# American Institute of Mathematical Sciences

September  2020, 13(9): 2443-2463. doi: 10.3934/dcdss.2020194

## Oscillations induced by quiescent adult female in a model of wild aedes aegypti mosquitoes

 1 Ibn Zohr University, CST Campus Universitaire Ait Melloul, Agadir, Morocco 2 Laboratory of Analysis, Geometry and Applications (LAGA), Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Campus Universitaire, BP 133, Kenitra, Morocco 3 Normandie Univ, France; ULH, LMAH, F-76600 Le Havre; FR-CNRS-3335, ISCN, 25 rue Ph. Lebon 76600, Le Havre, France 4 Department of Mathematical Sciences, United Arab Emirates University Al Ain, Abu Dhabi, United Arab Emirates

Received  November 2018 Revised  April 2019 Published  September 2020 Early access  December 2019

Aedes aegypti (Ae. aegypti: mosquito) is a known vector of several viruses including yellow fever, dengue, chikungunya and zika. In the current paper, we present a delayed mathematical model describing the dynamics of Ae. aegypti. Our model is governed by a system of three delay differential equations modeling the interactions between three compartments of the Ae. aegypti life cycle (females, eggs and pupae). By using time delay as a parameter of bifurcation, we prove stability/switch stability of the possible equilibrium points and the existence of bifurcating branch of small amplitude periodic solutions when time delay crosses some critical value. We establish an algorithm determining the direction of bifurcation and stability of bifurcating periodic solutions. In the end, some numerical simulations are carried out to support theoretical results..

