Nonlinear dynamics of a new seasonal epidemiological model with age-structure and nonlinear incidence rate. (English) Zbl 1476.34099
Summary: In this article, we study the dynamics of a new proposed age-structured population mathematical model driven by a seasonal forcing function that takes into account the variability of the climate. We introduce a generalized force of infection function to study different potential disease outcomes. Using nonlinear analysis tools and differential inequalities theorems, we obtain sufficient conditions that guarantee the existence of a positive periodic solution. Moreover, we provide sufficient conditions that assure the global attractivity of the positive periodic solution. Numerical results corroborate the theoretical results in the sense that the solutions are positive and the periodic solution is a global attractor. This type of models are important, since they take into account the variability of the weather and the impact on some epidemics such as the current COVID-19 pandemic.
MSC:
| 34C25 | Periodic solutions to ordinary differential equations |
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 37N25 | Dynamical systems in biology |
Keywords:
seasonal epidemic model; age-structured; mathematical modeling; global attraction; positive periodic solutionReferences:
| [1] | Arenas, AJ; González, G.; Jódar, L., Existence of periodic solutions in a model of respiratory syncytial virus RSV, J Math Anal Appl, 344, 2, 969-980 (2008) · Zbl 1137.92318 |
| [2] | Arenas, AJ; González-Parra, G.; Moraño, JA, Stochastic modeling of the transmission of respiratory syncytial virus RSV in the region of Valencia, Spain, Biosystems, 96, 3, 206-212 (2009) |
| [3] | Ávila-Vales, E.; Rivero-Esquivel, E.; García-Almeida, GE, Global dynamics of a periodic SEIRS model with general incidence rate, Int J Differ Equ, 2017, 1-14 (2017) · Zbl 1487.92033 |
| [4] | Azuma, K.; Kagi, N.; Kim, H.; Hayashi, M., Impact of climate and ambient air pollution on the epidemic growth during COVID-19 outbreak in Japan, Environ Res, 190, 110042 (2020) |
| [5] | Bashir MF, Ma B, Komal B, Bashir MA, Tan D, Bashir M., et al. (2020) Correlation between climate indicators and COVID-19 pandemic in New York, USA. Science of The Total Environment p. 138835 |
| [6] | Bezruchko, BP; Smirnov, DA, Constructing nonautonomous differential equations from experimental time series, Phys Rev E, 63, 1, 016207 (2000) |
| [7] | Blackwood, JC; Childs, LM, An introduction to compartmental modeling for the budding infectious disease modeler, Lett Biomath, 5, 1, 195-221 (2018) |
| [8] | Brauer, F., Mathematical epidemiology: past, present, and future, Infect Dis Model, 2, 2, 113-127 (2017) |
| [9] | Calatayud, J.; Jornet, M., Mathematical modeling of adulthood obesity epidemic in Spain using deterministic, frequentist and Bayesian approaches, Chaos Solitons Fractals, 140, 110179 (2020) |
| [10] | Capasso, V.; Serio, G., A generalization of the Kermack-Mckendrick deterministic epidemic model, Math Biosci, 42, 1-2, 43-61 (1978) · Zbl 0398.92026 |
| [11] | Caraballo T, Han X (2017) Applied nonautonomous and random dynamical systems: applied dynamical systems. Springer · Zbl 1370.37001 |
| [12] | Chen, F., Periodicity in a ratio-dependent predator-prey system with stage structure for predator, J Appl Math, 2005, 2, 153-169 (2005) · Zbl 1103.34060 |
| [13] | Codeço, CT, Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir, BMC Infect Dis, 1, 1, 1 (2001) |
| [14] | Coleman, BD; Hsieh, YH; Knowles, GP, On the optimal choice of r for a population in a periodic environment, Math Biosci, 46, 1-2, 71-85 (1979) · Zbl 0429.92022 |
| [15] | Corberán-Vallet A, Santonja FJ, Jornet-Sanz M, Villanueva RJ (2018) Modeling chickenpox dynamics with a discrete time bayesian stochastic compartmental model. Complexity 2018 · Zbl 1376.62082 |
