Infectious diseases have been one of the major causes of human mortality, and mathematical models have been playing significant roles in understanding the spread mechanism and controlling contagious diseases. In this paper, we propose a delayed SEIR epidemic model with intervention strategies and recovery under the low availability of resources. Non-delayed and delayed models both possess two equilibria: the disease-free equilibrium and the endemic equilibrium. When the basic reproduction number $ R_0 = 1 $, the non-delayed system undergoes a transcritical bifurcation. For the delayed system, we incorporate two important time delays: $ \tau_1 $ represents the latent period of the intervention strategies, and $ \tau_2 $ represents the period for curing the infected individuals. Time delays change the system dynamics via Hopf-bifurcation and oscillations. The direction and stability of delay induced Hopf-bifurcation are established using normal form theory and center manifold theorem. Furthermore, we rigorously prove that local Hopf bifurcation implies global Hopf bifurcation. Stability switching curves and crossing directions are analyzed on the two delay parameter plane, which allows both delays varying simultaneously. Numerical results demonstrate that by increasing the intervention strength, the infection level decays; by increasing the limitation of treatment, the infection level increases. Our quantitative observations can be useful for exploring the relative importance of intervention and medical resources. As a timing application, we parameterize the model for COVID-19 in Spain and Italy. With strict intervention policies, the infection numbers would have been greatly reduced in the early phase of COVID-19 in Spain and Italy. We also show that reducing the time delays in intervention and recovery would have decreased the total number of cases in the early phase of COVID-19 in Spain and Italy. Our work highlights the necessity to consider the time delays in intervention and recovery in an epidemic model.
Citation: Sarita Bugalia, Jai Prakash Tripathi, Hao Wang. Mathematical modeling of intervention and low medical resource availability with delays: Applications to COVID-19 outbreaks in Spain and Italy[J]. Mathematical Biosciences and Engineering, 2021, 18(5): 5865-5920. doi: 10.3934/mbe.2021295
Infectious diseases have been one of the major causes of human mortality, and mathematical models have been playing significant roles in understanding the spread mechanism and controlling contagious diseases. In this paper, we propose a delayed SEIR epidemic model with intervention strategies and recovery under the low availability of resources. Non-delayed and delayed models both possess two equilibria: the disease-free equilibrium and the endemic equilibrium. When the basic reproduction number $ R_0 = 1 $, the non-delayed system undergoes a transcritical bifurcation. For the delayed system, we incorporate two important time delays: $ \tau_1 $ represents the latent period of the intervention strategies, and $ \tau_2 $ represents the period for curing the infected individuals. Time delays change the system dynamics via Hopf-bifurcation and oscillations. The direction and stability of delay induced Hopf-bifurcation are established using normal form theory and center manifold theorem. Furthermore, we rigorously prove that local Hopf bifurcation implies global Hopf bifurcation. Stability switching curves and crossing directions are analyzed on the two delay parameter plane, which allows both delays varying simultaneously. Numerical results demonstrate that by increasing the intervention strength, the infection level decays; by increasing the limitation of treatment, the infection level increases. Our quantitative observations can be useful for exploring the relative importance of intervention and medical resources. As a timing application, we parameterize the model for COVID-19 in Spain and Italy. With strict intervention policies, the infection numbers would have been greatly reduced in the early phase of COVID-19 in Spain and Italy. We also show that reducing the time delays in intervention and recovery would have decreased the total number of cases in the early phase of COVID-19 in Spain and Italy. Our work highlights the necessity to consider the time delays in intervention and recovery in an epidemic model.
