American Institute of Mathematical Sciences

Advanced
June  2019, 24(6): 2837-2863. doi: 10.3934/dcdsb.2018289

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

Hongxiao Hu 1,, , Liguang Xu 2, and  Kai Wang 3,

1. 

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
2. 

Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China
3. 

Department of Medical Engineering and Technology, Xinjiang Medical University, Urumqi 830011, China

* Corresponding author: Hongxiao Hu

Received  July 2017 Revised  June 2018 Published  October 2018

Fund Project: 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.

Keywords: Predator-prey species, uniform persistence, species invasion, continuous-time Markov chain, Kolmogorov-Smirnov test, Shapiro-Wilk normality test.
Mathematics Subject Classification: Primary: 65C30, 60H10; Secondary: 46N60.
Citation: Hongxiao Hu, Liguang Xu, Kai Wang. A comparison of deterministic and stochastic predator-prey models with disease in the predator. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2837-2863. doi: 10.3934/dcdsb.2018289
References:
[1]

L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, 2$^{nd}$ edition, CRC Press, Boca Raton, FL, 2011.  Google Scholar

[2]

L. J. S. Allen and V. A. Bokil, Stochastic models for competing species with a shared pathogen, Math. Biosci. Eng., 9 (2012), 461-485.  doi: 10.3934/mbe.2012.9.461.  Google Scholar

[3]

L. J. S. Allen and N. Kirupaharan, Asymptotic dynamics of deterministic and stochastic epidemic models with multiple pathogens, Int. J. Numer. Anal. Modeling, 2 (2005), 329-344.   Google Scholar

[4]

L. J. S. Allen and G. E. Lahodny, Extinction thresholds in deterministic and stochastic epidemic models, J. Biol. Dyn., 6 (2012), 590-611.   Google Scholar

[5]

R. M. Anderson and R. M. May, The invasion, persistence, and spread of iufectious diseases within animal and plant communites, Phil. Trans. R. Soc. London B, 314 (1986), 533-570.   Google Scholar

[6]

Y. L. Cai, Y. Cai, M. Banerjee and W.M. Wang, A stochastic sirs epidemic model with infectious force under intervention strategies, J. Differential Equaitons, 259 (2015), 7463-7502.  doi: 10.1016/j.jde.2015.08.024.  Google Scholar

[7]

Y. L. Cai, Y. Kang and W. M. Wang, A stochastic sirs epidemic model with nonlinear incidence rate, Appl. Math. Comput., 305 (2017), 221-240.  doi: 10.1016/j.amc.2017.02.003.  Google Scholar

[8]

J. Chattopadhyay and O. Arino, A predator-prey model with disease in the prey, Nonlinear Anal., 36 (1999), 747-766.  doi: 10.1016/S0362-546X(98)00126-6.  Google Scholar

[9]

K. P. Das, A study of chaotic dynamics and its possible control in a predator-prey model with disease in the predator, J. Dyn. Control Syst., 21 (2015), 605-624.  doi: 10.1007/s10883-015-9283-6.  Google Scholar

[10]

K. P. Das, A study of harvesting in a predator-prey model with disease in both populations, Math. Methods Appl. Sci., 39 (2016), 2853-2870.  doi: 10.1002/mma.3735.  Google Scholar

[11]

K. S. Dorman, J. S. Sinsheimer and K. Lange, In the garden of branching processes, SIAM Rev., 46 (2004), 202-229.  doi: 10.1137/S0036144502417843.  Google Scholar

[12]

R. Durrett, Special invited paper: Coexistence in stochastic spatial models, Ann. Appl. Probab., 19 (2009), 477-496.  doi: 10.1214/08-AAP590.  Google Scholar

[13]

D. T. Gillespie, Markov Processes: An Introduction for Physical Scientists, Academic Press, Inc., Boston, MA, 1992.  Google Scholar

[14]

B. S. Goh, Management and Analysis of Biological Populations, Elsevier Sci. Pub. Com., Amsterdam, 1980. Google Scholar

[15]

B. S. Goh, Global stability in two species interactions, J. Math. Biol., 3 (1976), 313-318.  doi: 10.1007/BF00275063.  Google Scholar

[16]

F. M. D. Gulland, The impact of infectious diseases on wild animal populations–a review, Ecology of Infectious Diseases in Natural Populations. (B. T. Grenfell and A. P. Dobson, eds). Cambridge: Cambridge University Press, 1995, 20–51. Google Scholar

[17]