Citation: Ahmed Aghriche, Radouane Yafia, M. A. Aziz Alaoui, Abdessamad Tridane. Oscillations induced by quiescent adult female in a model of wild aedes aegypti mosquitoes. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2443-2463. doi: 10.3934/dcdss.2020194
##### References:
 [1] R. Barrera, M. Amador and A. J. MacKay, Population dynamics of Aedes aegypti and dengue as influenced by weather and human behavior in San Juan, Puerto Rico, PLoS Neglected Tropical Diseases, 5 (2011). doi: 10.1371/journal.pntd.0001378. [2] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York-London, 1963. [3] S. Bhatt, W. G. Peter, J. B. Oliver and P. M. Jane, The global distribution and burden of dengue, Nature, 496 (2013), 504-507.  doi: 10.1038/nature12060. [4] F. G. Boese, Stability with respect to the delay: On a paper of K. L. Cooke and P. van den Driessche, J. Math. Anal. Appl., 228 (1998), 293-321.  doi: 10.1006/jmaa.1998.6109. [5] L. Cai, S. Ai and G. Fan, Dynamics of delayed mosquitoes populations models with two different strategies of releasing sterile mosquitoes, Math. Biosci. Eng., 15 (2018), 1181-1202.  doi: 10.3934/mbe.2018054. [6] D. D. Chadee and R. Martinez, Landing periodicity of Aedes aegypti with implications for dengue transmission in Trinidad, West Indies, J. Vector Ecology, 25 (2000), 158-163. [7] N. Chitnis, J. M. Hyman and J. M. Cusching, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272-1296.  doi: 10.1007/s11538-008-9299-0. [8] K. L. Cooke and P. van den Driessche, On the zeroes of some transcendental equations, Funkcial. Ekvac., 29 (1986), 77-90. [9] J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Lecture Notes in Biomathematics, 20, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/978-3-642-93073-7. [10] H. V. Danks, Insect Dormancy: An Ecological Perspective, Biological Survey of Canada Monograph Series, Entomological Society of Canada, Broadway, 1987. [11] J. Dieudonn$\acute{e}$, Foundations of Modern Analysis, Pure and Applied Mathematics, 10, Academic Press, New York-London, 1960. [12] D. F. A. Diniz, M. R. A. Cleide, O. O. Luciana, A. V. M. Maria and F. J. A. Constancia, Diapause and quiescence: Dormancy mechanisms that contribute to the geographical expansion of mosquitoes and their evolutionary success, Parasites & Vectors, 10 (2017), 310pp. doi: 10.1186/s13071-017-2235-0. [13] C. Dye, Models for the population dynamics of the yellow fever mosquito, Aedes aegypti, J. Animal Ecology, 53 (1984), 247-268.  doi: 10.2307/4355. [14] K. Gopalsamy, Stability on the Oscillations in Delay Differential Equations of Population Dynamics, Mathematics and its Applications, 74, Kluwer Academic Publishers Group, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9. [15] D. J. Gubler and G. G. Clark, Dengue/dengue hemorrhagic fever: The emergence of a global health problem, Emerging Infectious Diseases, 1 (1995), 55-57.  doi: 10.3201/eid0102.952004. [16] D. J. Gubler, Dengue and dengue hemorrhagic fever, Clinical Microbiology Reviews, 11 (1998), 480-496.  doi: 10.1128/CMR.11.3.480. [17] B. Hassard, N. Kazarinoff and Y. Wan, Theory and Applications of Hopf Bifurcation, London Mathematical Society Lecture Note Series, 41, Cambridge University Press, Cambridge-New York, 1981. [18] T. Huraux, R. Misslin, A. Cebeillac, A. Vaguet and E. Daudè, Modélisation de l'impact des ȋlots de chaleur urbains sur les dynamiques de population d'Aedes aegypti, vecteur de la dengue et du virus Zika., Available from: https://halshs.archives-ouvertes.fr/halshs-01650033/document. [19] Y. Kuang, Delay Differential Equations, with Applications in Population Dynamics, Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993. [20] R. Li, L. Xu, O. N. Bjornstad and K. Liu, Climate-driven variation in mosquito density predicts the spatiotemporal dynamics of dengue, PNAS, 116 (2019), 3624-3629.  doi: 10.1073/pnas.1806094116. [21] N. MacDonald, Time Lags in Biological Models, Lecture Notes in Biomathematics, 27, Springer-Verlag, Berlin-New York, 1978. doi: 10.1007/978-3-642-93107-9. [22] J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences, 19, Springer-Verlag, New York, 1976. doi: 10.1007/978-1-4612-6374-6. [23] R. Martins, T. Lewinsohn and M. Barbeiros, Ecologia e comportamento dos insetos, Oecologia Bras, 8 (2000), 149-192. [24] D. Musso, V. M. Cao-Lormeau and D. J. Gubler, Zika virus: Following the path of dengue and chikungunya?, Lancet, 386 (2015), 243-244.  doi: 10.1016/S0140-6736(15)61273-9. [25] F. N. Ngoteya and Y. N. Gyekye, Sensitivity analysis of parameters in a competition model, Appl. Comput. Math., 4 (2015), 363-368.  doi: 10.11648/j.acm.20150405.15. [26] L. O. Oliva, R. La Corte, M. O. Santana and C. M. R. Albuquerque, Quiescence in Aedes aegypti: Interpopulation differences contribute to population dynamics and vectorial capacity, phInsects, 9 (2018), 111pp. doi: 10.3390/insects9030111. [27] S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 863-874.  doi: 10.1093/imammb/18.1.41. [28] L. M. Rueda, K. J. Patel, R. C. Axtell and R. E. Stinner, Temperature-dependent development and survival rates of Culex quinquefasciatus and Aedes aegypti (Diptera: Culicidae), J. Medical Entomology, 27 (1990), 892-898.  doi: 10.1093/jmedent/27.5.892. [29] A. C. T. Saulo, A. E. Bermudez and A. M. Loaiza, Controlling Aedes aegypti mosquitoes by using ovitraps: A mathematical model, Appl. Math. Sci., 11 (2017), 1123-1131.  doi: 10.12988/ams.2017.712. [30] Y. Song and S. Yuan, Bifurcation analysis in a predator prey system with time delay, Nonlinear Anal. Real World Appl., 7 (2006), 265-284.  doi: 10.1016/j.nonrwa.2005.03.002. [31] G. Vacus, Expansion G$\acute{e}$ographique d'Aedes Albopictus. Quel Risque de Maladies $\acute{E}$mergentes en France M$\acute{e}$tropolitaine?, Ph.D. thesis, Institut National de M$\acute{e}$decine Agricole, 2012. [32] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [33] R. K. Walsh, C. L. Aguilar, L. Facchinelli and L. Valerio, Assessing the impact of direct and delayed density dependence in natural larval populations of Aedes aeygpti, Amer. J. Tropical Medicine and Hygiene, 89 (2013), 68-77. [34] World Health Organization, Dengue Haemorrhagic Fever: Diagnosis, Treatment, Prevention and Control, Geneva, Switzerland, 1997. [35] World Health Organization, Dengue guidelines for diagnosis, treatment, prevention and control, Geneva, Switzerland, 2009. [36] X. Zhou, Y. Wu, Y. Li and X. Yao, Stability and Hopf bifurcation analysis on a two-neuron network with discrete and distributed delays, Chaos Solitons Fractals, 40 (2009), 1493-1505.  doi: 10.1016/j.chaos.2007.09.034.