[1] | F. Brauer, C. Castillo-Chavez, Mathematical models in population biology and epidemiology, New York: Springer, 2 (2012). |
[2] | K. Dietz, J. A. Heesterbeek, Daniel Bernoulli's epidemiological model revisited, Math. Biosci., 180 (2002), 1–21. doi: 10.1016/S0025-5564(02)00122-0 |
[3] | W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A Math. Phys. Sci., 115 (1927), 700–721. |
[4] | H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev. Soc. Ind. Appl. Math., 42 (2000), 599–653. |
[5] | R. M. Anderson, B. Anderson, R. M. May, Infectious diseases of humans: dynamics and control, Oxford University Press, 1992. |
[6] | S. Ruan, W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differ. Equ., 188 (2003), 135–163. doi: 10.1016/S0022-0396(02)00089-X |
[7] | J. P. Tripathi, S. Abbas, Global dynamics of autonomous and nonautonomous SI epidemic models with nonlinear incidence rate and feedback controls, Nonlinear Dyn, 86 (2016), 337–351. doi: 10.1007/s11071-016-2892-0 |
[8] | L. J. Gross, Mathematical models in plant biology: An overview, Applied Mathematical Ecology, (1989), 385–407. |
[9] | W. M. Liu, H. W. Hethcote, S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359–380. doi: 10.1007/BF00277162 |
[10] | Y. Cai, Y. Kang, M. Banerjee, W. Wang, A stochastic SIRS epidemic model with infectious force under intervention strategies, J. Differ. Equ., 259 (2015), 7463–7502. doi: 10.1016/j.jde.2015.08.024 |
[11] | M. Zhou, T. Zhang, Global Analysis of an SEIR Epidemic Model with Infectious Force under Intervention Strategies, J. Appl. Math. Phys., 7 (2019), 1706. doi: 10.4236/jamp.2019.78117 |
[12] | J. Cui, Y. Sun, H. Zhu, The impact of media on the control of infectious diseases, J. Dyn. Differ. Equ., 20 (2008), 31–53. doi: 10.1007/s10884-007-9075-0 |
[13] | J.A. Cui, X. Tao, H. Zhu, An SIS infection model incorporating media coverage, Rocky Mt. J. Math., (2008), 1323–1334. |
[14] | D. Xiao, S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419–429. doi: 10.1016/j.mbs.2006.09.025 |
[15] | Y. Xiao, T. Zhao, S. Tang, Dynamics of an infectious diseases with media/psychology induced non-smooth incidence, Math. Biosci. Eng., 10 (2013), 445. doi: 10.3934/mbe.2013.10.445 |
[16] | S. Tang, Y. Xiao, L. Yuan, R. A. Cheke, J. Wu, Campus quarantine (Fengxiao) for curbing emergent infectious diseases: Lessons from mitigating A/H1N1 in Xi'an, China, J. Theor. Biol., 295 (2012), 47–58. doi: 10.1016/j.jtbi.2011.10.035 |
[17] | Y. Xiao, S. Tang, J. Wu, Media impact switching surface during an infectious disease outbreak, Sci. Rep., 5 (2015), 7838. doi: 10.1038/srep07838 |
[18] | A. B. Gumel, S. Ruan, T. Day, J. Watmough, F. Brauer, P. Van den Driessche, et al., Modelling strategies for controlling SARS outbreaks, Proc. R. Soc. Lond. B Biol. Sci., 271 (2004), 2223–2232. doi: 10.1098/rspb.2004.2800 |
[19] | J. Zhang, J. Lou, Z. Ma, J. Wu, A compartmental model for the analysis of SARS transmission patterns and outbreak control measures in China, Appl. Math. Comput., 162 (2005), 909–924. |
[20] | S. Bugalia, V. P. Bajiya, J. P. Tripathi, M. T. Li, G. Q. Sun, Mathematical modeling of COVID-19 transmission: The roles of intervention strategies and lockdown, Math. Biosci. Eng., 17 (2020), 5961–5986. doi: 10.3934/mbe.2020318 |
[21] | V. P. Bajiya, S. Bugalia, J. P. Tripathi, Mathematical modeling of COVID-19: Impact of non-pharmaceutical interventions in India, Chaos Interdiscipl. J. Nonlinear Sci., 30 (2020), 113143. doi: 10.1063/5.0021353 |
[22] | M. T. Li, G. Q. Sun, J. Zhang, Y. Zhao, X. Pei, L. Li, et al., Analysis of COVID-19 transmission in Shanxi Province with discrete time imported cases, Math. Biosci. Eng., 17 (2020), 3710. doi: 10.3934/mbe.2020208 |