| [16] | Derrick, W.; Van den Driessche, P., A disease transmission model in a nonconstant population, J Math Biol, 31, 5, 495-512 (1993) · Zbl 0772.92015 |
| [17] | Ding, X.; Zhao, G., Periodic solutions for a semi-ratio-dependent predator-prey system with delays on time scales, Discrete Dyn Nat Soc, 2012, 15 (2012) · Zbl 1248.34140 |
| [18] | Donaldson, GC, Climate change and the end of the respiratory syncytial virus season, Clin Infect Dis, 42, 5, 677-679 (2006) |
| [19] | Fagan, WF; Bewick, S.; Cantrell, S.; Cosner, C.; Varassin, IG; Inouye, DW, Phenologically explicit models for studying plant-pollinator interactions under climate change, Theor Ecol, 7, 3, 289-297 (2014) |
| [20] | Ferrero, F.; Torres, F.; Abrutzky, R.; Ossorio, MF; Marcos, A.; Ferrario, C.; Rial, MJ, Seasonality of respiratory syncytial virus in Buenos Aires. Relationship with global climate change, Arch Argent Pediatr, 114, 1, 52-5 (2016) |
| [21] | Gaines, RE; Mawhin, JL, Coincidence degree and nonlinear differential equations (2006), Berlin: Springer, Berlin · Zbl 0339.47031 |
| [22] | Gao, Dp; Huang, Nj; Kang, SM; Zhang, C., Global stability analysis of an SVEIR epidemic model with general incidence rate, Bound Value Probl, 2018, 1, 42 (2018) · Zbl 1499.92107 |
| [23] | Garrett Birkhoff, GCR, Ordinary differential equations (1989), New York: Wiley, New York · JFM 63.0436.03 |
| [24] | Gölgeli M, Atay FM (2020) Analysis of an epidemic model for transmitted diseases in a group of adults and an extension to two age classes. Hacettepe J Math Stat 49(3):921-934 · Zbl 1488.92067 |
| [25] | González-Parra, G.; Arenas, AJ; Jódar, L., Piecewise finite series solutions of seasonal diseases models using multistage adomian method, Commun Nonlinear Sci Numer Simul, 14, 11, 3967-3977 (2009) · Zbl 1221.65201 |
| [26] | Gopalsamy K (2013) Stability and oscillations in delay differential equations of population dynamics, vol 74. Springer · Zbl 0752.34039 |
| [27] | Guerrero-Flores, S.; Osuna, O.; Vargas-DeLeon, C., Periodic solutions for seasonal SIQRS models with nonlinear infection terms, Electron J Differ Equ, 2019, 92, 1-13 (2019) · Zbl 1419.37079 |
| [28] | Guo, H.; Chen, X., Existence and global attractivity of positive periodic solution for a Volterra model with mutual interference and Beddington-Deangelis functional response, Appl Math Comput, 217, 12, 5830-5837 (2011) · Zbl 1217.34081 |
| [29] | Hethcote, HW; van den Driessche, P., Some epidemiological models with nonlinear incidence, J Math Biol, 29, 271-287 (1991) · Zbl 0722.92015 |
| [30] | Hogan AB, Glass K, Moore HC, Anderssen RS (2016a) Age structures in mathematical models for infectious diseases, with a case study of respiratory syncytial virus. In: Anderssen R, et al (eds) Applications + practical conceptualization + mathematics = fruitful innovation. Mathematics for industry, vol 11. Springer, Tokyo. doi:10.1007/978-4-431-55342-7_9 |
| [31] | Hogan AB, Glass K, Moore HC, Anderssen RS (2016b) Exploring the dynamics of respiratory syncytial virus RSV transmission in children. Theor Popul Biol 110:78-85 · Zbl 1365.92122 |
| [32] | Hogan, AB; Campbell, PT; Blyth, CC; Lim, FJ; Fathima, P.; Davis, S.; Moore, HC; Glass, K., Potential impact of a maternal vaccine for RSV: a mathematical modelling study, Vaccine, 35, 45, 6172-6179 (2017) |
| [33] | Hurtado, LA; Cáceres, L.; Chaves, LF; Calzada, JE, When climate change couples social neglect: malaria dynamics in Panamá, Emerg Microbes Infect, 3, 1, 1-11 (2014) |
| [34] | Jia, J.; Zhang, H., Existence and global attractivity of periodic solutions for chemostat model with delayed nutrients recycling, Differ Equ Appl, 6, 275-286 (2014) · Zbl 1300.92081 |
