# American Institute of Mathematical Sciences

November  2019, 24(11): 6297-6323. doi: 10.3934/dcdsb.2019140

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

 Department of Mathematics & Center for Computational and Applied Mathematics, California State University, Fullerton, Fullerton, CA 92831, USA

Received  August 2018 Revised  February 2019 Published  November 2019 Early access  July 2019

Is it possible to break the host-vector chain of transmission when there is an influx of infectious hosts into a naïve population and competent vector? To address this question, a class of vector-borne disease models with an arbitrary number of infectious stages that account for immigration of infective individuals is formulated. The proposed model accounts for forward and backward progression, capturing the mitigation and aggravation to and from any stages of the infection, respectively. The model has a rich dynamic, which depends on the patterns of infected immigrant influx into the host population and connectivity of the transfer between infectious classes. We provide conditions under which the answer of the initial question is positive.

Citation: Derdei Mahamat Bichara. Effects of migration on vector-borne diseases with forward and backward stage progression. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6297-6323. doi: 10.3934/dcdsb.2019140
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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
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 $\delta_{43} = 0.03$
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 $\delta_{43} = 0.03$
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))
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
Description of the parameters used in System (1)
 Parameters Description $\pi_h$ Recruitment of the host $\pi_v$ Recruitment of vectors $p_0$ Proportion of latent immigrants $p_i$ Proportion of infectious immigrants at stage $i$ $a$ Biting rate $\mu_h$ Host's natural death rate $\beta_{v, h}$ Host's infectiousness by mosquitoes per biting $\beta_{i}$ Vector's infectiousness by host at stage $i$ per biting $\nu_h$ Host's rate at which the exposed individuals become infectious $\eta_i$ Per capita recovery rate of an infected host at stage $i$ $\gamma_{ij}$ Host's per capita progression rate from stage $i$ to $j$ $\delta_{ij}$ Host's per capita regression rate from stage $i$ to $j$ $\mu_v$ Vectors' natural mortality rate $\delta_v$ Vectors' control-induced mortality rate $\nu_v$ Rate at which the exposed vectors become infectious
 Parameters Description $\pi_h$ Recruitment of the host $\pi_v$ Recruitment of vectors $p_0$ Proportion of latent immigrants $p_i$ Proportion of infectious immigrants at stage $i$ $a$ Biting rate $\mu_h$ Host's natural death rate $\beta_{v, h}$ Host's infectiousness by mosquitoes per biting $\beta_{i}$ Vector's infectiousness by host at stage $i$ per biting $\nu_h$ Host's rate at which the exposed individuals become infectious $\eta_i$ Per capita recovery rate of an infected host at stage $i$ $\gamma_{ij}$ Host's per capita progression rate from stage $i$ to $j$ $\delta_{ij}$ Host's per capita regression rate from stage $i$ to $j$ $\mu_v$ Vectors' natural mortality rate $\delta_v$ Vectors' control-induced mortality rate $\nu_v$ Rate at which the exposed vectors become infectious
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