JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2025, Vol. 60 ›› Issue (4): 1-19.doi: 10.6040/j.issn.1671-9352.0.2024.289
WANG Yi, HAN Zhimin, LI Siqi
CLC Number:
[1] 马知恩,周义仓,王稳地,等. 传染病动力学的数学建模与研究[M]. 北京:科学出版社,2004. MA Zhien, ZHOU Yicang, WANG Wendi, et al. Mathematical modeling and research on infectious disease dynamics[M]. Beijing: Science Press, 2004. [2] World Health Organation(WHO). Coronavirus disease(COVID-19)pandemic[EB/OL].(2019-11-12)[2025-02-18]. https://www.who.int/zh/emergencies/diseases/novel-coronavirus-2019. [3] CHENG Xinxin, WANG Yi, HUANG Gang. Edge-based compartmental modeling for the spread of cholera on random networks: a case study in Somalia[J]. Mathematical Biosciences, 2023, 366:109092. [4] JIN Xiulei, JIN Shuwan, GAO Daozhou. Mathematical analysis of the Ross-Macdonald model with quarantine[J]. Bulletin of Mathematical Biology, 2020, 82(4):47. [5] LAN Yuqiong, LI Yanqiu, ZHENG Dongmei. Global dynamics of an age-dependent multiscale hepatitis C virus model[J]. Journal of Mathematical Biology, 2022, 85(3):21. [6] KHAN M A, PARVEZ M, ISLAM S, et al. Mathematical analysis of typhoid model with saturated incidence rate[J]. Advanced Studies in Biology, 2015, 7(2):65-78. [7] BUONOMO B, DONOFRIO A, LACITIGNOLA D. Modeling of pseudo-rational exemption to vaccination for SEIR diseases[J]. Journal of Mathematical Analysis and Applications, 2013, 404(2):385-398. [8] SAHU G P, DHAR J. Analysis of an SVEIS epidemic model with partial temporary immunity and saturation incidence rate[J]. Applied Mathematical Modelling, 2012, 36(3):908-923. [9] LIU Qun, CHEN Qingmei, JIANG Daqing. The threshold of a stochastic delayed SIR epidemic model with temporary immunity[J]. Physica A: Statistical Mechanics and its Applications, 2016, 450:115-125. [10] LIU Qun, JIANG Daqing, HAYAT T, et al. Analysis of a delayed vaccinated SIR epidemic model with temporary immunity and Lévy jumps[J]. Nonlinear Analysis: Hybrid Systems, 2018, 27:29-43. [11] ZHAO H, FENG Z L. Staggered release policies for COVID-19 control: costs and benefits of relaxing restrictions by age and risk[J]. Mathematical Biosciences, 2020, 326:108405. [12] MUSA S S, WANG X Y, ZHAO S, et al. The heterogeneous severity of COVID-19 in African countries: a modeling approach[J]. Bulletin of Mathematical Biology, 2022, 84(3):32. [13] OLABODE D, CULP J, FISHER A, et al. Deterministic and stochastic models for the epidemic dynamics of COVID-19 in Wuhan, China[J]. Mathematical Biosciences and Engineering, 2021, 18(1):950-967. [14] LIU Yang, YAN Limeng, WAN Lagen, et al. Viral dynamics in mild and severe cases of COVID-19[J]. The Lancet Infectious Diseases, 2020, 20(6):656-657. [15] HE X, LAU E H Y, WU P, et al. Temporal dynamics in viral shedding and transmissibility of COVID-19[J]. Nature Medicine, 2020, 26(5):672-675. [16] GILCHRIST M A, SASAKI A. Modeling host-parasite coevolution: a nested approach based on mechanistic models[J]. Journal of Theoretical Biology, 2002, 218(3):289-308. [17] WEBB G F, DAGATA E M C, MAGAL P, et al. A model of antibiotic-resistant bacterial epidemics in hospitals[J]. Proceedings of the National Academy of Sciences of the United States of America, 2005, 102(37):13343-13348. [18] FRASER C, HOLLINGSWORTH T D, CHAPMAN R, et al. Variation in HIV-1 set-point viral load: epidemiological analysis and an evolutionary hypothesis[J]. Proceedings of the National Academy of Sciences of the United States of America, 2007, 104(44):17441-17446. [19] SAENZ R A, BONHOEFFER S. Nested model reveals potential amplification of an HIV epidemic due to drug resistance[J]. Epidemics, 2013, 5(1):34-43. [20] NUMFOR E, BHATTACHARYA S, LENHART S, et al. Optimal control in coupled within-host and between-host models[J]. Mathematical