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A comparison of deterministic and stochastic predator-prey models with disease in the predator
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 |
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.
References:
[1] |
L. J. S. Allen,
An Introduction to Stochastic Processes with Applications to Biology, 2$^{nd}$ edition, CRC Press, Boca Raton, FL, 2011. |
[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. |
[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.
|
[4] |
L. J. S. Allen and G. E. Lahodny,
Extinction thresholds in deterministic and stochastic epidemic models, J. Biol. Dyn., 6 (2012), 590-611.
|
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
R. Durrett,
Special invited paper: Coexistence in stochastic spatial models, Ann. Appl. Probab., 19 (2009), 477-496.
doi: 10.1214/08-AAP590. |
[13] |
D. T. Gillespie,
Markov Processes: An Introduction for Physical Scientists, Academic Press, Inc., Boston, MA, 1992. |
[14] |
B. S. Goh,
Management and Analysis of Biological Populations, Elsevier Sci. Pub. Com., Amsterdam, 1980. |
[15] |
B. S. Goh,
Global stability in two species interactions, J. Math. Biol., 3 (1976), 313-318.
doi: 10.1007/BF00275063. |
[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. |
[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. |
[18] |
P. Haccou, P. Jagers and V. A. Vatutin,
Branching Processes Variation, Growth, and Extinction of Populations, Cambridge University Press, Cambridge; IIASA, Laxenburg, 2007. |
[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. |
[20] |
J. K. Hale and P. Waltman,
Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.
doi: 10.1137/0520025. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[27] |
D. J. Higham,
Modeling and simulating chemical reactions, SIAM Rev., 50 (2008), 347-368.
doi: 10.1137/060666457. |
[28] |
M. Kimmel and D. Axelrod,
Branching Processes in Biology, Springer-Verlag, NewYork, 2002.
doi: 10.1007/b97371. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[35] |
R. K. McCormack and L. J. S. Allen,
Disease emergence in multi-host epidemic models, Math. Med. Biol., 24 (2007), 17-34.
|
[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.
|
[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. |
[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. |
[39] |
E. Venturino,
The influence of diseases on lotka-volterra systems, Rocky Mountain J. Math, 24 (1994), 381-402.
doi: 10.1216/rmjm/1181072471. |
[40] |
E. Venturino,
Epidemics in predator-prey models: disease in the predators, IMA J. Math. Appl. Med. Biol., 19 (2002), 185-205.
|
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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.
|
[4] |
L. J. S. Allen and G. E. Lahodny,
Extinction thresholds in deterministic and stochastic epidemic models, J. Biol. Dyn., 6 (2012), 590-611.
|
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
R. Durrett,
Special invited paper: Coexistence in stochastic spatial models, Ann. Appl. Probab., 19 (2009), 477-496.
doi: 10.1214/08-AAP590. |
[13] |
D. T. Gillespie,
Markov Processes: An Introduction for Physical Scientists, Academic Press, Inc., Boston, MA, 1992. |
[14] |
B. S. Goh,
Management and Analysis of Biological Populations, Elsevier Sci. Pub. Com., Amsterdam, 1980. |
[15] |
B. S. Goh,
Global stability in two species interactions, J. Math. Biol., 3 (1976), 313-318.
doi: 10.1007/BF00275063. |
[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. |
[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. |
[18] |
P. Haccou, P. Jagers and V. A. Vatutin,
Branching Processes Variation, Growth, and Extinction of Populations, Cambridge University Press, Cambridge; IIASA, Laxenburg, 2007. |
[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. |
[20] |
J. K. Hale and P. Waltman,
Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.
doi: 10.1137/0520025. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[27] |
D. J. Higham,
Modeling and simulating chemical reactions, SIAM Rev., 50 (2008), 347-368.
doi: 10.1137/060666457. |
[28] |
M. Kimmel and D. Axelrod,
Branching Processes in Biology, Springer-Verlag, NewYork, 2002.
doi: 10.1007/b97371. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[35] |
R. K. McCormack and L. J. S. Allen,
Disease emergence in multi-host epidemic models, Math. Med. Biol., 24 (2007), 17-34.
|
[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.
|
[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. |
[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. |
[39] |
E. Venturino,
The influence of diseases on lotka-volterra systems, Rocky Mountain J. Math, 24 (1994), 381-402.
doi: 10.1216/rmjm/1181072471. |
[40] |
E. Venturino,
Epidemics in predator-prey models: disease in the predators, IMA J. Math. Appl. Med. Biol., 19 (2002), 185-205.
