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

• * Corresponding author: Kun Lu

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.

Mathematics Subject Classification: Primary: 92D30; Secondary: 34D20.

 Citation: • • Figure 1.  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

Figure 2.  The existence of equilibria of system (2)

Figure 3.  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 $R_{21} = 1$

Figure 4.  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

Figure 5.  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

Figure 6.  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

Figure 7.  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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