W. J. Guo, Y. L. Cai, Q. M. Zhang and W. M. Wang, Stochastic persistence and stationary distribution in an sis epidemic model with media coverage, Physica A: Statistical Mechanics and its Applications, 492 (2018), 2220-2236.  doi: 10.1016/j.physa.2017.11.137.  Google Scholar

[18]

P. Haccou, P. Jagers and V. A. Vatutin, Branching Processes Variation, Growth, and Extinction of Populations, Cambridge University Press, Cambridge; IIASA, Laxenburg, 2007.  Google Scholar

[19]

K. P. Hadeler and H. I. Freedman, Predator-prey populations with parasitic infection, J. Math. Biol., 27 (1989), 609-631.  doi: 10.1007/BF00276947.  Google Scholar

[20]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.  doi: 10.1137/0520025.  Google Scholar

[21]

L. T. Han and Z. E. Ma, Four predator prey models with infectious diseases, Math. Comput. Modelling, 34 (2001), 849-858.  doi: 10.1016/S0895-7177(01)00104-2.  Google Scholar

[22]

L. T. Han, Z. E. Ma and T. Shi, An sirs epidemic model of two competitive species, Math. Comput. Modelling, 37 (2003), 87-108.  doi: 10.1016/S0895-7177(03)80008-0.  Google Scholar

[23]

L. T. Han and A. Pugliese, Epidemics in two competing species, Nonlinear Anal., 10 (2009), 723-744.  doi: 10.1016/j.nonrwa.2007.11.005.  Google Scholar

[24]

M. Haque, A predator-prey model with disease in the predator species only, Nonlinear Anal.: Real World Appl., 11 (2010), 2224-2236.  doi: 10.1016/j.nonrwa.2009.06.012.  Google Scholar

[25]

M. Haque and E. Venturino, An ecoepidemiological model with disease in predator: The ratio-dependent case, Math. Meth. Appl. Sci., 30 (2007), 1791-1809.  doi: 10.1002/mma.869.  Google Scholar

[26]

H. W. Hethcote, W. D. Wang, L. T. Han and Z. E. Ma, A predator-prey model with infected prey, Theor. Popul. Biol., 66 (2004), 259-268.   Google Scholar

[27]

D. J. Higham, Modeling and simulating chemical reactions, SIAM Rev., 50 (2008), 347-368.  doi: 10.1137/060666457.  Google Scholar

[28]

M. Kimmel and D. Axelrod, Branching Processes in Biology, Springer-Verlag, NewYork, 2002. doi: 10.1007/b97371.  Google Scholar

[29]

N. Lanchier and C. Neuhauser, A spatially explicit model for competition among specialists and generalists in a heterogeneous environment, Ann. Appl. Probab., 16 (2006), 1385-1410.  doi: 10.1214/105051606000000394.  Google Scholar

[30]

N. Lanchier and C. Neuhauser, Stochastic spatial models of host-pathogen and host-mutualist interactions. i, Ann. Appl. Probab., 16 (2006), 448-474.  doi: 10.1214/105051605000000782.  Google Scholar

[31]

Q. Liu, D. Q. Jiang, N. Z. Shi, T. Hayat and A. Alsaedi, The threshold of a stochastic sis epidemic model with imperfect vaccination, Math. Comput. Simulation, 144 (2018), 78-90.  doi: 10.1016/j.matcom.2017.06.004.  Google Scholar

[32]

M. Liu, C. Bai and Y. Jin, Population dynamical behavior of a two-predator one-prey stochastic model with time delay, Discrete Contin. Dyn. Syst., 37 (2017), 2513-2538.  doi: 10.3934/dcds.2017108.  Google Scholar

[33]

M. Liu, X. He and J. Yu, Dynamics of a stochastic regime-switching predator-prey model with harvesting and distributed delays, Nonlinear Anal. Hybrid Syst., 28 (2018), 87-104.  doi: 10.1016/j.nahs.2017.10.004.  Google Scholar

[34]

M. Liu and M. Fan, Stability in distribution of a three-species stochastic cascade predator-prey system with time delays, IMA J. Appl. Math., 82 (2017), 396-423.  doi: 10.1093/imamat/hxw057.  Google Scholar

[35]

R. K. McCormack and L. J. S. Allen, Disease emergence in multi-host epidemic models, Math. Med. Biol., 24 (2007), 17-34.   Google Scholar

[36]

