# American Institute of Mathematical Sciences

April  2018, 15(2): 441-460. doi: 10.3934/mbe.2018020

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

 1 Department of Mathematics, Indian Institute of Technology Roorkee, Uttarakhand 247667, India 2 Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, 76600 Le Havre, France

* Corresponding author

Received  December 15, 2016 Accepted  March 31, 2017 Published  June 2017

Fund Project: 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.

Citation: Arti Mishra, Benjamin Ambrosio, Sunita Gakkhar, M. A. Aziz-Alaoui. A network model for control of dengue epidemic using sterile insect technique. Mathematical Biosciences & Engineering, 2018, 15 (2) : 441-460. doi: 10.3934/mbe.2018020
##### References:
 [1] D. L. Chao and D. T. Dimitrov, Seasonality and the effectiveness of mass vaccination, Math Biosci Eng, 13 (2016), 249-259.  doi: 10.3934/mbe.2015001.  Google Scholar [2] G. Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse and J. M. Hyman, The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile, Math Biosci Eng, 10 (2013), 1455-1474.  doi: 10.3934/mbe.2013.10.1455.  Google Scholar [3] L. Esteva and H. M. Yang, Mathematical model to acess the control of aedes aegypti mosquitoes by sterile insect technique, Math. Biosci, 198 (2005), 132-147.  doi: 10.1016/j.mbs.2005.06.004.  Google Scholar [4] T. P. O. Evans and S. R. Bishop, A spatial model with pulsed releases to compare strategies for the sterile insect technique applied to the mosquito aedes aegypti, Math. Biosci, 254 (2014), 6-27.  doi: 10.1016/j.mbs.2014.06.001.  Google Scholar [5] Z. Feng and J. X. Velasco-Hernandez, Competitive exclusion in a vector-host model for the dengue fever, J. Math. Biol, 35 (1997), 523-544.  doi: 10.1007/s002850050064.  Google Scholar [6] D. J. Gubler, Dengue and dengue hemorrhagic fever: Its history and resurgence as a global public health problem, (eds. D. J. Gubler, G. Kuno), Dengue and Dengue Hemorrhagic Fever, New York: CAB International, (1997), 1-22.   Google Scholar [7] D. J. Gubler, Epidemic dengue/dengue hemorrhagic fever as a public health, social and economic problem in the 21st century, Trends Microbiol, 10 (2002), 100-103.  doi: 10.1016/S0966-842X(01)02288-0.  Google Scholar [8] R.-W. S. Hendron and M. B. Bonsall, The interplay of vaccination and vector control on small dengue networks, J. Theor. Biol, 407 (2016), 349-361.  doi: 10.1016/j.jtbi.2016.07.034.  Google Scholar [9] H. Hughes and N. F. Britton, Modelling the use of wolbachia to control dengue fever transmission, Bull Math Biol, 75 (2013), 796-818.  doi: 10.1007/s11538-013-9835-4.  