| [35] | Jódar, L.; Villanueva, RJ; Arenas, A., Modeling the spread of seasonal epidemiological diseases: theory and applications, Math Comput Model, 48, 3, 548-557 (2008) · Zbl 1145.92336 |
| [36] | Jornet-Sanz, M.; Corberán-Vallet, A.; Santonja, F.; Villanueva, R., A Bayesian stochastic SIRS model with a vaccination strategy for the analysis of respiratory syncytial virus, SORT Stat Oper Res Trans, 41, 1, 159-176 (2017) · Zbl 1376.62082 |
| [37] | Korobeinikov, A., Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull Math Biol, 71, 75-83 (2009) · Zbl 1169.92041 |
| [38] | Li, B.; Xiong, X., Existence and global attractivity of periodic solution for a discrete prey-predator model with sex structure, Nonlinear Anal Real World Appl, 11, 3, 1986-2000 (2010) · Zbl 1205.39004 |
| [39] | Li, Y.; Qin, J., Existence and global exponential stability of periodic solutions for quaternion-valued cellular neural networks with time-varying delays, Neurocomputing, 292, 91-103 (2018) |
| [40] | Li, Y.; Qin, J.; Li, B., Existence and global exponential stability of anti-periodic solutions for delayed quaternion-valued cellular neural networks with impulsive effects, Math Methods Appl Sci, 42, 1, 5-23 (2019) · Zbl 1440.92005 |
| [41] | Li, Y.; Teng, Z.; Ruan, S.; Li, M.; Feng, X., A mathematical model for the seasonal transmission of schistosomiasis in the lake and marshland regions of China, Math Biosci Eng, 14, 1279 (2017) · Zbl 1382.37100 |
| [42] | De Luca G, Van Kerckhove K, Coletti P, Poletto C, Bossuyt N, Hens N, Colizza V (2018) The impact of regular school closure on seasonal influenza epidemics: a data-driven spatial transmission model for Belgium. BMC Infect Diseases 18(1):1-16 |
| [43] | Malki, Z.; Atlam, ES; Hassanien, AE; Dagnew, G.; Elhosseini, MA; Gad, I., Association between weather data and COVID-19 pandemic predicting mortality rate: machine learning approaches, Chaos Solitons Fractals, 138, 110137 (2020) |
| [44] | Mateus, JP; Silva, CM, Existence of periodic solutions of a periodic SEIRS model with general incidence, Nonlinear Anal Real World Appl, 34, 379-402 (2017) · Zbl 1354.34085 |
| [45] | Mawhin, J.; Ferrera, J.; López-Gómez, J.; del Portal, FR, Periodic solutions in the golden sixties: the birth of a continuation theorem, 10 mathematical essays on approximation in analysis and topology, 199-214 (2005), Amsterdam: Elsevier Science, Amsterdam · Zbl 1078.34022 |
| [46] | Mitchell, C.; Kribs, C., Invasion reproductive numbers for periodic epidemic models, Infect Dis Model, 4, 124-141 (2019) |
| [47] | Noyola, D.; Mandeville, P., Effect of climatological factors on respiratory syncytial virus epidemics, Epidemiol Infect, 136, 10, 1328-1332 (2008) |
| [48] | Okuneye, K.; Abdelrazec, A.; Gumel, AB, Mathematical analysis of a weather-driven model for the population ecology of mosquitoes, Math Biosci Eng, 15, 1, 57-93 (2018) · Zbl 1375.92055 |
| [49] | Paiva, TM; Ishida, MA; Benega, MA; Constantino, CR; Silva, DB; Santos, KC; Oliveira, MI; Barbosa, HA; Carvalhanas, TR; Schuck-Paim, C., Shift in the timing of respiratory syncytial virus circulation in a subtropical megalopolis: Implications for immunoprophylaxis, J Med Virol, 84, 11, 1825-1830 (2012) |
| [50] | Ponciano, JM; Capistran, MA, First principles modeling of nonlinear incidence rates in seasonal epidemics, PLoS Comput Biol, 7, 1-14 (2011) |
| [51] | Posny, D.; Wang, J., Modelling cholera in periodic environments, J Biol Dyn, 8, 1, 1-19 (2014) · Zbl 1448.92330 |
| [52] | Ren, X.; Zhang, T., Global analysis of an SEIR epidemic model with a ratio-dependent nonlinear incidence rate, J Appl Math Phys, 5, 2311-2319 (2017) |
| [53] | Rogovchenko, SP; Rogovchenko, YV, Effect of periodic environmental fluctuations on the Pearl-Verhulst model, Chaos Solitons Fractals, 39, 3, 1169-1181 (2009) · Zbl 1197.34062 |