S. Sarwardi, M. Haque and E. Venturino, Global stability and persistence in lg-holling type ii diseased predator ecosystems, J. Biol. Phys., 37 (2011), 91-106.   Google Scholar

[37]

H. R. Thieme, Covergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.  Google Scholar

[38]

H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435.  doi: 10.1137/0524026.  Google Scholar

[39]

E. Venturino, The influence of diseases on lotka-volterra systems, Rocky Mountain J. Math, 24 (1994), 381-402.  doi: 10.1216/rmjm/1181072471.  Google Scholar

[40]

E. Venturino, Epidemics in predator-prey models: disease in the predators, IMA J. Math. Appl. Med. Biol., 19 (2002), 185-205.   Google Scholar

[41]

P. Whittle, The outcome of a stochastic epidemic: A note on bailey's paper, Biometrika, 42 (1955), 116-122.  doi: 10.1093/biomet/42.1-2.116.  Google Scholar

[42]

Y. N. Xiao and L. S. Chen, Modeling and analysis of a predator-prey model with disease in the prey, Math. Biosci., 171 (2001), 59-82.  doi: 10.1016/S0025-5564(01)00049-9.  Google Scholar

[43]

R. Xu and S. H. Zhang, Modelling and analysis of a delayed predator-prey model with disease in the predator, Appl. Math. Comput., 224 (2013), 372-386.  doi: 10.1016/j.amc.2013.08.067.  Google Scholar

[44]

Y. Yuan and L. J. S. Allen, Stochastic models for virus and immune system dynamics, Math. Biosci, 234 (2011), 84-94.  doi: 10.1016/j.mbs.2011.08.007.  Google Scholar

show all references

References:
[1]

L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, 2$^{nd}$ edition, CRC Press, Boca Raton, FL, 2011.  Google Scholar

[2]

L. J. S. Allen and V. A. Bokil, Stochastic models for competing species with a shared pathogen, Math. Biosci. Eng., 9 (2012), 461-485.  doi: 10.3934/mbe.2012.9.461.  Google Scholar

[3]

L. J. S. Allen and N. Kirupaharan, Asymptotic dynamics of deterministic and stochastic epidemic models with multiple pathogens, Int. J. Numer. Anal. Modeling, 2 (2005), 329-344.   Google Scholar

[4]

L. J. S. Allen and G. E. Lahodny, Extinction thresholds in deterministic and stochastic epidemic models, J. Biol. Dyn., 6 (2012), 590-611.   Google Scholar

[5]

R. M. Anderson and R. M. May, The invasion, persistence, and spread of iufectious diseases within animal and plant communites, Phil. Trans. R. Soc. London B, 314 (1986), 533-570.   Google Scholar

[6]

Y. L. Cai, Y. Cai, M. Banerjee and W.M. Wang, A stochastic sirs epidemic model with infectious force under intervention strategies, J. Differential Equaitons, 259 (2015), 7463-7502.  doi: 10.1016/j.jde.2015.08.024.  Google Scholar

[7]

Y. L. Cai, Y. Kang and W. M. Wang, A stochastic sirs epidemic model with nonlinear incidence rate, Appl. Math. Comput., 305 (2017), 221-240.  doi: 10.1016/j.amc.2017.02.003.  Google Scholar

[8]

J. Chattopadhyay and O. Arino, A predator-prey model with disease in the prey, Nonlinear Anal., 36 (1999), 747-766.  doi: 10.1016/S0362-546X(98)00126-6.  Google Scholar

[9]

K. P. Das, A study of chaotic dynamics and its possible control in a predator-prey model with disease in the predator, J. Dyn. Control Syst., 21 (2015), 605-624.  doi: 10.1007/s10883-015-9283-6.  Google Scholar

[10]

K. P. Das, A study of harvesting in a predator-prey model with disease in both populations, Math. Methods Appl. Sci., 39 (2016), 2853-2870.  doi: 10.1002/mma.3735.  Google Scholar

[11]

K. S. Dorman, J. S. Sinsheimer and K. Lange, In the garden of branching processes, SIAM Rev., 46 (2004), 202-229.  doi: 10.1137/S0036144502417843.  Google Scholar

[12]

R. Durrett, Special invited paper: Coexistence in stochastic spatial models, Ann. Appl. Probab., 19 (2009), 477-496.  doi: 10.1214/08-AAP590.  Google Scholar

[13]

D. T. Gillespie, Markov Processes: An Introduction for Physical Scientists, Academic Press, Inc., Boston, MA, 1992.  Google Scholar