[23] | C. N. Ngonghala, E. Iboi, S. Eikenberry, M. Scotch, C. R. MacIntyre, M. H. Bonds, et al., Mathematical assessment of the impact of non-pharmaceutical interventions on curtailing the 2019 novel Coronavirus, Math. Biosci., 325 (2020), 108364. doi: 10.1016/j.mbs.2020.108364 |
[24] | S. Batabyal, A. Batabyal, Mathematical computations on epidemiology: A case study of the novel coronavirus (SARS-CoV-2), Theory Biosci., (2021), 1–16. |
[25] | S. Batabyal, COVID-19: Perturbation dynamics resulting chaos to stable with seasonality transmission, Chaos Solitons Fractals, 145 (2021), 110772. doi: 10.1016/j.chaos.2021.110772 |
[26] | W. Wang, S. Ruan, Bifurcations in an epidemic model with constant removal rate of the infectives, J. Math. Anal. Appl., 291 (2004), 775–793. doi: 10.1016/j.jmaa.2003.11.043 |
[27] | W. Wang, Epidemic models with nonlinear infection forces, Math. Biosci. Eng., 3 (2006), 267. doi: 10.3934/mbe.2006.3.267 |
[28] | C. Shan, H. Zhu, Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds, J. Differ. Equ., 257 (2014), 1662–1688. doi: 10.1016/j.jde.2014.05.030 |
[29] | G. H. Li, Y. X. Zhang, Dynamic behaviors of a modified SIR model in epidemic diseases using nonlinear incidence and recovery rates, PLoS One, 12 (2017), e0175789. doi: 10.1371/journal.pone.0175789 |
[30] | R. Mu, A. Wei, Y. Yang, Global dynamics and sliding motion in A (H7N9) epidemic models with limited resources and Filippov control, J. Math. Anal. Appl., 477 (2019), 1296–1317. doi: 10.1016/j.jmaa.2019.05.013 |
[31] | H. Zhao, L. Wang, S. M. Oliva, H. Zhu, Modeling and Dynamics Analysis of Zika Transmission with Limited Medical Resources, Bull. Math. Biol., 82 (2020), 1–50. doi: 10.1007/s11538-019-00680-3 |
[32] | A. Sirijampa, S. Chinviriyasit, W. Chinviriyasit, Hopf bifurcation analysis of a delayed SEIR epidemic model with infectious force in latent and infected period, Adv. Differ. Equ., 1 (2018), 348. |
[33] | E. Beretta, D. Breda, An SEIR epidemic model with constant latency time and infectious period, Math. Biosci. Eng., 8 (2011), 931. doi: 10.3934/mbe.2011.8.931 |
[34] | Z. Zhang, S. Kundu, J. P. Tripathi, S. Bugalia, Stability and Hopf bifurcation analysis of an SVEIR epidemic model with vaccination and multiple time delays, Chaos Solitons Fractals, 131 (2020), 109483. doi: 10.1016/j.chaos.2019.109483 |
[35] | P. Song, Y. Xiao, Global hopf bifurcation of a delayed equation describing the lag effect of media impact on the spread of infectious disease, J. Math. Biol., 76 (2018), 1249–1267. doi: 10.1007/s00285-017-1173-y |
[36] | A. K. Misra, A. Sharma, V. Singh, Effect of awareness programs in controlling the prevalence of an epidemic with time delay, J. Biol. Sys., 19 (2011), 389–402. doi: 10.1142/S0218339011004020 |
[37] | F. Al Basir, S. Adhurya, M. Banerjee, E. Venturino, S. Ray, Modelling the Effect of Incubation and Latent Periods on the Dynamics of Vector-Borne Plant Viral Diseases, Bull. Math. Biol., 82 (2020), 1–22. doi: 10.1007/s11538-019-00680-3 |
[38] | S. Liao, W. Yang, Cholera model incorporating media coverage with multiple delays, Math. Methods Appl. Sci., 42 (2019), 419–439. doi: 10.1002/mma.5175 |
[39] | K. A. Pawelek, S. Liu, F. Pahlevani, L. Rong, A model of HIV-1 infection with two time delays: mathematical analysis and comparison with patient data, Math. Biosci., 235 (2012), 98–109. doi: 10.1016/j.mbs.2011.11.002 |
[40] | D. Greenhalgh, S. Rana, S. Samanta, T. Sardar, S. Bhattacharya, J. Chattopadhyay, Awareness programs control infectious disease–multiple delay induced mathematical model, Appl. Math. Comput., 251 (2015), 539–563. |
[41] | H. Zhao, M. Zhao, Global Hopf bifurcation analysis of an susceptible-infective-removed epidemic model incorporating media coverage with time delay, J. Biol. Dyn., 11 (2017), 8–24. doi: 10.1080/17513758.2016.1229050 |
