[1] |
Bernoulli, D.: Essai d’une nouvelle analyse de la mortalité cause par la petite vérole et des avantages de l’inoculation pour la prévenir. Mémoires de l’Académie Roy. des Sci. de Paris. (1760) |
[2] |
Harmer, W., Epidemic disease in England-The evidence of variability and of persistency of type, Lancet., 167, 4306, 733-739 (1906) |
[3] |
Ross, SR, The prevention of malaria (1911), London: London Murray, London |
[4] |
Kermack, WO; McKendrick, AG, A contribution to the mathematical theory of epidemics-I, B. Math. Biol., 53, 1-2, 33-55 (1991) |
[5] |
Kermack, WO; McKendrick, AG, Contributions to the mathematical theory of epidemics-II. The problem of endemicity, B. Math. Biol., 53, 1-2, 57-87 (1991) |
[6] |
May, RM; Anderson, RM; Irwin, ME, The transmission dynamics of human immunodeficiency virus (HIV), Philos. Trans. Roy. Soc. Lond. B., 321, 1207, 565-607 (1988) · doi:10.1098/rstb.1988.0108 |
[7] |
Chattopadhyay, J.; Arino, O., A predator-prey model with disease in the prey, Nonlinear Anal-Theor., 36, 6, 747-766 (1999) · Zbl 0922.34036 · doi:10.1016/S0362-546X(98)00126-6 |
[8] |
Getz, WM; Pickering, J., Epidemic models: Thresholds and population regulation, Am. Nat., 121, 6, 892-898 (1983) · doi:10.1086/284112 |
[9] |
Bairagi, N.; Roy, PK; Chattopadhyay, J., Role of infection on the stability of a predator-prey system with several response functions-a comparative study, J. Theor. Biol., 248, 1, 10-25 (2007) · Zbl 1451.92274 · doi:10.1016/j.jtbi.2007.05.005 |
[10] |
Lu, C.; Ding, XH, Periodic solutions and stationary distribution for a stochastic predator-prey system with impulsive perturbations, Appl. Math. Comput., 350, 313-322 (2019) · Zbl 1428.34056 |
[11] |
Xu, CH; Yu, YG; Ren, GJ, Dynamic analysis of a stochastic predator-prey model with Crowley-Martin functional response, disease in predator, and saturation incidence, J. Comput. Nonlin. Dyn., 15, 7, 071004 (2020) · doi:10.1115/1.4047085 |
[12] |
Liu, GD; Wang, X.; Meng, XZ; Gao, SJ, Extinction and persistence in mean of a novel delay impulsive stochastic infected predator-prey system with jumps, Complexity., 2017, 1950970 (2017) · Zbl 1407.92112 · doi:10.1155/2017/1950970 |
[13] |
Xu, CH; Yu, YG; Ren, GJ; Hai, XD; Lu, ZZ, Extinction and permanence analysis of stochastic predator-prey model with disease, ratio-dependent type functional response and nonlinear incidence rate, J. Comput. Nonlin. Dyn., 16, 11, 111004 (2021) · doi:10.1115/1.4051996 |
[14] |
Levi, T.; Kilpatrick, AM; Mangel, M.; Wilmers, CC, Deer, predators, and the emergence of Lyme disease, P. Natl. Acad. Sci. U.S.A., 109, 27, 10942-10947 (2012) · doi:10.1073/pnas.1204536109 |
[15] |
Biswas, S.; Sasmal, S.; Samanta, S.; Saifuddin, M.; Khan, QJA; Chattopadhyay, J., A delayed eco-epidemiological system with infected prey and predator subject to the weak Allee effect, Math. Biosci., 263, 198-208 (2015) · Zbl 1348.92145 · doi:10.1016/j.mbs.2015.02.013 |
[16] |
Zhang, XL; Huang, YH; Weng, PX, Permanence and stability of a diffusive predator-prey model with disease in the prey, Comput. Math. Appl., 68, 10, 1431-1445 (2014) · Zbl 1367.92112 · doi:10.1016/j.camwa.2014.09.011 |
[17] |
Abhijit, M.; Debadatta, A.; Nandadulal, B., Persistence and extinction of species in a disease-induced ecological system under environmental stochasticity, Phys. Rev. E., 103, 3, 032412 (2021) · doi:10.1103/PhysRevE.103.032412 |
[18] |
Deng, ML; Fan, YB, Invariant measure of a stochastic hybrid predator-prey model with infected prey, Appl. Math. Lett., 124, 107670 (2022) · Zbl 1478.92153 · doi:10.1016/j.aml.2021.107670 |
[19] |
Foryś, U.; Qiao, MH, Asymptotic dynamics of a deterministic and stochastic predator-prey model with disease in the prey species, Math. Method. Appl. Sci., 37, 3, 306-320 (2014) · Zbl 1288.34044 · doi:10.1002/mma.2783 |
[20] |
Ji, CY; Jiang, DQ, Dynamics of a stochastic density dependent predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 381, 1, 441-453 (2011) · Zbl 1232.34072 · doi:10.1016/j.jmaa.2011.02.037 |
[21] |
