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## Dynamical behavior of a rotavirus disease model with two strains and homotypic protection

 1 Department of Mathematics, Shaanxi University of Science and Technology, Xi'an 710021, China 2 School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: Kun Lu

Received  June 2019 Revised  September 2019 Published  January 2020

Fund Project: The first author is supported by Natural Science Foundation of Shaanxi Provincial Department of Education grant 18JK0092, the second author is supported by National Natural Science Foundation of China grant 11571284, and the third author is supported by National Natural Science Foundation of China grant 11971281.

A two-strain rotavirus model with vaccination and homotypic protection is proposed to study the survival of the two strains of rotavirus within the host. Corresponding to the different efficacy of monovalent vaccine against different strains, the vaccination reproduction numbers of the two strains and the reproduction numbers of their mutual invasion are found. Based on the existence and local stability of equilibria, our results suggest that the obtained reproduction numbers determine together the dynamics of the model, and that the two-strain rotavirus dies out as both the numbers is less than unity. The coexistence of two strains, one of which is dominant, is also related to the two reproduction numbers.

Citation: Kun Lu, Wendi Wang, Jianquan Li. Dynamical behavior of a rotavirus disease model with two strains and homotypic protection. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020066
##### References:
 [1] C. Atchison, B. Lopman and W. J. Edmunds, Modelling the seasonality of rotavirus disease and the impact of vaccination in England and Wales, Vaccine, 28 (2010), 3118-3126.  doi: 10.1016/j.vaccine.2010.02.060.  Google Scholar [2] R. F. Bishop, G. L. Barnes, E. Cipriani and J. S. Lund, Clinical immunity after neonatal rotavirus infection — A prospective longitudinal study in young children, New England J. Medicine, 309 (1983), 72-76.  doi: 10.1056/NEJM198307143090203.  Google Scholar [3] P. H. Dennehy, Rotavirus vaccines - An update, Pediatric Infectious Disease J., 17 (2005), 88-92.   Google Scholar [4] E. Kindler, E. Trojnar, G. Heckel, P. H. Otto and R. Johne, Analysis of rotavirus species diversity and evolution including the newly determined full-length genome sequences of rotavirus F and G, Infection, Genetics Evolution, 14 (2013), 58-67.  doi: 10.1016/j.meegid.2012.11.015.  Google Scholar [5] J. P. LaSalle, The stability of dynamical systems, in Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1976.  Google Scholar [6] A. C. Linhares, Y. B. Gabbay, J. D. P. Mascarenhas, R. B. Freitas and G. M. Beards, Epidemiology of rotavirus subgroups and serotypes in Belem, Brazil: A three-year study, Ann. Inst. Pasteur Virol., 139 (1988), 89-99.  doi: 10.1016/S0769-2617(88)80009-1.  Google Scholar [7] A. C. Linhares, Y. B. Gabbay, R. B. Freitas and E. S. Travassos da Rosa, et al., Longitudinal study of rotavirus infections among children from Belém, Brazil, Epidemiology Infection, 102 (1989), 129–145. doi: 10.1017/S0950268800029769.  Google Scholar [8] T. Nakagomi and O. Nakagomi, A critical review on a globally-licensed, live, orally-administrable, monovalent human rotavirus vaccine: Rotarix, Expert Opinion Biol. Therapy, 9 (2009), 1073-1086.  doi: 10.1517/14712590903103787.  