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# A comparison of deterministic and stochastic predator-prey models with disease in the predator

• * Corresponding author: Hongxiao Hu

This work was supported by the National Science Foundation of China under grants 11401382, 11501518 and 11461073

• In this paper, we study the dynamics of deterministic and stochastic models for a predator-prey, where the predator species is subject to an SIS form of parasitic infection. The deterministic model is a system of ordinary differential equations for a predator-prey model with disease in the predator only. The existence and local stability of the boundary equilibria and the uniform persistence for the ODE model are investigated. Based on these results, some threshold values for successful invasion of disease or prey species are obtained. A new stochastic model is derived in the form of continuous-time Markov chains. Branching process theory is applied to the continuous-time Markov chain models to estimate the probabilities for disease outbreak or prey species invasion. The deterministic and stochastic threshold theories are compared and some relationships between the deterministic and stochastic thresholds are derived. Finally, some numerical simulations are introduced to illustrate the main results and to highlight some of the differences between the deterministic and stochastic models.

Mathematics Subject Classification: Primary: 65C30, 60H10; Secondary: 46N60.

 Citation:

• Figure 1.

Figure 2.  Comparison of ODE and CTMC (3 sample paths) solutions for Case 1. We start the simulation at $x_1(0) = 255$, $S(0) = 355$, $I(0) = 1$

Figure 3.  Qqplot (see R statistical software) of an outbreak sample paths for $x_1$, $S$ and $I$ of Case 1 when $I(0) = 1$ based on 5000 sample paths at $t = 50$

Figure 4.  Solution of ODE model for Case 2. We start the simulation at $x_1(0) = 1$, $S(0) = 92$, $I(0) = 218$

Figure 5.  Comparison of ODE and CTMC (3 sample paths) solutions for Case 2. We start the simulation at $x_1(0) = 1$, $S(0) = 92$, $I(0) = 218$

Figure 6.  Comparison of ODE and CTMC (3 sample paths) solutions for Case 3. We start the simulation at $x_1(0) = 1$, $S(0) = 92$, $I(0) = 218(P_0' = 0.335)$

Figure 7.  Qqplot of successful invasion sample paths for $x_1$, $S$ and $I$ of Case 3 when $x_1(0) = 1$ based on 5000 sample paths at $t = 50$

Figure 8.  Case 1: Boxplot of an outbreak sample paths for prey species $x_1$, susceptible predator species $S$ and infected predator species $I$ with initial value $I(0) = 1$ (first line) and $I(0) = 2$ (second line) based on 5000 sample paths at time $t = 30, 40, 50$

Figure 9.  Case 3: Boxplot of successful invasion sample paths for prey species $x_1$, susceptible predator species $S$ and infected predator species $I$ with initial value $x_1(0) = 1$ (first line) and $x_2(0) = 2$ (second line) based on 5000 sample paths at time $t = 30, 40, 50$

Figure 10.  Case 1: Histogram of the probability density function for an outbreak sample paths for prey species $x_1$ (first and second lines), susceptible predator species $S$ (third and fourth lines) and infected predator species $I$ (fifth and sixth lines) of Case 1 when $I(0) = 1$ (odd number lines) and when $I(0) = 2$ (even number lines) based on 5000 sample paths at $t = 30$ (left column), $t = 40$ (middle column), $t = 50$(right column)

Figure 11.  Case 3: Histogram of the probability density function for successful invasion sample paths for prey species $x_1$ (first and second lines), susceptible predator species $S$ (third and fourth lines) and infected predator species $I$ (fifth and sixth lines) of Case 3 when $x_1(0) = 1$ (odd number lines) and when $x_1(0) = 2$ (even number lines) based on 5000 sample paths at $t = 30$ (left column), $t = 40$ (middle column), $t = 50$(right column)

Table 1.  State transitions and the infinitesimal probabilities for the CTMC epidemic model.

 Description State transition $a \to b$ Rate $P(a, b)$ 1 Birth of $S$ $(x_1, S, I) \to (x_1, S+1, I)$ $P_1=x_2(b_2-c_2a_2x_2)$ 2 Death of $S$ $(x_1, S, I) \to (x_1, S-1, I)$ $P_2=S(d_2+(1-c_2)a_2x_2-\varepsilon\eta x_1)$ 3 Infection $(x_1, S, I) \to (x_1, S-1, I+1)$ $P_3=\beta SI$ 4 Death of $I$ $(x_1, S, I) \to (x_1, S, I-1)$ $P_4=I(d_2+(1-c_2)a_2x_2+\mu-pq\varepsilon\eta x_1)$ 5 Recover of $I$ $(x_1, S, I) \to (x_1, S+1, I-1)$ $P_5=\gamma I$ 6 Birth of $x_1$ $(x_1, S, I) \to (x_1+1, S, I)$ $P_6=x_1(b_1-c_1a_1x_1)$ 7 Death of $x_1$ $(x_1, S, I) \to (x_1-1, S, I)$ $P_7=x_1(d_1+(1-c_1)a_1x_1+\eta(S+qI))$