| [54] | Rosa, S.; Torres, DF, Optimal control of a fractional order epidemic model with application to human respiratory syncytial virus infection, Chaos Solitons Fractals, 117, 142-149 (2018) · Zbl 1442.92180 |
| [55] | Rui, X.; Lan-sun, C.; Fei-long, H., Periodic solutions of a delayed predator-prey model with stage structure for prey, Acta Math Appl Sin Engl Ser, 20, 2, 323-332 (2004) · Zbl 1067.34073 |
| [56] | Safi, MA; Garba, SM, Global stability analysis of SEIR model with holling type II incidence function, Comput Math Methods Med, 2012, 1-8 (2012) · Zbl 1256.92041 |
| [57] | Samanta, S.; Alquran, M.; Chattopadhyay, J., Existence and global stability of positive periodic solution of tri-trophic food chain with middle predator migratory in nature, Appl Math Model, 39, 15, 4285-4299 (2015) · Zbl 1443.92167 |
| [58] | Sekerci, Y.; Petrovskii, S., Mathematical modelling of plankton-oxygen dynamics under the climate change, Bull Math Biol, 77, 12, 2325-2353 (2015) · Zbl 1332.92083 |
| [59] | Sengupta, S.; Das, P., Dynamics of two-prey one-predator non-autonomous type-III stochastic model with effect of climate change and harvesting, Nonlinear Dyn, 97, 4, 2777-2798 (2019) · Zbl 1430.37118 |
| [60] | Shobugawa, Y.; Takeuchi, T.; Hibino, A.; Hassan, MR; Yagami, R.; Kondo, H.; Odagiri, T.; Saito, R., Occurrence of human respiratory syncytial virus in summer in Japan, Epidemiol Infect, 145, 2, 272-284 (2017) |
| [61] | Traoré B, Sangaré B, Traoré S (2017) A mathematical model of malaria transmission with structured vector population and seasonality. J Appl Math 2017 · Zbl 1437.92081 |
| [62] | Valle, SYD; Hyman, JM; Chitnis, N., Mathematical models of contact patterns between age groups for predicting the spread of infectious diseases, Math Biosci Eng, 10, 1475 (2013) · Zbl 1273.92052 |
| [63] | Vega, YL; Ramirez, OV; Herrera, BA, Impact of climatic variability in the respiratory syncytial virus pattern in childrenunder 5 years-old using the Bulto climatic index in Cuba, Int J Virol Infect Dis, 2, 14-19 (2017) |
| [64] | Wang, L., Existence of periodic solutions of seasonally forced SIR models with impulse vaccination, Taiwan J Math, 19, 6, 1713-1729 (2015) · Zbl 1357.92078 |
| [65] | Weber, A.; Weber, M.; Milligan, P., Modeling epidemics caused by respiratory syncytial virus RSV, Math Biosci, 172, 95-113 (2001) · Zbl 0988.92025 |
| [66] | White, L.; Mandl, J.; Gomes, M.; Bodley-Tickell, A.; Cane, P.; Perez-Brena, P.; Aguilar, J.; Siqueira, M.; Portes, S.; Straliotto, S.; Waris, M.; Nokes, D.; Medley, G., Understanding the transmission dynamics of respiratory syncytial virus using multiple time series and nested models, Math Biosci, 209, 222-239 (2007) · Zbl 1120.92039 |
| [67] | White, L.; Waris, M.; Cane, P.; Nokes, D.; Medley, G., The transmission dynamics of groups A and B human respiratory syncytial virus hRSV in England and Wales and Finland: seasonality and cross-protection, Epidemiol Infect, 133, 279-289 (2005) |
| [68] | Yan, Y.; Sugie, J., Global asymptotic stability of a unique positive periodic solution for a discrete hematopoiesis model with unimodal production functions, Monatshefte Math, 191, 2, 325-348 (2020) · Zbl 1430.37120 |
| [69] | Zhang, T.; Liu, J.; Teng, Z., Existence of positive periodic solutions of an SEIR model with periodic coefficients, Appl Math, 57, 6, 601-616 (2012) · Zbl 1274.34150 |
| [70] | Zhu, Y.; Wang, K., Existence and global attractivity of positive periodic solutions for a predator-prey model with modified Leslie-Gower Holling-type II schemes, J Math Anal Appl, 384, 2, 400-408 (2011) · Zbl 1232.34077 |
| [71] | Zu, J.; Wang, L., Periodic solutions for a seasonally forced SIR model with impact of media coverage, Adv Differ Equ, 136, 1-10 (2015) · Zbl 1422.92182 |
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