Google Scholar [9] O. L. Omondi, C. C. Wang, X. P. Xue and O. G. Lawi, Modeling the effects of vaccination on rotavirus infection, Adv. Difference Equ., 2015 (2015), 381-392.  doi: 10.1186/s13662-015-0722-1.  Google Scholar [10] U. Parashar, Global illness and deaths caused by rotavirus disease in children, Emerg. Infect. Dis., 9 (2003), 565-572.  doi: 10.3201/eid0905.020562.  Google Scholar [11] G. M. Ruiz-Palacios, I. Pérez-Schael, F. Raúl Velázquez, H. Abate and M. O'Ryan, Safety and efficacy of an attenuated vaccine against severe rotavirus gastroenteritis, New England J. Medicine, 354 (2006), 11-22.  doi: 10.1056/NEJMoa052434.  Google Scholar [12] E. Shim, Z. Feng, M. Martcheva and C. Castillo-Chavez, An age-structured epidemic model of rotavirus with vaccination, J. Math. Biol., 53 (2006), 719-746.  doi: 10.1007/s00285-006-0023-0.  Google Scholar [13] A. Steele and B. Ivanoff, Rotavirus strains circulating in Africa during 1996–1999: Emergence of G9 strains and P[6] strains., Vaccine, 21 (2003), 361-367.   Google Scholar [14] J. E. Tate, M. M. Patel, A. D. Steele and J. R. Gentsch, et al., Global impact of rotavirus vaccines, Expert Rev. Vaccines, 9 (2010), 395–407. Google Scholar [15] J. E. Tate, A. H. Burton, C. Boschi-Pinto and A. D. Steele, et al., 2008 estimate of worldwide rotavirus-associated mortality in children younger than 5 years before the introduction of universal rotavirus vaccination programmes: A systematic review and meta-analysis, Lancet Infectious Diseases, 12 (2012), 136–141. doi: 10.1016/S1473-3099(11)70253-5.  Google Scholar [16] H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymtotically autonomous differential eqluations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.  Google Scholar [17] T. Van Effelterre, M. Soriano-Gabarró, S. Debrus, N. E. Claire and J. Gray, A mathematical model of the indirect effects of rotavirus vaccination, Epidemiol. Infect., 138 (2010), 884-897.  doi: 10.1017/S0950268809991245.  Google Scholar [18] F. R. Velázquez, D. O. Matson, J. J. Calva and M. L. Guerrero, et al., Rotavirus infections in infants as protection against subsequent infections, New England J. Medicine, 335 (1996), 1022–1028. Google Scholar [19] T. Vesikari, D. O. Matson, P. Dennehy, P. V. Damme and P. M. Heaton, Safety and efficacy of a pentavalent human-bovine (WC3) reassortant rotavirus vaccine, New England J. Medicine, 354 (2006), 23-33.  doi: 10.1056/NEJMoa052664.  Google Scholar [20] L. J. White, M. J. Cox and G. F. Medley, Cross immunity and vaccination against multiple microparasite strains, Ima J. Math. Appl. Medicine Biol., 15 (1998), 211-233.  doi: 10.1093/imammb/15.3.211.  Google Scholar [21] Y. Yan and W. Wang, Global stability of a five-dimensional model with immune responses and delay, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 401-416.  doi: 10.3934/dcdsb.2012.17.401.  Google Scholar [22] G. Young, E. Shim and G. B. Ermentrout, Qualitative effects of monovalent vaccination against rotavirus: A comparison of North America and South America, Bull. Math. Biol., 77 (2015), 1854-1885.  doi: 10.1007/s11538-015-0107-3.  Google Scholar [23] L. Yuan, S.-I. Ishida, S. Honma and J. Patton, et al., Homotypic and heterotypic serum isotype–specific antibody responses to rotavirus nonstructural protein 4 and viral protein (VP) 4, VP6, and VP7 in infants who received selected live oral rotavirus vaccines. J. Infect. Diseases, 189 (2004), 1833–1845. doi: 10.1086/383416.  Google Scholar