Table 2.  Parameter Values for Cases 1, 2 and 3

 Parameter Interpretation Case 1 Case2 Case3 $a_1$ Density dependent of prey $0.0005$ $0.0005$ $0.0005$ $a_2$ Density dependent of predator $0.0006$ $0.0006$ $0.0006$ $b_1$ Intrinsic birth rate of prey $2$ $0.7795$ $2$ $b_2$ Intrinsic birth rate of predator $1$ $1$ $1$ $c_1$ Density dependence effects of prey $0.5$ $0.5$ $0.5$ $c_2$ Density dependence effects of predator $0.5$ $0.5$ $0.5$ $d_1$ Natural mortality of prey $0.1$ $0.1$ $0.1$ $d_2$ Natural mortality of predator $0.8$ $0.8$ $0.8$ $\eta$ Predation rate of susceptible predator $0.005$ $0.005$ $0.005$ $\varepsilon$ Conversion rate of susceptible predator $0.01$ $0.01$ $0.01$ $p\eta$ Predation rate of infected predator $0.0005$ $0.0005$ $0.0005$ $q\varepsilon$ Conversion rate of infected predator $0.002$ 0.002 $0.001$ $\beta$ Transmission $0.01$ $0.01$ $0.01$ $\gamma$ Recover rate $0.01$ $0.01$ $0.01$ $\mu$ Disease related mortality $0.02$ $0.02$ $0.02$

Table 3.  Equilibria in the form $(x_1, S, I)$ and their local stability for the ODE model (2) with parameters given in Table 2, U = unstable, S = stable

 Case 1 Case 2 Case 3 Equilibria S/U Equilibria S/U Equilibria S/U $(0, 0, 0)$ U $(0, 0, 0)$ U $(0, 0, 0)$ U $(3800, 0, 0)$ U $(1359, 0, 0)$ U $(3800, 0, 0)$ U $(0, 333, 0)$ U $(0, 333, 0)$ U $(0, 333, 0)$ U $(255, 355, 0)$ U $(0, 92.298, 217.629)$ U $(255, 355, 0)$ U $(0, 92, 218)$ U $(0.724, 92.298, 217.647)$ S (0, 92, 218) U $(2327, 94, 268)$ S (2590, 94, 272) S

Table 4.  Case 1: Probability of an outbreak $(1-P_0)$ computed from the theory of branching processes, and based on 5000 sample paths of the CTMC model for initial values of $I(0) = 1$ and $I(0) = 2$ at $t = 50$

 Cases Initial value $1-P_0$ CTMC 1 $I(0)=1$ 0.7359 0.7440 $I(0)=2$ 0.9303 0.9288

Table 5.  Case 1: Using the Kolmogorov-Smirnov test to compare the distributions of an outbreak sample paths of $x_1$, $S$ and $I$ at $t = 30, 40, 50$ based on 5000 sample paths, where $(n_i, m_j)$ means test whether the sample paths at $t = n$ with $I(0) = i$ and $t = m$ with $I(0) = j$ are from the same distribution

 $(30_1, 40_1)$ $(40_1, 50_1)$ $(30_1, 30_2)$ $(40_1, 40_2)$ $(50_1, 50_2)$ $p$ $x_1$ 0.9571 0.8605 0.9115 0.81 0.7619 $S$ 0.3127 0.1434 0.9989 0.2852 0.9961 $I$ 0.3411 0.5824 0.5220 0.4625 1 $D$ $x_1$ 0.0118 0.014 0.0123 0.014 0.0147 $S$ 0.0223 0.0266 0.0083 0.0217 0.009 $I$ 0.0218 0.0118 0.0179 0.0187 0.0068

Table 6.  Case 1: Using the Shapiro-Wilk normality test to verify whether an outbreak sample paths of $x_1$, $S$ and $I$ with initial value $I(0) = 1$ at time $t = 50$ follow the normal distribution

 $x_1$ $S$ $I$ W 0.9996 0.9974 0.9995 p-value 0.6331 4.672e-06 0.454

Table 7.  Case 3: Probability of prey species invasion $(1-P'_0)$ computed from the theory of branching processes, and based on 5000 sample paths of the CTMC model for initial values of $x_1(0) = 1$ and $x_1(0) = 2$ at $t = 50$

 Cases Initial value $1-P_0'$ CTMC 3 $x_1(0)=1$ 0.6647 0.6784 $x_1(0)=2$ 0.8876 0.8922

Table 8.  Case 3: Using the Kolmogorov-Smirnov test to compare the distributions of an outbreak sample paths of $x_1$, $S$ and $I$ at $t = 30, 40, 50$ based on 5000 sample paths, where $(n_i, m_j)$ means test whether the sample paths at $t = n$ with $x_1(0) = i$ and $t = m$ with $x_1(0) = j$ are from the same distribution

 $(30_1, 40_1)$ $(40_1, 50_1)$ $(30_1, 30_2)$ $(40_1, 40_2)$ $(50_1, 50_2)$ $p$ $x_1$ 0.9481 0.7244 0.5647 0.8352 0.7135 $S$ 0.3127 0.1434 0.9989 0.2852 0.9961 $I$ 0.0684 0.6225 0.0022 0.6726 0.3696 $D$ $x_1$ 0.0127 0.0168 0.0179 0.0141 0.0159 $S$ 0.0223 0.0266 0.0083 0.0217 0.009 $I$ 0.0315 0.0183 0.0421 0.0165 0.0209

Table 9.  Case 3: Using the Shapiro-Wilk normality test to verify whether the sample paths of $x_1$, $S$ and $I$ with initial value $x_1(0) = 1$ at time $t = 50$ follow the normal distribution

 $x_1$ $S$ $I$ W 0.9996 0.9972 0.9992 p-value 0.7315 6.957e-06 0.1244
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