Jana, S.; Kar, TK, Modeling and analysis of a prey-predator system with disease in the prey, Chaos Solitons Fractals, 47, 42-53 (2013) · Zbl 1258.92038 · doi:10.1016/j.chaos.2012.12.002 |
[22] |
Chakraborty, K.; Das, K.; Haldar, S.; Kar, TK, A mathematical study of an eco-epidemiological system on disease persistence and extinction perspective, Appl. Math. Comput., 254, 99-112 (2015) · Zbl 1410.92094 |
[23] |
Anderson, R.; May, R., Population biological of infectious disease (1982), Springer, Berlin: Heidelberg. Germany, Springer, Berlin · doi:10.1007/978-3-642-68635-1 |
[24] |
Capasso, V.; Serio, G., Generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42, 1-2, 43-61 (1978) · Zbl 0398.92026 · doi:10.1016/0025-5564(78)90006-8 |
[25] |
Wei, CJ; Chen, LS, A delayed epidemic model with pulse vaccination, Discrete Dyn. Nat. Soc., 2008, 1, 746951 (2008) · Zbl 1149.92329 |
[26] |
Kaddar, A., On the dynamics of a delayed SIR epidemic model with a modified saturated incidence rate, Electron. J. Differ. Eq., 2009, 133, 1-7 (2009) · Zbl 1183.37092 |
[27] |
Liu, ZJ, Dynamics of positive solutions to SIR and SEIR epidemic models with saturated incidence rates, Nonlinear Anal-Real., 14, 3, 1286-1299 (2013) · Zbl 1261.34040 · doi:10.1016/j.nonrwa.2012.09.016 |
[28] |
Suryanto, A.; Kusumawinahyu, WM; Darti, I.; Yanti, I., Dynamically consistent discrete epidemic model with modified saturated incidence rate, Comput. Appl. Math., 32, 2, 373-383 (2013) · Zbl 1293.37035 · doi:10.1007/s40314-013-0026-6 |
[29] |
Tan, RH; Liu, ZJ; Guo, SL; Xiang, HL, On a nonautonomous competitive system subject to stochastic and impulsive perturbations, Appl. Math. Comput., 256, 702-714 (2015) · Zbl 1338.92113 |
[30] |
Liu, M.; Wang, K., On a stochastic logistic equation with impulsive perturbations, Comput. Math. Appl., 63, 5, 871-886 (2012) · Zbl 1247.60085 · doi:10.1016/j.camwa.2011.11.003 |
[31] |
Liu, M.; Wang, K., Persistence, extinction and global asymptotical stability of a non-autonomous predator-prey model with random perturbation, Appl. Math. Model., 36, 11, 5344-5353 (2012) · Zbl 1254.34074 · doi:10.1016/j.apm.2011.12.057 |
[32] |
Mao, X.; Yuan, C., Stochastic differential equations with Markovian switching (2006), London: Imperial College Press, London · Zbl 1126.60002 · doi:10.1142/p473 |
[33] |
Mao, X., Stochastic differential equations and applications (1997), Chichester: Horwood Publishing, Chichester · Zbl 0892.60057 |
[34] |
Liu, M.; Wang, K., Dynamics and simulations of a logistic model with impulsive perturbations in a random environment, Math. Comput. Simulat., 92, 53-75 (2013) · Zbl 1499.34294 · doi:10.1016/j.matcom.2013.04.011 |
[35] |
Wu, RH; Zou, XL; Wang, K., Asymptotic behavior of a stochastic non-autonomous predator-prey model with impulsive perturbations, Commun. Nonlinear Sci., 20, 3, 965-974 (2015) · Zbl 1304.92117 · doi:10.1016/j.cnsns.2014.06.023 |
[36] |
Shi, RQ; Jiang, XW; Chen, LS, A predator-prey model with disease in the prey and two impulses for integrated pest management, Appl. Math. Model., 33, 5, 2248-2256 (2008) · Zbl 1185.34015 · doi:10.1016/j.apm.2008.06.001 |
[37] |
Liu, ZJ; Wu, JH; Chen, YP; Haque, M., Impulsive perturbations in a periodic delay differential equation model of plankton allelopathy, Nonlinear Anal-Real., 11, 1, 432-445 (2010) · Zbl 1190.34084 · doi:10.1016/j.nonrwa.2008.11.017 |
[38] |
Wei, CJ; Chen, LS, Periodic solution and heteroclinic bifurcation in a predator-prey system with Allee effect and impulsive harvesting, Nonlinear Dynam., 76, 2, 1109-1117 (2014) · Zbl 1306.92052 · doi:10.1007/s11071-013-1194-z |
[39] |
Hardy, GH; Littlewood, JE; Polya, G., Inequalities (1952), Cambridge: Cambridge University Press, Cambridge · Zbl 0047.05302 |
[40] |
Karatzas, I.; Shreve, SE, Brownian motion and stochastic calculus (1998), Berlin: Springer-Verlag, Berlin · Zbl 0734.60060 · doi:10.1007/978-1-4612-0949-2 |
[41] |
Barblart, I., Systemes d’équations différentielles d’oscillations non linéaires, RevueRoumaine de Mathematiques Pures et Appliquees., 4, 2, 267-270 (1959) · Zbl 0090.06601 |