Modelling of Natural Phenomena, 2014, 9(4):171-203. [21] MARTCHEVA M. An introduction to mathematical epidemiology[M]. New York: Springer, 2015. [22] WANG Xueying, WANG Sunpeng, WANG Jin, et al. A multiscale model of COVID-19 dynamics[J]. Bulletin of Mathematical Biology, 2022, 84(9):99. [23] SOUZA M O. Multiscale analysis for a vector-borne epidemic model[J]. Journal of Mathematical Biology, 2014, 68(5):1269-1293. [24] BLOWER S, BERNOULLI D. An attempt at a new analysis of the mortality caused by smallpox and of the advantages of inoculation to prevent it[J]. Reviews in Medical Virology, 2004, 14(5):275. [25] ENKO P D. On the course of epidemics of some infectious diseases[J]. International Journal of Epidemiology, 1989, 18(4):749-755. [26] HAMER W H. Epidemic disease in England: the evidence of variability and of persistency of type[M]. [S.l.] : Bedford Press, 1906. [27] BROWNLEE J. Statistical studies in immunity: the theory of an epidemic[J]. Proceedings of the Royal Society of Edinburgh, 1906, 26(1):484-521. [28] ROSS R. The prevention of malaria[M]. [S.l.] : John Murray, 1911. [29] KERMACK W O, MCKENDRICK A G. A contribution to the mathematical theory of epidemics[J]. Proceedings of the Royal Society of London. Series A, 1927, 115(772):700-721. [30] KERMACK W O, MCKENDRICK A G. Contributions to the mathematical theory of epidemics, part II[J]. Proceedings of the Royal Society of London. Series A, 1932, 138:55-83. [31] KERMACK W O, MCKENDRICK A G. Contributions to the mathematical theory of epidemics, part III[J]. Proceedings of the Royal Society of London. Series A, 1933, 141:94-112. [32] TURKYILMAZOGLU M. Explicit formulae for the peak time of an epidemic from the SIR model[J]. Physica D: Nonlinear Phenomena, 2021, 422:132902. [33] NG K Y, GUI M M. COVID-19: development of a robust mathematical model and simulation package with consideration for ageing population and time delay for control action and resusceptibility[J]. Physica D: Nonlinear Phenomena, 2020, 411: 132599. [34] CUI Jingan, WU Yucui, GUO Songbai. Effect of non-homogeneous mixing and asymptomatic individuals on final epidemic size and basic reproduction number in a meta-population model[J]. Bulletin of Mathematical Biology, 2022, 84(3):38. [35] XIAO Y N, XIANG C C, CHEKE R A, et al. Coupling the macroscale to the microscale in a spatiotemporal context to examine effects of spatial diffusion on disease transmission[J]. Bulletin of Mathematical Biology, 2020, 82(5):58. [36] FENG Z L, CASTILLO-CHAVEZ C, CAPURRO A F. A model for tuberculosis with exogenous reinfection[J]. Theoretical Population Biology, 2000, 57(3): 235-247. [37] WANG Xia, WU Hulin, TANG Sanyi. Assessing age-specific vaccination strategies and post-vaccination reopening policies for COVID-19 control using SEIR modeling approach[J]. Bulletin of Mathematical Biology, 2022, 84(10):108. [38] BUGALIA S, TRIPATHI J P, WANG H. Mutations make pandemics worse or better: modeling SARS-CoV-2 variants and imperfect vaccination[J]. Journal of Mathematical Biology, 2024, 88(4):45. [39] CUEVAS-MARAVER J, KEVREKIDIS P G, CHEN Q Y, et al. Vaccination compartmental epidemiological models for the delta and omicron SARS-CoV-2 variants[J]. Mathematical Biosciences, 2024, 367:109109. [40] PANT B, SAFDAR S, SANTILLANA M, et al. Mathematical assessment of the role of human behavior changes on SARS-CoV-2 transmission dynamics in the United States[J]. Bulletin of Mathematical Biology, 2024, 86(8):92. [41] 唐三一,肖燕妮,彭志行,等.新型冠状病毒肺炎疫情预测建模、数据融合与防控策略分析[J].