Google Scholar [10] [11] [12] J. H. Jones, Notes on $R_{0}$ Department of Anthropological Sciences, Stanford University, 2007. Google Scholar [13] G. Knerer, C. S. M. Currie and S. C. Brailsford, Impact of combined vector-control and vaccination strategies on transmission dynamics of dengue fever: A model-based analysis, Health Care Manag Sci, 18 (2015), 205-217.  doi: 10.1007/s10729-013-9263-x.  Google Scholar [14] E. F. Knipling, Possibilities of insect control or eradication through the use of sexually sterile males, J. Econ. Entomol, 48 (1955), 459-462.  doi: 10.1093/jee/48.4.459.  Google Scholar [15] E. F. Knipling, The Basic Principles of Insect Population and Suppression and Management USDA handbook. Washington, D. C. , USDA, 1979. Google Scholar [16] E. F. Knipling, Sterile insect technique as a screwworm control measure: The concept and its development, Misc. Pub. Entomol. Soc. Am, 62 (1985), 4-7.   Google Scholar [17] J. P. LaSalle, The Stability of Dynamical Systems Regional Conf. Series Appl. Math. , 25, SIAM, Philadelphia, 1976.  Google Scholar [18] A. Mishra and S. Gakkhar, The effects of awareness and vector control on two strains dengue dynamics, Appl. Math. Comput, 246 (2014), 159-167.  doi: 10.1016/j.amc.2014.07.115.  Google Scholar [19] A. M. P. Montoya, A. M. Loaiza and O. T. Gerard, Simulation model for dengue fever transmission with integrated control, Appl. Math. Sci, 10 (2016), 175-185.  doi: 10.12988/ams.2016.510661.  Google Scholar [20] D. Moulay, M. A. Aziz-Alaoui and Hee-Dae Kwon, Optimal control of chikungunya disease: Larvae reduction, treatment and prevention, Math Biosci Eng, 9 (2012), 369-392.  doi: 10.3934/mbe.2012.9.369.  Google Scholar [21] D. Moulay, M. A. Aziz-Alaoui and M. Cadivel, The chikungunya disease: Modeling, vector and transmission global dynamics, Math. Biosci, 229 (2011), 50-63.  doi: 10.1016/j.mbs.2010.10.008.  Google Scholar [22] L. Perko, Differential Equations and Dynamical Systems Texts in Applied Mathematics, 7. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4684-0392-3.  Google Scholar [23] H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres, Vaccination models and optimal control strategies to dengue, Math. Biosci, 247 (2014), 1-12.  doi: 10.1016/j.mbs.2013.10.006.  Google Scholar [24] S. Syafruddin and M. S. M. Noorani, SEIR model for transmission of dengue fever in Selangor Malaysia, International Journal of Modern Physics: Conference Series, 9 (2012), 380-389.   Google Scholar [25] R. C. A. Thomé, H. M. Yang and L. Esteva, Optimal control of aedes aegypti mosquitoes by the sterile insect technique and insecticide, Math. Biosci, 223 (2010), 12-23.  doi: 10.1016/j.mbs.2009.08.009.  Google Scholar [26] World Health Organization, Dengue: Guidelines for Diagnosis, Treatment, Prevention and Control, Geneva: World Health Organization and the Special Programme for Research and Training in Tropical Diseases, 2009. Google Scholar [27]