show all references

##### References:
 [1] C. Atchison, B. Lopman and W. J. Edmunds, Modelling the seasonality of rotavirus disease and the impact of vaccination in England and Wales, Vaccine, 28 (2010), 3118-3126.  doi: 10.1016/j.vaccine.2010.02.060.  Google Scholar [2] R. F. Bishop, G. L. Barnes, E. Cipriani and J. S. Lund, Clinical immunity after neonatal rotavirus infection — A prospective longitudinal study in young children, New England J. Medicine, 309 (1983), 72-76.  doi: 10.1056/NEJM198307143090203.  Google Scholar [3] P. H. Dennehy, Rotavirus vaccines - An update, Pediatric Infectious Disease J., 17 (2005), 88-92.   Google Scholar [4] E. Kindler, E. Trojnar, G. Heckel, P. H. Otto and R. Johne, Analysis of rotavirus species diversity and evolution including the newly determined full-length genome sequences of rotavirus F and G, Infection, Genetics Evolution, 14 (2013), 58-67.  doi: 10.1016/j.meegid.2012.11.015.  Google Scholar [5] J. P. LaSalle, The stability of dynamical systems, in Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1976.  Google Scholar [6] A. C. Linhares, Y. B. Gabbay, J. D. P. Mascarenhas, R. B. Freitas and G. M. Beards, Epidemiology of rotavirus subgroups and serotypes in Belem, Brazil: A three-year study, Ann. Inst. Pasteur Virol., 139 (1988), 89-99.  doi: 10.1016/S0769-2617(88)80009-1.  Google Scholar [7] A. C. Linhares, Y. B. Gabbay, R. B. Freitas and E. S. Travassos da Rosa, et al., Longitudinal study of rotavirus infections among children from Belém, Brazil, Epidemiology Infection, 102 (1989), 129–145. doi: 10.1017/S0950268800029769.  Google Scholar [8] T. Nakagomi and O. Nakagomi, A critical review on a globally-licensed, live, orally-administrable, monovalent human rotavirus vaccine: Rotarix, Expert Opinion Biol. Therapy, 9 (2009), 1073-1086.  doi: 10.1517/14712590903103787.  Google Scholar [9] O. L. Omondi, C. C. Wang, X. P. Xue and O. G. Lawi, Modeling the effects of vaccination on rotavirus infection, Adv. Difference Equ., 2015 (2015), 381-392.  doi: 10.1186/s13662-015-0722-1.  Google Scholar [10] U. Parashar, Global illness and deaths caused by rotavirus disease in children, Emerg. Infect. Dis., 9 (2003), 565-572.  doi: 10.3201/eid0905.020562.  Google Scholar [11] G. M. Ruiz-Palacios, I. Pérez-Schael, F. Raúl Velázquez, H. Abate and M. O'Ryan, Safety and efficacy of an attenuated vaccine against severe rotavirus gastroenteritis, New England J. Medicine, 354 (2006), 11-22.  doi: 10.1056/NEJMoa052434.  Google Scholar [12] E. Shim, Z. Feng, M. Martcheva and C. Castillo-Chavez, An age-structured epidemic model of rotavirus with vaccination, J. Math. Biol., 53 (2006), 719-746.  doi: 10.1007/s00285-006-0023-0.  Google Scholar [13] A. Steele and B. Ivanoff, Rotavirus strains circulating in Africa during 1996–1999: Emergence of G9 strains and P[6] strains., Vaccine, 21 (2003), 361-367.   Google Scholar [14] J. E. Tate, M. M. Patel, A. D. Steele and J. R. Gentsch, et al., Global impact of rotavirus vaccines, Expert Rev. Vaccines, 9 (2010), 395–407. Google Scholar [15] J. E. Tate, A. H. Burton, C. Boschi-Pinto and A. D. Steele, et al., 2008 estimate of worldwide rotavirus-associated mortality in children younger than 5 years before the introduction of universal rotavirus vaccination programmes: A systematic review and meta-analysis, Lancet Infectious Diseases, 12 (2012), 136–141. doi: 10.1016/S1473-3099(11)70253-5.  