JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (10): 84-96.doi: 10.6040/j.issn.1671-9352.0.2022.543

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Dynamics of a two-strain co-infection epidemic model with vaccination

Gang CHEN(),Rui ZHANG*()   

  1. School of Mathematics and Physics, Lanzhou Jiaotong University, Gansu 730070, Lanzhou, China
  • Received:2022-10-10 Online:2023-10-20 Published:2023-10-17
  • Contact: Rui ZHANG E-mail:chencom@163.com;zhr639066@163.com

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Abstract:

To explore the dynamics of co-infection of multiple strains in the same host population, a mathematical model of co-transmission dynamics of two strains after continuous inoculation with strain 1 vaccine is established and analyzed. Firstly, the sufficient conditions for the existence of four equilibrium points are obtained by calculating and analyzing the model. In addition to the disease-free equilibrium point and the two single endemic equilibrium points, the model also has an endemic equilibrium point where both strains 1 and 2 coexist. Secondly, Lyapunov stability theorem is used to prove that the disease-free equilibrium is globally stable when the basic reproduction number of two strains is less than 1. The invasion-reproduction number is introduced to determine the stability of the single-strain endemic equilibrium point. When the corresponding invasion reproduction number is less than 1, the endemic equilibrium point of the strain is locally stable. Then, using Castillo-Chavez and Song?s bifurcation theorem, it is proved that the model does not have backward bifurcation phenomenon, and then it is proved that the coexistence equilibrium point is locally asymptotically stable when the basic reproduction number of the two strains is greater than 1. Finally, the above conclusions are verified by numerical simulation.

Key words: multiple strains, co-infection, basic reproduction number, global stability, endemic equilibrium, invasion reproduction number

CLC Number: 

  • O175

Fig.1

Flowchart of a two-strain co-infection model"

Table 1

Parameters description of Model (1)"

参数 参数描述 参数 参数描述
Λ 外来人口输入率 γ3 共感者同时对菌株1和菌株2的恢复率
μ 区域内人口自然死亡率 δ1β1 共感者对菌株2感染者的传播率
ρ 输入人口菌株1疫苗的接种率(ρ < 1) δ2β2 共感者对菌株1感染者的传播率
ν 单位时间菌株1疫苗持续的接种率 α1、α2 菌株1、2感染者的恢复率
β1、β2 菌株1/2感染者对易感人群的有效接触率 d1、d2 菌株1、2感染者的因病致死率
β3 共感者同时对易感人群的菌株1，2的传播率 d3 共感者的因病致死率
γ1、γ2 共感者对菌株1、2的恢复率 σ 恢复人群对免疫能力的消失率

Table 2

Parameters values of Model (1)"

参数 图 2(a)取值 图 2(b)取值 图 2(c)取值 图 2(d)取值
Λ 0.070 0.070 0.070 0.070
μ 3.5×10-5 3.5×10-5 3.5×10-5 3.5×10-5
ρ 5.0×10-3 5.0×10-3 5.0×10-3 5.0×10-3
ν 1.0×10-5 1.0×10-5 1.0×10-5 1.0×10-5
β1 0.050 0.040 0.060 0.013
β2 0.050 0.080 0.030 0.075
β3 0.050 0.015 0.020 0.050
γ1 0.045 0.030 0.030 0.030
γ2 0.030 0.025 0.025 0.038
γ3 0.020 0.050 0.050 0.050
δ1 0.800 0.800 1.100 0.800
δ2 0.600 0.400 1.300 0.400
α1 0.060 0.018 0.020 0.020
α2 0.060 0.025 0.035 0.030
d1 5.5×10-6 5.5×10-6 5.5×10-6 5.5×10-6
d2 5.5×10-6 5.5×10-6 5.5×10-6 5.5×10-6
d3 9.0×10-6 9.0×10-6 9.0×10-6 9.0×10-6
σ 0.080 0.080 0.080 0.080

Fig.2

Numerical simulations of Model (1)"

