Effects of migration on vector-borne diseases with forward and backward stage progression

Figure 1.  Flow diagram of Model 1. Note that, to unclutter the figure, we did not display the arrows that represent the recruitments for $ I_2 $, $ I_3 $ and $ I_4 $. Similarly, the arrows representing the death, $ \mu_i $, and recovery rates, $ \eta_i, $ in all host classes are not displayed

Figure 2.  Effects of host-vector transmission on the dynamics of Model (2) with $ n = 4 $. The proportions of infectious influx are $ p_1 = 0.2 $, $ p_3 = 0.1 $, $ p_4 = 0 $ and $ p_{5} = 0.3 $. The transfer matrix $ M $ is such as $ \gamma_{13} = \gamma_{24} = 0.1 $, $ \gamma_{14} = 0.2 $, $ \delta_{21} = 0.01 $, $ \delta_{31} = 0.02 $, $ \delta_{41} = 0.001 $, $ \delta_{32} = 0.03 $, $ \delta_{42} = 0.01 $ and $ \delta_{43} = 0.03 $

Figure 3.  Dynamics of infected hosts and vectors when the hypotheses of Theorem 3.1, Item 3 are satisfied. The proportions of infectious influx are $ p_0 = 0 $, $ {\bf p} = (0, 0, p_3, p_4)^T = (0, 0, 0.2, 0.0001)^T $ and $ p_{5} = 0.3 $. The transfer matrix $ M $ is such as $ \gamma_{13} = 0.1 $, $ \gamma_{14} = \gamma_{24} = 0 $, $ \delta_{21} = 0.01 $, $ \delta_{31} = \delta_{32} = \delta_{42} = 0 $, $ \delta_{41} = 0.035 $, and $ \delta_{43} = 0.03 $

Figure 4.  Dynamics of infected hosts and vectors when the hypotheses of Theorem 3.1, Item 4 are satisfied. The proportions of infectious influx are $ p_0 = 0 $, $ {\bf p} = (0, 0, p_3, p_4)^T = (0, 0, 0.2, 0.0001)^T $ and $ p_{5} = 0.3 $. The transfer matrix $ M $ is such as $ \gamma_{13} = 0.1 $, $ \gamma_{14} = \gamma_{24} = 0 $, $ \delta_{21} = 0.01 $, $ \delta_{31} = \delta_{32} = \delta_{41} = \delta_{42} = 0 $, and $ \delta_{43} = 0.03 $. With these parameters, $ {\mathcal N}_0^2 = 0.3237\leq1 $. As expected, the vector population will be disease-free (Figure 4(b) and Figure 4(d)) and the infectious hosts are generated only through influx of infectious immigrants at stage 3 and 4 (Figure 4(a) and Figure 4(c))

Figure 5.  Dynamics of infected hosts and vectors when the hypotheses of Theorem 3.1, Item 4 are satisfied. The proportions of infectious influx are $ p_0 = 0 $, $ {\bf p} = (0, 0, p_3, p_4)^T = (0, 0, 0.2, 0.0001)^T $ and $ p_{5} = 0.3 $. The transfer matrix $ M $ is such as $ \gamma_{13} = 0.1 $, $ \gamma_{14} = \gamma_{24} = 0 $, $ \delta_{21} = 0.01 $, $ \delta_{31} = \delta_{32} = \delta_{41} = \delta_{42} = 0 $, and $ \delta_{43} = 0.03 $. Using the values $ a = 0.9 $ and $ \beta_{vh} = 0.9 $, $ {\mathcal N}_0^2 = 1.6051>1 $, and thus the trajectories of the system converge towards an interior equilibrium