|
[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. |
[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. |
[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. |
[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. |










Description | State transition | Rate | |
1 | Birth of | | |
2 | Death of | | |
3 | Infection | | |
4 | Death of | | |
5 | Recover of | | |
6 | Birth of | | |
7 | Death of | |
Description | State transition | Rate | |
1 | Birth of | | |
2 | Death of | | |
3 | Infection | | |
4 | Death of | | |
5 | Recover of | | |
6 | Birth of | | |
7 | Death of | |
Parameter | Interpretation | Case 1 | Case2 | Case3 |
| Density dependent of prey | | | |
| Density dependent of predator | | | |
| Intrinsic birth rate of prey | | | |
| Intrinsic birth rate of predator | | | |
| Density dependence effects of prey | | | |
| Density dependence effects of predator | | | |
| Natural mortality of prey | |||
| Natural mortality of predator | |||
| Predation rate of susceptible predator | | ||
| Conversion rate of susceptible predator | |||
| Predation rate of infected predator | |||
| Conversion rate of infected predator | 0.002 | ||
| Transmission | |||
| Recover rate | |||
| Disease related mortality |
Parameter | Interpretation | Case 1 | Case2 | Case3 |
| Density dependent of prey | | | |
| Density dependent of predator | | | |
| Intrinsic birth rate of prey | | | |
| Intrinsic birth rate of predator | | | |
| Density dependence effects of prey | | | |
| Density dependence effects of predator | | | |
| Natural mortality of prey | |||
| Natural mortality of predator | |||
| Predation rate of susceptible predator | | ||
| Conversion rate of susceptible predator | |||
| Predation rate of infected predator | |||
| Conversion rate of infected predator | 0.002 | ||
| Transmission | |||
| Recover rate | |||
| Disease related mortality |
Case 1 | Case 2 | Case 3 | |||
Equilibria | S/U | Equilibria | S/U | Equilibria | S/U |
| U | U | U | ||
| U | U | U | ||
| U | U | U | ||
| U | U | U | ||
| U | S | (0, 92, 218) | U | |
| S | (2590, 94, 272) | S |
Case 1 | Case 2 | Case 3 | |||
Equilibria | S/U | Equilibria | S/U | Equilibria | S/U |
| U | U | U | ||
| U | U | U | ||
| U | U | U | ||
| U | U | U | ||
| U | S | (0, 92, 218) | U | |
| S | (2590, 94, 272) | S |
Cases | Initial value | CTMC | |
1 | 0.7359 | 0.7440 | |
0.9303 | 0.9288 |
Cases | Initial value | CTMC | |
1 | 0.7359 | 0.7440 | |
0.9303 | 0.9288 |
| 0.9571 | 0.8605 | 0.9115 | 0.81 | 0.7619 | |
0.3127 | 0.1434 | 0.9989 | 0.2852 | 0.9961 | ||
0.3411 | 0.5824 | 0.5220 | 0.4625 | 1 | ||
| 0.0118 | 0.014 | 0.0123 | 0.014 | 0.0147 | |
0.0223 | 0.0266 | 0.0083 | 0.0217 | 0.009 | ||
0.0218 | 0.0118 | 0.0179 | 0.0187 | 0.0068 |
| 0.9571 | 0.8605 | 0.9115 | 0.81 | 0.7619 | |
0.3127 | 0.1434 | 0.9989 | 0.2852 | 0.9961 | ||
0.3411 | 0.5824 | 0.5220 | 0.4625 | 1 | ||
| 0.0118 | 0.014 | 0.0123 | 0.014 | 0.0147 | |
0.0223 | 0.0266 | 0.0083 | 0.0217 | 0.009 | ||
0.0218 | 0.0118 | 0.0179 | 0.0187 | 0.0068 |
| |||
W | 0.9996 | 0.9974 | 0.9995 |
p-value | 0.6331 | 4.672e-06 | 0.454 |
| |||
W | 0.9996 | 0.9974 | 0.9995 |
p-value | 0.6331 | 4.672e-06 | 0.454 |
Cases | Initial value | CTMC | |
3 | 0.6647 | 0.6784 | |
0.8876 | 0.8922 |
Cases | Initial value | CTMC | |
3 | 0.6647 | 0.6784 | |
0.8876 | 0.8922 |
| 0.9481 | 0.7244 | 0.5647 | 0.8352 | 0.7135 | |
0.3127 | 0.1434 | 0.9989 | 0.2852 | 0.9961 | ||
0.0684 | 0.6225 | 0.0022 | 0.6726 | 0.3696 | ||
| 0.0127 | 0.0168 | 0.0179 | 0.0141 | 0.0159 | |
0.0223 | 0.0266 | 0.0083 | 0.0217 | 0.009 | ||
0.0315 | 0.0183 | 0.0421 | 0.0165 | 0.0209 |
| 0.9481 | 0.7244 | 0.5647 | 0.8352 | 0.7135 | |
0.3127 | 0.1434 | 0.9989 | 0.2852 | 0.9961 | ||
0.0684 | 0.6225 | 0.0022 | 0.6726 | 0.3696 | ||
| 0.0127 | 0.0168 | 0.0179 | 0.0141 | 0.0159 | |
0.0223 | 0.0266 | 0.0083 | 0.0217 | 0.009 | ||
0.0315 | 0.0183 | 0.0421 | 0.0165 | 0.0209 |
| |||
W | 0.9996 | 0.9972 | 0.9992 |
p-value | 0.7315 | 6.957e-06 | 0.1244 |
| |||
W | 0.9996 | 0.9972 | 0.9992 |
p-value | 0.7315 | 6.957e-06 | 0.1244 |
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