中华流行病学杂志,2020,41(4): 480-484. TANG Sanyi, XIAO Yanni, PENG Zhihang, et al. Prediction modeling with data fusion and prevention strategy analysis for the COVID-19 outbreak[J]. Chinese Journal of Epidemiology, 2020, 41(4):480-484. [42] WANG Sunpeng, PAN Yang, WANG Quanyi, et al. Modeling the viral dynamics of SARS-CoV-2 infection[J]. Mathematical Biosciences, 2020, 328:108438. [43] FENG Z L, VELASCO-HERNANDEZ J, TAPIA-SANTOS B, et al. A model for coupling within-host and between-host dynamics in an infectious disease[J]. Nonlinear Dynamics, 2012, 68(3):401-411. [44] CAI L M, TUNCER N, MARTCHEVA M. How does within-host dynamics affect population-level dynamics? Insights from an immuno-epidemiological model of malaria[J]. Mathematical Methods in the Applied Sciences, 2017, 40(18):6424-6450. [45] COOMBS D, GILCHRIST M A, BALL C L. Evaluating the importance of within-and between-host selection pressures on the evolution of chronic pathogens[J]. Theoretical Population Biology, 2007, 72(4):576-591. [46] DANG Y X, LI X Z, MARTCHEVA M. Competitive exclusion in a multi-strain immuno-epidemiological influenza model with environmental transmission[J]. Journal of Biological Dynamics, 2016, 10(1):416-456. [47] MARTCHEVA M, LI X Z. Linking immunological and epidemiological dynamics of HIV: the case of super-infection[J]. Journal of Biological Dynamics, 2013, 7(1):161-182. [48] CANDELA M G, SERRANO E, MARTINEZ-CARRASCO C, et al. Coinfection is an important factor in epidemiological studies: the first serosurvey of the aoudad(Ammotragus lervia)[J]. European Journal of Clinical Microbiology & Infectious Diseases, 2009, 28(5):481-489. [49] LI Xuezhi, GAO Shasha, FU Yike, et al. Modeling and research on an immuno-epidemiological coupled system with coinfection[J]. Bulletin of Mathematical Biology, 2021, 83(11):116. [50] BALL C L, GILCHRIST M A, COOMBS D. Modeling within-host evolution of HIV: mutation, competition and strain replacement[J]. Bulletin of Mathematical Biology, 2007, 69(7):2361-2385. [51] GILCHRIST M A, COOMBS D. Evolution of virulence: interdependence, constraints, and selection using nested models[J]. Theoretical Population Biology, 2006, 69(2):145-153. [52] DUAN Xichao, SUN Xiaosa, YUAN Sanling. Dynamics of an immune-epidemiological model with virus evolution and super infection[J]. Journal of the Franklin Institute, 2022, 359(7):3210-3237. [53] XUE Y Y, CHEN D P, SMITH S R, et al. Coupling the within-host process and between-host transmission of COVID-19 suggests vaccination and school closures are critical[J]. Bulletin of Mathematical Biology, 2022, 85(1):6. [54] SHEN Mingwang, XIAO Yanni, RONG Libin. Global stability of an infection-age structured HIV-1 model linking within-host and between-host dynamics[J]. Mathematical Biosciences, 2015, 263:37-50. [55] SUN Xiaodan, XIAO Yanni, TANG Sanyi, et al. Early HAART initiation may not reduce actual reproduction number and prevalence of MSM infection: perspectives from coupled within-and between-host modelling studies of Chinese MSM populations[J]. PLoS One, 2016, 11(3):e0150513. [56] SHEN Mingwang, XIAO Yanni, RONG Libin, et al. Conflict and accord of optimal treatment strategies for HIV infection within and between hosts[J]. Mathematical Biosciences, 2019, 309:107-117. [57] SHEN Mingwang, XIAO Yanni, RONG Libin, et al. Global dynamics and cost-effectiveness analysis of HIV pre-exposure prophylaxis and structured treatment interruptions based on a multi-scale model[J]. Applied Mathematical Modelling, 2019, 75:162-200. [58] ALMOCERA A E S, NGUYEN V K, HERNANDEZ-VARGAS E A. Multiscale model within-host and between-host for viral infectious diseases[J]. Journal of Mathematical Biology, 2018, 77(4):1035-1057. [59] ALMOCERA A E S, HERNANDEZ-VARGAS E A. Coupling multiscale within-host dynamics and between-host transmission with recovery(SIR)dynamics[J]. Mathematical Biosciences, 2019, 309:34-41. [60] AILI A, TENG Z D, ZHANG L. Dynamics in a disease transmission