Google Scholar [16] H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymtotically autonomous differential eqluations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.  Google Scholar [17] T. Van Effelterre, M. Soriano-Gabarró, S. Debrus, N. E. Claire and J. Gray, A mathematical model of the indirect effects of rotavirus vaccination, Epidemiol. Infect., 138 (2010), 884-897.  doi: 10.1017/S0950268809991245.  Google Scholar [18] F. R. Velázquez, D. O. Matson, J. J. Calva and M. L. Guerrero, et al., Rotavirus infections in infants as protection against subsequent infections, New England J. Medicine, 335 (1996), 1022–1028. Google Scholar [19] T. Vesikari, D. O. Matson, P. Dennehy, P. V. Damme and P. M. Heaton, Safety and efficacy of a pentavalent human-bovine (WC3) reassortant rotavirus vaccine, New England J. Medicine, 354 (2006), 23-33.  doi: 10.1056/NEJMoa052664.  Google Scholar [20] L. J. White, M. J. Cox and G. F. Medley, Cross immunity and vaccination against multiple microparasite strains, Ima J. Math. Appl. Medicine Biol., 15 (1998), 211-233.  doi: 10.1093/imammb/15.3.211.  Google Scholar [21] Y. Yan and W. Wang, Global stability of a five-dimensional model with immune responses and delay, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 401-416.  doi: 10.3934/dcdsb.2012.17.401.  Google Scholar [22] G. Young, E. Shim and G. B. Ermentrout, Qualitative effects of monovalent vaccination against rotavirus: A comparison of North America and South America, Bull. Math. Biol., 77 (2015), 1854-1885.  doi: 10.1007/s11538-015-0107-3.  Google Scholar [23] L. Yuan, S.-I. Ishida, S. Honma and J. Patton, et al., Homotypic and heterotypic serum isotype–specific antibody responses to rotavirus nonstructural protein 4 and viral protein (VP) 4, VP6, and VP7 in infants who received selected live oral rotavirus vaccines. J. Infect. Diseases, 189 (2004), 1833–1845. doi: 10.1086/383416.  Google Scholar
Progression of infection from the breast-fed infants $X$ through the susceptible $S$, the vaccinated $V$, the infected $I_1$, $I_2$ and recovered $R_1$, $R_2$ for the compartments of the model
The existence of equilibria of system (2)
The local stability of the boundary equilibria of system (2), where LAS denotes locally asymptotically stable, $R_{20} = R_{20}^{(2)}$ is equivalent to $R_{12} = 1$, $R_{20} = R_{20}^{(1)}$ is equivalent to $R_{21} = 1$
The trajectories of $I_1$ and $I_2$ for the case that $R_{10}\geq 1$ and $R_{20}>R_{20}^{(2)}$. Here, $\beta_1 = 0.075$ and $\beta_2 = 0.2$. Correspondingly, $R_{10} = 1.66$, $R_{20} = 4.54$, $R_{20}^{(2)} = 1.64$, $E_2$ is globally stable
The trajectories of $I_1$ and $I_2$ for the case that $R_{10}> 1$ and $R_{20}<1$. Here, $\beta_1 = 0.08$ and $\beta_2 = 0.04$. Correspondingly, $R_{10} = 1.77$, $R_{20} = 0.68$, $E_1$ is globally stable
The trajectories of $I_1$ and $I_2$ for the case that $R_{10}>1$ and $1<R_{20}<R_{20}^{(2)}$. Here, $\beta_1 = 2$ and $\beta_2 = 0.2$. Correspondingly, $R_{10} = 44.28$, $R_{20} = 4.54$, $R_{20}^{(1)} = 5.26$, $E_1$ is globally stable
The trajectories of $I_1$ and $I_2$ for the case that $R_{10}>1$ and $R_{20}^{(1)}<R_{20}<R_{20}^{(2)}$. Here, $\beta_1 = 0.8$ and $\beta_2 = 0.4$. Correspondingly, $R_{10} = 17.71$, $R_{20} = 9.08$, $R_{20}^{(1)} = 4.58$, $R_{20}^{(2)} = 13.62$, $E^*$ is globally stable
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