[14]

B. S. Goh, Management and Analysis of Biological Populations, Elsevier Sci. Pub. Com., Amsterdam, 1980. Google Scholar

[15]

B. S. Goh, Global stability in two species interactions, J. Math. Biol., 3 (1976), 313-318.  doi: 10.1007/BF00275063.  Google Scholar

[16]

F. M. D. Gulland, The impact of infectious diseases on wild animal populations–a review, Ecology of Infectious Diseases in Natural Populations. (B. T. Grenfell and A. P. Dobson, eds). Cambridge: Cambridge University Press, 1995, 20–51. Google Scholar

[17]

W. J. Guo, Y. L. Cai, Q. M. Zhang and W. M. Wang, Stochastic persistence and stationary distribution in an sis epidemic model with media coverage, Physica A: Statistical Mechanics and its Applications, 492 (2018), 2220-2236.  doi: 10.1016/j.physa.2017.11.137.  Google Scholar

[18]

P. Haccou, P. Jagers and V. A. Vatutin, Branching Processes Variation, Growth, and Extinction of Populations, Cambridge University Press, Cambridge; IIASA, Laxenburg, 2007.  Google Scholar

[19]

K. P. Hadeler and H. I. Freedman, Predator-prey populations with parasitic infection, J. Math. Biol., 27 (1989), 609-631.  doi: 10.1007/BF00276947.  Google Scholar

[20]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.  doi: 10.1137/0520025.  Google Scholar

[21]

L. T. Han and Z. E. Ma, Four predator prey models with infectious diseases, Math. Comput. Modelling, 34 (2001), 849-858.  doi: 10.1016/S0895-7177(01)00104-2.  Google Scholar

[22]

L. T. Han, Z. E. Ma and T. Shi, An sirs epidemic model of two competitive species, Math. Comput. Modelling, 37 (2003), 87-108.  doi: 10.1016/S0895-7177(03)80008-0.  Google Scholar

[23]

L. T. Han and A. Pugliese, Epidemics in two competing species, Nonlinear Anal., 10 (2009), 723-744.  doi: 10.1016/j.nonrwa.2007.11.005.  Google Scholar

[24]

M. Haque, A predator-prey model with disease in the predator species only, Nonlinear Anal.: Real World Appl., 11 (2010), 2224-2236.  doi: 10.1016/j.nonrwa.2009.06.012.  Google Scholar

[25]

M. Haque and E. Venturino, An ecoepidemiological model with disease in predator: The ratio-dependent case, Math. Meth. Appl. Sci., 30 (2007), 1791-1809.  doi: 10.1002/mma.869.  Google Scholar

[26]

H. W. Hethcote, W. D. Wang, L. T. Han and Z. E. Ma, A predator-prey model with infected prey, Theor. Popul. Biol., 66 (2004), 259-268.   Google Scholar

[27]

D. J. Higham, Modeling and simulating chemical reactions, SIAM Rev., 50 (2008), 347-368.  doi: 10.1137/060666457.  Google Scholar

[28]

M. Kimmel and D. Axelrod, Branching Processes in Biology, Springer-Verlag, NewYork, 2002. doi: 10.1007/b97371.  Google Scholar

[29]

N. Lanchier and C. Neuhauser, A spatially explicit model for competition among specialists and generalists in a heterogeneous environment, Ann. Appl. Probab., 16 (2006), 1385-1410.  doi: 10.1214/105051606000000394.  Google Scholar

[30]

N. Lanchier and C. Neuhauser, Stochastic spatial models of host-pathogen and host-mutualist interactions. i, Ann. Appl. Probab., 16 (2006), 448-474.  doi: 10.1214/105051605000000782.  Google Scholar

[31]

Q. Liu, D. Q. Jiang, N. Z. Shi, T. Hayat and A. Alsaedi, The threshold of a stochastic sis epidemic model with imperfect vaccination, Math. Comput. Simulation, 144 (2018), 78-90.  doi: 10.1016/j.matcom.2017.06.004.  Google Scholar

[32]

M. Liu, C. Bai and Y. Jin, Population dynamical behavior of a two-predator one-prey stochastic model with time delay, Discrete Contin. Dyn. Syst., 37 (2017), 2513-2538.  doi: 10.3934/dcds.2017108.  Google Scholar

[33]

M. Liu, X. He and J. Yu, Dynamics of a stochastic regime-switching predator-prey model with harvesting and distributed delays, Nonlinear Anal. Hybrid Syst., 28 (2018), 87-104.  doi: 10.1016/j.nahs.2017.10.004.  Google Scholar

