mu=0$ $s$ cottontails reproduction rate $c_{SS}$ cottontails intraspecific competition $n$ cottontails natural mortality rate $u$ hares reproduction rate $c_{LL}$ hares intraspecific competition $p$ hares natural mortality rate $\xi$ hares infection, i.e. direct mortality rate, by cottontails environment pollution $\eta$ hares infection, i.e. direct mortality rate, by infected foxes environment pollution
Equilibria feasibility
 Equilibrium Feasibility conditions $E_0$ $-$ $E_1$ $r \ge m$ $E_2^-$ (23), (24), (25) $E_2^{\pm}$ sufficent (26) $E_3$ $s \ge n$, $E_4$ numerical $E_5$ $u \ge p$ $E_6$ (27) $E_7$ numerical $E_8^+$ sufficient: $u\ge p+\xi$ $E_8^{\pm}$ $u<\xi$, $(u-p)^2 \ge 4 c_{LL} S_8 (p+\xi-u)$ $E_9$ numerical
 Equilibrium Feasibility conditions $E_0$ $-$ $E_1$ $r \ge m$ $E_2^-$ (23), (24), (25) $E_2^{\pm}$ sufficent (26) $E_3$ $s \ge n$, $E_4$ numerical $E_5$ $u \ge p$ $E_6$ (27) $E_7$ numerical $E_8^+$ sufficient: $u\ge p+\xi$ $E_8^{\pm}$ $u<\xi$, $(u-p)^2 \ge 4 c_{LL} S_8 (p+\xi-u)$ $E_9$ numerical
Equilibria stability
 Equilibria Stability conditions $E_0$ $m>r$, $s>n$, $u>p$ $E_1$ $\max \left\{ { \frac {\zeta r_U - m - \mu} {c_{UV} } , \frac {s - n} a , \frac {u - p} b } \right\}< V_1$ $E_2$ $s< n+aV_2+cU_2, \quad u< p+bV_2+hU_2+\eta \frac {U_2}{U_2+V_2}$, (39) $E_3$ $n< s$, $u< p + \xi$, (41) $E_4$ $u< p+bV_4+hU_4 + \frac{\xi S_4 + \eta U_4}{V_4+U_4+S_4}$, (43) $E_5$ $r+r_H ebL_5< m$, $\zeta r_U + \zeta r_I he L_5< m+\mu$, $s< n$ $E_6$ $\zeta r_U-c_{UV}V_6+\zeta r_I he L_6< m+\mu$, $s< n + a V_6$ $E_7$ $s< n + aV_7 + cU_7$, (47) $E_8$ $\frac{\xi S_8}{(S_8+L_8)^2}< c_{LL}$, $n< s$, (49) $E_9$ numerical
 Equilibria Stability conditions $E_0$ $m>r$, $s>n$, $u>p$ $E_1$ $\max \left\{ { \frac {\zeta r_U - m - \mu} {c_{UV} } , \frac {s - n} a , \frac {u - p} b } \right\}< V_1$ $E_2$ $s< n+aV_2+cU_2, \quad u< p+bV_2+hU_2+\eta \frac {U_2}{U_2+V_2}$, (39) $E_3$ $n< s$, $u< p + \xi$, (41) $E_4$ $u< p+bV_4+hU_4 + \frac{\xi S_4 + \eta U_4}{V_4+U_4+S_4}$, (43) $E_5$ $r+r_H ebL_5< m$, $\zeta r_U + \zeta r_I he L_5< m+\mu$, $s< n$ $E_6$ $\zeta r_U-c_{UV}V_6+\zeta r_I he L_6< m+\mu$, $s< n + a V_6$ $E_7$ $s< n + aV_7 + cU_7$, (47) $E_8$ $\frac{\xi S_8}{(S_8+L_8)^2}< c_{LL}$, $n< s$, (49) $E_9$ numerical
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