[42] | F. Al Basir, S. Ray, E. Venturino, Role of media coverage and delay in controlling infectious diseases: A mathematical model, Appl. Math. Comput., 337 (2018), 372–385. |
[43] | J. Liu, Bifurcation analysis for a delayed SEIR epidemic model with saturated incidence and saturated treatment function, J. Biol. Dyn., 13 (2019), 461–480. doi: 10.1080/17513758.2019.1631965 |
[44] | G. H. Li, Y. X. Zhang, Dynamic behaviors of a modified SIR model in epidemic diseases using nonlinear incidence and recovery rates, PLoS One, 12 (2017), e0175789. doi: 10.1371/journal.pone.0175789 |
[45] | D. P. Gao, N. J. Huang, S. M. Kang, C. Zhang, Global stability analysis of an SVEIR epidemic model with general incidence rate, Bound. Value Probl., (2018), 42. |
[46] | G. Huang, Y. Takeuchi, W. Ma, D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192–1207. doi: 10.1007/s11538-009-9487-6 |
[47] | E. Beretta, Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144–1165. doi: 10.1137/S0036141000376086 |
[48] | Q. An, E. Beretta, Y. Kuang, C. Wang, H. Wang, Geometric stability switch criteria in delay differential equations with two delays and delay dependent parameters, J. Differ. Equ., 266 (2019), 7073–7100. doi: 10.1016/j.jde.2018.11.025 |
[49] | Y. Kuang, editor, Delay differential equations: with applications in population dynamics, Academic Press, 1993. |
[50] | X. Yang, Generalized form of Hurwitz-Routh criterion and Hopf bifurcation of higher order, Appl. Math. Lett., 15 (2002), 615–621. doi: 10.1016/S0893-9659(02)80014-3 |
[51] | C. Castillo-Chavez, B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361. doi: 10.3934/mbe.2004.1.361 |
[52] | C. Castillo-Chavez, Z. Feng, W. Huang, On the computation of $R_0$ and its role on global stability, IMA Volumes in Mathematics and Its Applications, 1 (2002), 229. |
[53] | Y. Y. Min, Y. G. Liu, Barbalat lemma and its application in analysis of system stability, Journal of Shandong University (engineering science), 37 (2007), 51–55. |
[54] | S. Ruan, J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynamics of Continuous Discrete and Impulsive Systems Series A, 10 (2003), 863–874. |
[55] | H. L. Freedman, V. S. Rao, The trade-off between mutual interference and time lags in predator-prey systems, Bull. Math. Biol., 45 (1983), 991–1004. doi: 10.1016/S0092-8240(83)80073-1 |
[56] | L. H. Erbe, H. I. Freedman, V. S. Rao, Three-species food-chain models with mutual interference and time delays, Math. Biosci., 80 (1986), 57–80. doi: 10.1016/0025-5564(86)90067-2 |
[57] | B. D. Hassard, N. D. Kazarinoff, Y. H. Wan, Y. W. Wan, Theory and applications of Hopf bifurcation, CUP Archive, 41 (1981). |
[58] | J. J. Benedetto, W. Czaja, Riesz Representation Theorem, In Integration and Modern Analysis, Birkhäuser Boston, (2009), 321–357. |
[59] | J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Am. Math. Soc., 350 (1998), 4799–4838. doi: 10.1090/S0002-9947-98-02083-2 |
[60] | H. I. Freedman, S. Ruan, M. Tang, Uniform persistence and flows near a closed positively invariant set, J. Dyn. Differ. Equ., 6 (1994), 583–600. doi: 10.1007/BF02218848 |
[61] | Worldometer, coronavirus, available from: https://www.worldometers.info/coronavirus/. |
[62] | The print news, available from: https://theprint.in/world/this-is-how-france\-italy-and-spain-are-easing-their-lockdowns-one-step-at-a-time/413060/. |
[63] | World Health Organization, COVID-19. Available from: https://covid19.who.int/. |
[64] | X. Lin, H. Wang, Stability analysis of delay differential equations with two discrete delays, Canadian Appl. Math. Quart., 20 (2012), 519–533. |
[65] | L. Wang, J. Wang, H. Zhao, Y. Shi, K. Wang, P. Wu, et al., Modelling and assessing the effects of medical resources on transmission of novel coronavirus (COVID-19) in Wuhan, China, Math. Biosci. Eng., 17 (2020), 2936–2949. doi: 10.3934/mbe.2020165 |