[34]

M. Liu and M. Fan, Stability in distribution of a three-species stochastic cascade predator-prey system with time delays, IMA J. Appl. Math., 82 (2017), 396-423.  doi: 10.1093/imamat/hxw057.  Google Scholar

[35]

R. K. McCormack and L. J. S. Allen, Disease emergence in multi-host epidemic models, Math. Med. Biol., 24 (2007), 17-34.   Google Scholar

[36]

S. Sarwardi, M. Haque and E. Venturino, Global stability and persistence in lg-holling type ii diseased predator ecosystems, J. Biol. Phys., 37 (2011), 91-106.   Google Scholar

[37]

H. R. Thieme, Covergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.  Google Scholar

[38]

H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435.  doi: 10.1137/0524026.  Google Scholar

[39]

E. Venturino, The influence of diseases on lotka-volterra systems, Rocky Mountain J. Math, 24 (1994), 381-402.  doi: 10.1216/rmjm/1181072471.  Google Scholar

[40]

E. Venturino, Epidemics in predator-prey models: disease in the predators, IMA J. Math. Appl. Med. Biol., 19 (2002), 185-205.   Google Scholar

[41]

P. Whittle, The outcome of a stochastic epidemic: A note on bailey's paper, Biometrika, 42 (1955), 116-122.  doi: 10.1093/biomet/42.1-2.116.  Google Scholar

[42]

Y. N. Xiao and L. S. Chen, Modeling and analysis of a predator-prey model with disease in the prey, Math. Biosci., 171 (2001), 59-82.  doi: 10.1016/S0025-5564(01)00049-9.  Google Scholar

[43]

R. Xu and S. H. Zhang, Modelling and analysis of a delayed predator-prey model with disease in the predator, Appl. Math. Comput., 224 (2013), 372-386.  doi: 10.1016/j.amc.2013.08.067.  Google Scholar

[44]

Y. Yuan and L. J. S. Allen, Stochastic models for virus and immune system dynamics, Math. Biosci, 234 (2011), 84-94.  doi: 10.1016/j.mbs.2011.08.007.  Google Scholar

