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Article Contents

# A network model for control of dengue epidemic using sterile insect technique

• * Corresponding author
The first author is supported by MHRD grant.
• In this paper, a network model has been proposed to control dengue disease transmission considering host-vector dynamics in $n$ patches. The control of mosquitoes is performed by SIT. In SIT, the male insects are sterilized in the laboratory and released into the environment to control the number of offsprings. The basic reproduction number has been computed. The existence and stability of various states have been discussed. The bifurcation diagram has been plotted to show the existence and stability regions of disease-free and endemic states for an isolated patch. The critical level of sterile male mosquitoes has been obtained for the control of disease. The basic reproduction number for $n$ patch network model has been computed. It is evident from numerical simulations that SIT control in one patch may control the disease in the network having two/three patches with suitable coupling among them.

Mathematics Subject Classification: Primary: 97M10, 37C75.

 Citation:

• Figure 1.  The transfer diagram representing the system (1)-(11) dynamics. The births, deaths and migration are not included.

Figure 2.  The existence of the state $P_2$ with respect to parameters $\beta^2$ and $\phi$.

Figure 3.  Bifurcation diagram for the stability of the state $P_2$

Figure 4.  (a) A 3D phase plot showing the unstable behavior for the state $P^-_3$ for the four initial conditions $Y_1$=(14, 0.007, 0.001, 34, 75, 3.5, 2.5, 32, 0.004, 39,600), $Y_2$=(19, 0.009, 0.003, 32, 70, 3.0, 3.0, 36, 0.009, 35,615), $Y_3$=(20, 0.009, 0.08, 30, 78, 3.9, 2.0, 38, 0.09, 42,620) and $Y_4$=(15, 0.006, 0.05, 37, 72, 3.2, 1.5, 30, 0.02, 45,630). (b) A 3D phase plot showing the stable behavior for the state $P^+_3$ for the four initial conditions $Z_1$=(0.4, 0.013, 0.005, 40,400, 18, 50,140, 0.15,200,600), $Z_2$=(1.8, 0.025, 0.009, 51,410, 21, 43,145, 0.09,210,630), $Z_3$=(1.5, 0.022, 0.008, 48,398, 14, 45,132, 0.11,194,615) and $Z_4$=(0.8, 0.018, 0.007, 45,390, 17, 48,138, 0.18,198,610).

Figure 5.  Bifurcation plot of system (12)-(22) with respect to $\omega^{1}$.

Figure 6.  The network topology for $n$=2.

Figure 7.  The network topology for $n$=3.

Figure 8.  (a) Time series for $I_1$ (black colour) and $I_2$ (grey colour) converge to zero. (b) Time series for $I_1$ and $I_2$ converge to endemic state. (c) Time series for $I_1$ and $I_2$ converge to endemic state. (d) Time series for $I_1$ and $I_2$ converge to disease-free state. (e)Time series for $I_1$ (black colour), $I_2$ (dotted line) and $I_3$ (grey colour) for different patches converge to disease-free state. (f)Time series for $I_1$, $I_2$ and $I_3$ converge to endemic state.

Table 1.  Parameters of the Model

 Parameters Description of parameters $\alpha$ Human recovery rate $\beta^1$ Transmission rate of infection from female mosquitoes to human $\beta^2$ Mosquitoes mating rate $\beta^3$ Transmission rate of infection from human to female mosquito $\gamma$ Transition rate from aquatic stage to adult mosquito $\mu$ Natural death rate of human $\omega$ Birth rate of human $\omega^1$ Constant recruitment rate of sterile male mosquito $\phi$ Recruitment rate for aquatic mosquito $C$ Carrying capacity for aquatic/adult mosquito $d$ Natural death rate of mosquito at aquatic state $d^{1}$ Natural death rate of mosquito $k$ Rate at which exposed human become infectious $m_{ij}$ Migration rate from patch j to patch i $p$ Proportion of female mosquito

Table 2.  [24,3,5]

 Parameters Parameters values $\alpha$ 0.3 $\beta^1$ 0.02 $\beta^2$ 0.7 $\beta^3$ 0.03 $\gamma$ 0.075 $\omega$ 0.002 $\phi$ 5 $\mu$ 0.0000456 $C$ 450 $d$ 0.05 $d^{1}$ 0.0714 $k$ 0.1667 $p$ 0.5

Table 3.  [24,3,5]

 Parameters Parameters values $\alpha_i$ 0.5 $\beta_i^1$ 0.001 $\beta_i^2$ 0.7 $\beta_i^3$ 0.001 $\gamma_i$ 0.075 $\omega_i$ 0.029 $\phi_i$ 5 $\mu_i$ 0.0000456 $C_i$ 450 $d_i$ 0.05 $d_i^{1}$ 0.0714 $k_i$ 0.1667 $p_i$ 0.5

Table 4.  When SIT is applied only in the patch-1 i.e.($\omega^{1}_1=70$, $\omega^{1}_2=0$)

 Cases Migration in patch-1 Migration in patch-2 Conclusions (a) $m_{12}$=0 $m_{21}$=0 Isolated patches with $R_0$ of patch-1 is $0.2515<1$ (37). Disease is controlled in patch-1 only. (b) $m_{12}$=0.005 $m_{21}$=0.0006 $R_0^2$ for two patch network model is $0.7678<1$ (43). Two patch network model may be disease-free. The time-series confirms that the infection level tends to zero in both the patches Figure 8a. (c) $m_{12}$=0.009 $m_{21}$=0.0025 $R_0^2$ for two-patch network model is $1.4763>1$ (43). SIT method will not be able to control disease with this migration combination Figure 8b.

Table 5.  When SIT is applied only in the patch-1 i.e.($\omega^{1}_1=60$, $\omega^{1}_2=0$)

 Cases Migration in patch-1 Migration in patch-2 Conclusions (d) $m_{12}$=0 $m_{21}$=0 Isolated patches with $R_0$ of patch-1 is 0.2515$<1$. Disease is controlled in patch-1 only. (e) $m_{12}$=0.005 $m_{21}$=0.001 $R_0^2$ for two-patch network model is $1.1778>1$. SIT method will not be able to control disease with this migration combination. Disease may persist in the network Figure 8c. (f) $m_{12}$=0.009 $m_{21}$=0.001 $R_0^2$ for two-patch network model is $0.7273<1$. The network may now be disease free Figure 8d.

Table 6.  When SIT is applied only in the patch-1 i.e.($\omega^{1}_1=70$, $\omega^{1}_2=0$ and $\omega^{1}_3=0$)

 Cases Migration in patch-1 Migration in patch-2 Migration in patch-3 $R^n_0$ for network model (g) $m_{12}$=0.0001, $m_{13}$=0.002 $m_{21}$=0.0001 $m_{31}$=0.0001 $0.6762 (<1)$ (h) $m_{12}$=0.0002, $m_{13}$=0.0002 $m_{21}$=0.001 $m_{31}$=0.001 $2.6813 (>1)$
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Tables(6)