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##### References:
 [1] D. L. Chao and D. T. Dimitrov, Seasonality and the effectiveness of mass vaccination, Math Biosci Eng, 13 (2016), 249-259.  doi: 10.3934/mbe.2015001.  Google Scholar [2] G. Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse and J. M. Hyman, The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile, Math Biosci Eng, 10 (2013), 1455-1474.  doi: 10.3934/mbe.2013.10.1455.  Google Scholar [3] L. Esteva and H. M. Yang, Mathematical model to acess the control of aedes aegypti mosquitoes by sterile insect technique, Math. Biosci, 198 (2005), 132-147.  doi: 10.1016/j.mbs.2005.06.004.  Google Scholar [4] T. P. O. Evans and S. R. Bishop, A spatial model with pulsed releases to compare strategies for the sterile insect technique applied to the mosquito aedes aegypti, Math. Biosci, 254 (2014), 6-27.  doi: 10.1016/j.mbs.2014.06.001.  Google Scholar [5] Z. Feng and J. X. Velasco-Hernandez, Competitive exclusion in a vector-host model for the dengue fever, J. Math. Biol, 35 (1997), 523-544.  doi: 10.1007/s002850050064.  Google Scholar [6] D. J. Gubler, Dengue and dengue hemorrhagic fever: Its history and resurgence as a global public health problem, (eds. D. J. Gubler, G. Kuno), Dengue and Dengue Hemorrhagic Fever, New York: CAB International, (1997), 1-22.   Google Scholar [7] D. J. Gubler, Epidemic dengue/dengue hemorrhagic fever as a public health, social and economic problem in the 21st century, Trends Microbiol, 10 (2002), 100-103.  doi: 10.1016/S0966-842X(01)02288-0.  Google Scholar [8] R.-W. S. Hendron and M. B. Bonsall, The interplay of vaccination and vector control on small dengue networks, J. Theor. Biol, 407 (2016), 349-361.  doi: 10.1016/j.jtbi.2016.07.034.  Google Scholar [9] H. Hughes and N. F. Britton, Modelling the use of wolbachia to control dengue fever transmission, Bull Math Biol, 75 (2013), 796-818.  doi: 10.1007/s11538-013-9835-4.  Google Scholar [10] [11] [12] J. H. Jones, Notes on $R_{0}$ Department of Anthropological Sciences, Stanford University, 2007. Google Scholar [13] G. Knerer, C. S. M. Currie and S. C. Brailsford, Impact of combined vector-control and vaccination strategies on transmission dynamics of dengue fever: A model-based analysis, Health Care Manag Sci, 18 (2015), 205-217.  doi: 10.1007/s10729-013-9263-x.  Google Scholar [14] E. F. Knipling, Possibilities of insect control or eradication through the use of sexually sterile males, J. Econ. Entomol, 48 (1955), 459-462.  doi: 10.1093/jee/48.4.459.  Google Scholar [15] E. F. Knipling, The Basic Principles of Insect Population and Suppression and Management USDA handbook. Washington, D. C. , USDA, 1979. Google Scholar [16] E. F. Knipling, Sterile insect technique as a screwworm control measure: The concept and its development, Misc. Pub. Entomol. Soc. Am, 62 (1985), 4-7.   Google Scholar [17] J. P. LaSalle, The Stability of Dynamical Systems Regional Conf. Series Appl. Math. , 25, SIAM, Philadelphia, 1976.  Google Scholar [18] A. Mishra and S. Gakkhar, The effects of awareness and vector control on two strains dengue dynamics, Appl. Math. Comput, 246 (2014), 159-167.  doi: 10.1016/j.amc.2014.07.115.  Google Scholar [19] A. M. P. Montoya, A. M. Loaiza and O. T. Gerard, Simulation model for dengue fever transmission with integrated control, Appl. Math. Sci, 10 (2016), 175-185.  doi: 10.12988/ams.2016.510661.  Google Scholar [20] D. Moulay, M. A. Aziz-Alaoui and Hee-Dae Kwon, Optimal control of chikungunya disease: Larvae reduction, treatment and prevention, Math Biosci Eng, 9 (2012), 369-392.  doi: 10.3934/mbe.2012.9.369.  Google Scholar [21] D. Moulay, M. A. Aziz-Alaoui and M. Cadivel, The chikungunya disease: Modeling, vector and transmission global dynamics, Math. Biosci, 229 (2011), 50-63.  doi: 10.1016/j.mbs.2010.10.008.  Google Scholar [22] L. Perko, Differential Equations and Dynamical Systems Texts in Applied Mathematics, 7. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4684-0392-3.  Google Scholar [23] H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres, Vaccination models and optimal control strategies to dengue, Math. Biosci, 247 (2014), 1-12.  doi: 10.1016/j.mbs.2013.10.006.  Google Scholar [24] S. Syafruddin and M. S. M. Noorani, SEIR model for transmission of dengue fever in Selangor Malaysia, International Journal of Modern Physics: Conference Series, 9 (2012), 380-389.   Google Scholar [25] R. C. A. Thomé, H. M. Yang and L. Esteva, Optimal control of aedes aegypti mosquitoes by the sterile insect technique and insecticide, Math. Biosci, 223 (2010), 12-23.  doi: 10.1016/j.mbs.2009.08.009.  Google Scholar [26] World Health Organization, Dengue: Guidelines for Diagnosis, Treatment, Prevention and Control, Geneva: World Health Organization and the Special Programme for Research and Training in Tropical Diseases, 2009. Google Scholar [27]
The transfer diagram representing the system (1)-(11) dynamics. The births, deaths and migration are not included.
The existence of the state $P_2$ with respect to parameters $\beta^2$ and $\phi$.
Bifurcation diagram for the stability of the state $P_2$
(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).
Bifurcation plot of system (12)-(22) with respect to $\omega^{1}$.
The network topology for $n$=2.
The network topology for $n$=3.
(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.
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
 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
[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
 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
[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
 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
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.
 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.
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.
 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.
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 $2.6813 (>1)$
 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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