Figure 1.   
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Figure 2.  Comparison of ODE and CTMC (3 sample paths) solutions for Case 1. We start the simulation at $x_1(0) = 255$, $S(0) = 355$, $I(0) = 1$
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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 $t = 50$
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Figure 4.  Solution of ODE model for Case 2. We start the simulation at $x_1(0) = 1$, $S(0) = 92$, $I(0) = 218$
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Figure 5.  Comparison of ODE and CTMC (3 sample paths) solutions for Case 2. We start the simulation at $x_1(0) = 1$, $S(0) = 92$, $I(0) = 218$
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Figure 6.  Comparison of ODE and CTMC (3 sample paths) solutions for Case 3. We start the simulation at $x_1(0) = 1$, $S(0) = 92$, $I(0) = 218(P_0' = 0.335)$
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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 $t = 50$
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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 $t = 30, 40, 50$
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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 $t = 30, 40, 50$
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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)
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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)
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Table 1.  State transitions and the infinitesimal probabilities for the CTMC epidemic model.
DescriptionState transition $a \to b$Rate $P(a, b)$
1Birth of $S$ $(x_1, S, I) \to (x_1, S+1, I)$ $P_1=x_2(b_2-c_2a_2x_2)$
2Death of $S$ $(x_1, S, I) \to (x_1, S-1, I)$ $P_2=S(d_2+(1-c_2)a_2x_2-\varepsilon\eta x_1)$
3Infection $(x_1, S, I) \to (x_1, S-1, I+1)$$P_3=\beta SI$
4Death of $I$ $(x_1, S, I) \to (x_1, S, I-1)$$P_4=I(d_2+(1-c_2)a_2x_2+\mu-pq\varepsilon\eta x_1)$
5Recover of $I$ $(x_1, S, I) \to (x_1, S+1, I-1)$$P_5=\gamma I$
6Birth of $x_1$ $(x_1, S, I) \to (x_1+1, S, I)$$P_6=x_1(b_1-c_1a_1x_1)$
7Death of $x_1$ $(x_1, S, I) \to (x_1-1, S, I)$$P_7=x_1(d_1+(1-c_1)a_1x_1+\eta(S+qI))$
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DescriptionState transition $a \to b$Rate $P(a, b)$
1Birth of $S$ $(x_1, S, I) \to (x_1, S+1, I)$ $P_1=x_2(b_2-c_2a_2x_2)$
2Death of $S$ $(x_1, S, I) \to (x_1, S-1, I)$ $P_2=S(d_2+(1-c_2)a_2x_2-\varepsilon\eta x_1)$
3Infection $(x_1, S, I) \to (x_1, S-1, I+1)$$P_3=\beta SI$
4Death of $I$ $(x_1, S, I) \to (x_1, S, I-1)$$P_4=I(d_2+(1-c_2)a_2x_2+\mu-pq\varepsilon\eta x_1)$
5Recover of $I$ $(x_1, S, I) \to (x_1, S+1, I-1)$$P_5=\gamma I$
6Birth of $x_1$ $(x_1, S, I) \to (x_1+1, S, I)$$P_6=x_1(b_1-c_1a_1x_1)$
7Death of $x_1$ $(x_1, S, I) \to (x_1-1, S, I)$$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
ParameterInterpretationCase 1Case2Case3
$a_1$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$$0.02$
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ParameterInterpretationCase 1Case2Case3
$a_1$Density dependent of prey $0.0005$ $0.0005$ $0.0005$
$a_2$Density dependent of predator $0.0006$ $0.0006$ $0.0006$
$b_1$Intrinsic birth rate of prey $2$ $0.7795$ $2$
$b_2$Intrinsic birth rate of predator $1$ $1$ $1$
$c_1$Density dependence effects of prey $0.5$ $0.5$ $0.5$
$c_2$Density dependence effects of predator $0.5$ $0.5$ $0.5$
$d_1$Natural mortality of prey$0.1$$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 1Case 2Case 3
EquilibriaS/UEquilibriaS/UEquilibriaS/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
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Case 1Case 2Case 3
EquilibriaS/UEquilibriaS/UEquilibriaS/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 $t = 50$
CasesInitial value$1-P_0$CTMC
1$I(0)=1$0.73590.7440
$I(0)=2$0.93030.9288
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CasesInitial value$1-P_0$CTMC
1$I(0)=1$0.73590.7440
$I(0)=2$0.93030.9288
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$$x_1$0.95710.86050.91150.810.7619
$S$0.31270.14340.99890.28520.9961
$I$0.34110.58240.52200.46251
$D$$x_1$0.01180.0140.01230.0140.0147
$S$0.02230.02660.00830.02170.009
$I$0.02180.01180.01790.01870.0068
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$(30_1, 40_1)$$(40_1, 50_1)$$(30_1, 30_2)$$(40_1, 40_2)$$(50_1, 50_2)$
$p$$x_1$0.95710.86050.91150.810.7619
$S$0.31270.14340.99890.28520.9961
$I$0.34110.58240.52200.46251
$D$$x_1$0.01180.0140.01230.0140.0147
$S$0.02230.02660.00830.02170.009
$I$0.02180.01180.01790.01870.0068
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$$I$
W0.99960.99740.9995
p-value0.63314.672e-060.454
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$x_1$ $S$$I$
W0.99960.99740.9995
p-value0.63314.672e-060.454
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 $t = 50$
CasesInitial value$1-P_0'$CTMC
3$x_1(0)=1$0.66470.6784
$x_1(0)=2$0.88760.8922
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CasesInitial value$1-P_0'$CTMC
3$x_1(0)=1$0.66470.6784
$x_1(0)=2$0.88760.8922
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$$x_1$0.94810.72440.56470.83520.7135
$S$0.31270.14340.99890.28520.9961
$I$0.06840.62250.00220.67260.3696
$D$$x_1$0.01270.01680.01790.01410.0159
$S$0.02230.02660.00830.02170.009
$I$0.03150.01830.04210.01650.0209
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$(30_1, 40_1)$$(40_1, 50_1)$$(30_1, 30_2)$$(40_1, 40_2)$$(50_1, 50_2)$
$p$$x_1$0.94810.72440.56470.83520.7135
$S$0.31270.14340.99890.28520.9961
$I$0.06840.62250.00220.67260.3696
$D$$x_1$0.01270.01680.01790.01410.0159
$S$0.02230.02660.00830.02170.009
$I$0.03150.01830.04210.01650.0209
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$$I$
W0.99960.99720.9992
p-value0.73156.957e-060.1244
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$x_1$ $S$$I$
W0.99960.99720.9992
p-value0.73156.957e-060.1244
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