1 BRAUER F , CASTILLO- CHÁVEZ C . Mathematical models in population biology and epidemiology[M]. 2nd ed. New York: Springer, 2012: 18- 25.
2 UNAIDS . Report on the global AIDS epidemic[M]. Geneva: World Health Organization, 2008: 11- 62.
3 皮特·布鲁克史密斯. 未来的灾难: 瘟疫复活与人类的生存之战[M]. 马永波, 译. 海口: 海南出版社, 1999: 17-184.
BROOKESMITH Peter. Future disaster: the resurrection of plague and the battle for human survival[M]. MA Yongbo, Translation. Haikou: Hainan Press, 1999: 17-184.
4 World Health Organization . World malaria situation in 1993[J]. Releve Epidemiologique Hebdomadaire, 1996, 71 (3): 17- 22.
5 UNAIDS , World Health Organization . Epidemiological Fact Sheet on HIV/AIDS and sexually transmitted infections[J]. Age, 2000, 30 (34): 45- 49.
6 MARTCHEVA M . An introduction to mathematical epidemiology[M]. Boston: Springer, 2015: 173- 174.
7 GETAHUN H , GUNNEBERG C , GRANICH R , et al. HIV infection-associated tuberculosis: the epidemiology and the response[J]. Clinical Infectious Diseases, 2010, 50 (Suppl.3): 201- 207.
8 KANG M , HOLLABAUGH K , PHAM V , et al. Virologic and serologic outcomes of mono versus dual HBV therapy and characterization of HIV/HBV coinfection in a US cohort[J]. Journal of Acquired Immune Deficiency Syndromes, 2014, 66 (2): 172- 180.
doi: 10.1097/QAI.0000000000000149
9 SHARP G B , KAWAOKA Y , JONES D J , et al. Coinfection of wild ducks by influenza A viruses: distribution patterns and biological significance[J]. Journal of Virology, 1997, 71 (8): 6128- 6135.
doi: 10.1128/jvi.71.8.6128-6135.1997
10 CHATURVEDI A K , KATKI H A , HILDESHEIM A , et al. Human papillomavirus infection with multiple types: pattern of coinfection and risk of cervical disease[J]. The Journal of Infectious Diseases, 2011, 203 (7): 910- 920.
doi: 10.1093/infdis/jiq139
11 ALLEN L J S , LANGLAIS M , PHILLIPS C J . The dynamics of two viral infections in a single host population with applications to hantavirus[J]. Mathematical Biosciences, 2003, 186 (2): 191- 217.
doi: 10.1016/j.mbs.2003.08.002
12 FERGUSON N , ANDERSON R , GUPTA S . The effect of antibody-dependent enhancement on the transmission dynamics and persistence of multiple-strain pathogens[J]. Proceedings of the National Academy of Sciences of the United States of America, 1999, 96 (2): 790- 794.
13 KAWAGUCHI I , SASAKI A , BOOTS M . Why are dengue virus serotypes so distantly related? Enhancement and limiting serotype similarity between dengue virus strains[J]. Proceedings Biological Sciences, 2003, 270 (1530): 2241- 2247.
doi: 10.1098/rspb.2003.2440
14 ZHANG P , SANDLAND G J , FENG Z , et al. Evolutionary implications for interactions between multiple strains of host and parasite[J]. Journal of Theoretical Biology, 2007, 248 (2): 225- 240.
doi: 10.1016/j.jtbi.2007.05.011
15 HAUG S , LAKEW T , HABTEMARIAM G , et al. The decline of pneumococcal resistance after cessation of mass antibiotic distributions for trachoma[J]. Clinical Infectious Diseases, 2010, 51 (5): 571- 574.
doi: 10.1086/655697
16 LAKSHMIKANTHAM V , LEELA S , MARTYN I UK A A . Stability analysis of nonlinear systems[M]. New York: New York Dekker, 1989: 42- 43.
17 SMITH H L , WALTMAN P E . The theory of the chemostat: dynamics of microbial competition[M]. Cambridge: Cambridge University Press, 1995: 120.
18 VAN DEN DRIESSCHE P , WATMOUGH J . Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission[J]. Mathematical Biosciences, 2002, 180 (1/2): 29- 48.
19 DIEKMANN O , HEESTERBEEK J A P , ROBERTS M G . The construction of next-generation matrices for compartmental epidemic models[J]. Journal of the Royal Society Interface, 2010, 7 (47): 873- 885.
doi: 10.1098/rsif.2009.0386
20 林支桂. 数学生态学导引[M]. 北京: 科学出版社, 2013: 19.
LIN Zhigui . Guidance of mathematical ecology[M]. Beijing: Science Press, 2013: 19.
21 CASTILLO-CHAVEZ C , SONG B J . Dynamical models of tuberculosis and their applications[J]. Mathematical Biosciences and Engineering, 2004, 1 (2): 361- 404.
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