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

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