model coupled virus infection in host with incubation delay and environmental effects[J]. Journal of Applied Mathematics and Computing, 2022, 68(6):4331-4359. [61] LI Jun, ZHAO Yulin, LI Shimin. Fast and slow dynamics of Malaria model with relapse[J]. Mathematical Biosciences, 2013, 246(1):94-104. [62] SULLIVAN A, AGUSTO F, BEWICK S, et al. A mathematical model for within-host Toxoplasma gondii invasion dynamics[J]. Mathematical Biosciences and Engineering, 2012, 9(3):647-662. [63] TURNER M, LENHART S, ROSENTHAL B, et al. Modeling effective transmission pathways and control of the worlds most successful parasite[J]. Theoretical Population Biology, 2013, 86:50-61. [64] FENG Z L, VELASCO-HERNANDEZ J, TAPIA-SANTOS B. A mathematical model for coupling within-host and between-host dynamics in an environmentally-driven infectious disease[J]. Mathematical Biosciences, 2013, 241(1):49-55. [65] CEN Xiuli, FENG Zhilan, ZHAO Yulin. Emerging disease dynamics in a model coupling within-host and between-host systems[J]. Journal of Theoretical Biology, 2014, 361:141-151. [66] FENG Zhilan, CEN Xiuli, ZHAO Yulin, et al. Coupled within-host and between-host dynamics and evolution of virulence[J]. Mathematical Biosciences, 2015, 270:204-212. [67] XUE Yuyi, XIAO Yanni. Analysis of a multiscale HIV-1 model coupling within-host viral dynamics and between-host transmission dynamics[J]. Mathematical Biosciences and Engineering, 2020, 17(6):6720-6736. [68] LIU Qiutong, XIAO Yanni. A coupled evolutionary model of the viral virulence in an SIS community[J]. Discrete and Continuous Dynamical Systems-B, 2023, 28(9):5012-5036. [69] IWASA Y, POMIANKOWSKI A, NEE S. The evolution of costly mate preferences II: the "handicap" principle[J]. Evolution, 1991, 45(6):1431-1442. [70] YANG Junyuan, JIA Peiqi, WANG Jin, et al. Rich dynamics of a bidirectionally linked immuno-epidemiological model for cholera[J]. Journal of Mathematical Biology, 2023, 87(5):71. [71] ERDÖS P, RÉNYI A. On random graphs[J]. Publicationes Mathematicae Debrecen, 2022, 6(314):290-297. [72] ERDÖS P, RÉNYI A. On the evolution of random graphs[J]. Publications of the Mathematical Institute of the Hungarian Academy of Sciences, 1960, 5:17-61. [73] ERDÖS P, RÉNYI A. On the strength of connectedness of a random graph[J]. Acta Mathematica Academiae Scientiarum Hungaricae, 1964, 12(1):261-267. [74] 汪小帆,李翔,陈关荣. 复杂网络理论及其应用[M]. 北京:清华大学出版社,2006. WANG Xiaofan, LI Xiang, CHEN Guanrong. Theory and applications of complex networks[M]. Beijing: Tsinghua University Press, 2006. [75] WATTS D J, STROGATZ S H. Collective dynamics of ‘small-world’ networks[J]. Nature, 1998, 393(6684):440-442. [76] KITSAK M, GALLOS L K, HAVLIN S, et al. Identification of influential spreaders in complex networks[J]. Nature Physics, 2010, 6(11):888-893. [77] PIQUEIRA J R C, NAVARRO B F, MONTEIRO L H A. Epidemiological models applied to viruses in computer networks[J]. Journal of Computer Science, 2005, 1(1):31-34. [78] DE SILVA E, STUMPF M P H. Complex networks and simple models in biology[J]. Journal of the Royal Society Interface, 2005, 2(5):419-430. [79] CATANZARO M, BUCHANAN M. Network opportunity[J]. Nature Physics, 2013, 9(3):121-123. [80] NEWMAN M J, PARK J. Why social networks are different from other types of networks[J]. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 2003, 68(3):036122. [81] KUMPULA J M, ONNELA J P, SARAMÄKI J, et al. Emergence of communities in weighted networks[J]. Physical Review Letters, 2007, 99(22):228701. [82] PASTOR-SATORRAS R, CASTELLANO C, VAN MIEGHEM P, et al. Epidemic processes in complex networks[J]. Reviews of Modern Physics, 2015, 87(3):925-979. [83] WANG Yi, CAO Jinde, JIN Zhen, et al. Impact of media coverage on epidemic spreading in complex networks[J]. Physica A, 2013, 392(23):5824-5835. [84] ZANETTE D H. Dynamics of rumor propagation on small-world networks[J]. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 2002, 65(4):041908. [85] HOLME P, SARAMÄKI J. Temporal networks[J]. Physics Reports, 2012, 519(3):97-125. [86] LIU Guirong, LIU Zhimei, JIN Zhen. Dynamics analysis of epidemic and information spreading in overlay networks[J]. Journal of Theoretical Biology, 2018, 444:28-37. [87] 汪小帆,李翔,陈关荣.网络科学导论[M]. 北京:高等教育出版社,2012. WANG Xiaofan, LI Xiang, CHEN Guanrong. Introduction to network science[M]. Beijing: Higher Education Press, 2012. [88] 刘茂省.网络传染病动力学新进展[M]. 北京:世界图书出版公司,2015. LIU Maoxing. Recent advances in the dynamics of network infectious diseases[M]. Beijing: World Publishing Corporation, 2015. [89] WANG Yi, WEI Zhouchao, CAO Jinde. Epidemic dynamics of influenza-like diseases spreading in complex networks[J]. Nonlinear Dynamics, 2020, 101(3):1801-1820. [90] KLOVDAHL A S. Social networks and the spread of infectious diseases: the AIDS example[J]. Social Science & Medicine, 1985, 21(11):1203-1216. [91] MAY R M, ANDERSON R M. Commentary transmission dynamics of HIV infection[J]. Nature, 1987, 326(6109):137-142. [92] NEWMAN M E J, WATTS D J. Scaling and percolation in the small-world network model[J]. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 1999, 60(6):7332-7342. [93] PASTOR-SATORRAS R, VESPIGNANI A. Epidemic spreading in scale-free networks[J]. Physical Review Letters, 2001, 86(14):3200-3203. [94] MORENO Y, PASTOR-SATORRAS R, VESPIGNANI A. Epidemic outbreaks in complex heterogeneous networks[J]. The European Physical Journal B, 2002, 26(4):521-529. [95] BOGUNÁ M, PASTOR-SATORRAS R. Epidemic spreading in correlated complex networks[J]. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 2002, 66(4):047104. [96] MIZUTAKA S, MORI K, HASEGAWA T. Synergistic epidemic spreading in correlated networks[J]. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 2022, 106(3):034305. [97] LIU Z H, HU B. Epidemic spreading in community networks[J]. Europhysics Letters, 2005, 72(2):315-321. [98] BONACCORSI S, OTTAVIANO S, DE PELLEGRINI F, et al. Epidemic outbreaks in two-scale community networks[J]. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 2014, 90(1):012810. [99] LIU Junli, ZHANG Tailei. Epidemic spreading of an SEIRS model in scale-free networks[J]. Communications in Nonlinear Science and Numerical Simulation, 2011, 16(8):3375-3384. [100] MENG Xueyu, CAI Zhiqiang, SI Shubin, et al. Analysis of epidemic vaccination strategies on heterogeneous networks: based on SEIRV model and evolutionary game[J]. Applied Mathematics and Computation, 2021, 403:126172. [101] STELLA L, MARTÍNEZ A P, BAUSO D, et al. The role of asymptomatic infections in the COVID-19 epidemic via complex networks and stability analysis[J]. SIAM Journal on Control and Optimization, 2022, 60(2):S119-S144. [102] KEELING M J, RAND D A, MORRIS A J. Correlation models for childhood epidemics[J]. Proceedings of the Royal Society of London. Series B, 1997, 264(1385):1149-1156. [103] EAMES K T D, KEELING M J. Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases[J]. Proceedings of the National Academy of Sciences, 2002, 99(20):13330-13335. [104] EAMES K T D, KEELING M J. Monogamous networks and the spread of sexually transmitted diseases[J]. Mathematical Biosciences, 2004, 189(2):115-130. [105] LINDQUIST J, MA J L, VAN DEN DRIESSCHE P, et al. Effective degree network disease models[J]. Journal of Mathematical Biology, 2011, 62(2):143-164. [106] NEWMAN M J. Spread of epidemic disease on networks[J]. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 2002, 66(1):016128. [107] VOLZ E. SIR dynamics in random networks with heterogeneous connectivity[J]. Journal of Mathematical Biology, 2008, 56(3):293-310. [108] MILLER J C. A note on a paper by Erik Volz: SIR dynamics in random networks[J]. Journal of Mathematical Biology, 2011, 62(3):349-358. [109] WANG Yi, MA Junling, CAO Jinde, et al. Edge-based epidemic spreading in degree-correlated complex networks[J]. Journal of Theoretical Biology, 2018, 454:164-181. [110] WANG Y, MA Junling, CAO Jinde. Basic reproduction number for the SIR epidemic in degree correlated networks[J]. Physica D: Nonlinear Phenomena, 2022, 433:133183. [111] GROSS T, DLIMA C J D, BLASIUS B. Epidemic dynamics on an adaptive network[J]. Physical Review Letters, 2006, 96(20):208701. [112] GROSS T, BLASIUS B. Adaptive coevolutionary networks: a review[J]. Journal of the Royal Society Interface, 2008, 5(20):259-271. [113] SHAW L B, SCHWARTZ I B. Fluctuating epidemics on adaptive networks[J]. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 2008, 77(6):066101. [114] SZABÓ-SOLTICZKY A, BERTHOUZE L, KISS I Z, et al. Oscillating epidemics in a dynamic network model: stochastic and mean-field analysis[J]. Journal of Mathematical Biology, 2016, 72(5):1153-1176. [115] PU Xiaojun, ZHU Jiaqi, WU Yunkun, et al. Dynamic adaptive spatio-temporal graph network for COVID-19 forecasting[J]. CAAI Transactions on Intelligence Technology, 2023, 9(3):769-786. [116] BARTHÉLEMY M. Spatial networks[J]. Physics Reports, 2011, 499(1/2/3):1-101. [117] HOLME P, SARAMÄKI J. Temporal networks[J]. Physics Reports, 2012, 519(3):97-125. [118] PUJARI B S, SHEKATKAR S. Multi-city modeling of epidemics using spatial networks: application to 2019-nCov(COVID-19)coronavirus in India[J]. Medrxiv, 2020: 2020.03.13.20035386. [119] FRIESWIJK K, ZINO L, CAO M. A polarized temporal network model to study the spread of recurrent epidemic diseases in a partially vaccinated population[J]. IEEE Transactions on Network Science and Engineering, 2023, 10(6):3732-3743. [120] PORTER M A. Nonlinearity+networks: a 2020 vision[M] //KEVREKIDIS P, CUEVAS-MARAVER J, SAXENA A. Emerging Frontiers in Nonlinear Science. Cham: Springer International Publishing, 2020:131-159. [121] GUPTA C, TUNCER N, MARTCHEVA M. A network immuno-epidemiological HIV model[J]. Bulletin of Mathematical Biology, 2021, 83(3):18. [122] SMITH H L, DE LEENHEER P. Virus dynamics: a global analysis[J]. SIAM Journal on Applied Mathematics, 2003, 63(4):1313-1327. [123] GUPTA C, TUNCER N, MARTCHEVA M. A network immuno-epidemiological model of HIV and opioid epidemics[J]. Mathematical Biosciences and Engineering, 2023, 20(2):4040-4068. [124] DUAN X C, Li X Z, MARTCHEVA M. Coinfection dynamics of heroin transmission and HIV infection in a single population[J]. Journal of Biological Dynamics, 2020, 14(1):116-142. [125] U.S. Overdose deaths in 2021 increased half as much as in 2020-but are still up 15 percent, 2022 [EB/OL].(2022-03-07)[2025-02-18]. https://www.cdc.gov/nchs/pressroom/nchs press releases/2022/202205.html [126] Centers for Disease Control and Prevention, HIV basic statistics, 2022[EB/OL].(2022-04-09)[2025-02-18]. https://www.cdc.gov/hiv/basics/statistics.html. [127] YANG Junyuan, DUAN Xinyi, LI Xuezhi. Dynamical analysis of an immumo-epidemiological coupled system on complex networks[J]. Communications in Nonlinear Science and Numerical Simulation, 2023, 119:107116. [128] HAN Zhimin, WANG Yi, JIN Zhen. Final and peak epidemic sizes of immuno-epidemiological SIR models[J]. Discrete and Continuous Dynamical Systems-B, 2024, 29(11):4432-4462. [129] GILCHRIST M A, COOMBS D. Evolution of virulence: interdependence, constraints, and selection using nested models[J]. Theoretical Population Biology, 2006, 69(2):145-153. |
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