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##### References:
 [1] R. Barrera, M. Amador and A. J. MacKay, Population dynamics of Aedes aegypti and dengue as influenced by weather and human behavior in San Juan, Puerto Rico, PLoS Neglected Tropical Diseases, 5 (2011). doi: 10.1371/journal.pntd.0001378. [2] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York-London, 1963. [3] S. Bhatt, W. G. Peter, J. B. Oliver and P. M. Jane, The global distribution and burden of dengue, Nature, 496 (2013), 504-507.  doi: 10.1038/nature12060. [4] F. G. Boese, Stability with respect to the delay: On a paper of K. L. Cooke and P. van den Driessche, J. Math. Anal. Appl., 228 (1998), 293-321.  doi: 10.1006/jmaa.1998.6109. [5] L. Cai, S. Ai and G. Fan, Dynamics of delayed mosquitoes populations models with two different strategies of releasing sterile mosquitoes, Math. Biosci. Eng., 15 (2018), 1181-1202.  doi: 10.3934/mbe.2018054. [6] D. D. Chadee and R. Martinez, Landing periodicity of Aedes aegypti with implications for dengue transmission in Trinidad, West Indies, J. Vector Ecology, 25 (2000), 158-163. [7] N. Chitnis, J. M. Hyman and J. M. Cusching, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272-1296.  doi: 10.1007/s11538-008-9299-0. [8] K. L. Cooke and P. van den Driessche, On the zeroes of some transcendental equations, Funkcial. Ekvac., 29 (1986), 77-90. [9] J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Lecture Notes in Biomathematics, 20, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/978-3-642-93073-7. [10] H. V. Danks, Insect Dormancy: An Ecological Perspective, Biological Survey of Canada Monograph Series, Entomological Society of Canada, Broadway, 1987. [11] J. Dieudonn$\acute{e}$, Foundations of Modern Analysis, Pure and Applied Mathematics, 10, Academic Press, New York-London, 1960. [12] D. F. A. Diniz, M. R. A. Cleide, O. O. Luciana, A. V. M. Maria and F. J. A. Constancia, Diapause and quiescence: Dormancy mechanisms that contribute to the geographical expansion of mosquitoes and their evolutionary success, Parasites & Vectors, 10 (2017), 310pp. doi: 10.1186/s13071-017-2235-0. [13] C. Dye, Models for the population dynamics of the yellow fever mosquito, Aedes aegypti, J. Animal Ecology, 53 (1984), 247-268.  doi: 10.2307/4355. [14] K. Gopalsamy, Stability on the Oscillations in Delay Differential Equations of Population Dynamics, Mathematics and its Applications, 74, Kluwer Academic Publishers Group, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9. [15] D. J. Gubler and G. G. Clark, Dengue/dengue hemorrhagic fever: The emergence of a global health problem, Emerging Infectious Diseases, 1 (1995), 55-57.  doi: 10.3201/eid0102.952004. [16] D. J. Gubler, Dengue and dengue hemorrhagic fever, Clinical Microbiology Reviews, 11 (1998), 480-496.  doi: 10.1128/CMR.11.3.480. [17] B. Hassard, N. Kazarinoff and Y. Wan, Theory and Applications of Hopf Bifurcation, London Mathematical Society Lecture Note Series, 41, Cambridge University Press, Cambridge-New York, 1981. [18] T. Huraux, R. Misslin, A. Cebeillac, A. Vaguet and E. Daudè, Modélisation de l'impact des ȋlots de chaleur urbains sur les dynamiques de population d'Aedes aegypti, vecteur de la dengue et du virus Zika., Available from: https://halshs.archives-ouvertes.fr/halshs-01650033/document. [19] Y. Kuang, Delay Differential Equations, with Applications in Population Dynamics, Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993. [20] R. Li, L. Xu, O. N. Bjornstad and K. Liu, Climate-driven variation in mosquito density predicts the spatiotemporal dynamics of dengue, PNAS, 116 (2019), 3624-3629.  doi: 10.1073/pnas.1806094116. [21] N. MacDonald, Time Lags in Biological Models, Lecture Notes in Biomathematics, 27, Springer-Verlag, Berlin-New York, 1978. doi: 10.1007/978-3-642-93107-9. [22] J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences, 19, Springer-Verlag, New York, 1976. doi: 10.1007/978-1-4612-6374-6. [23] R. Martins, T. Lewinsohn and M. Barbeiros, Ecologia e comportamento dos insetos, Oecologia Bras, 8 (2000), 149-192. [24] D. Musso, V. M. Cao-Lormeau and D. J. Gubler, Zika virus: Following the path of dengue and chikungunya?, Lancet, 386 (2015), 243-244.  doi: 10.1016/S0140-6736(15)61273-9. [25] F. N. Ngoteya and Y. N. Gyekye, Sensitivity analysis of parameters in a competition model, Appl. Comput. Math., 4 (2015), 363-368.  doi: 10.11648/j.acm.20150405.15. [26] L. O. Oliva, R. La Corte, M. O. Santana and C. M. R. Albuquerque, Quiescence in Aedes aegypti: Interpopulation differences contribute to population dynamics and vectorial capacity, phInsects, 9 (2018), 111pp. doi: 10.3390/insects9030111. [27] S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 863-874.  doi: 10.1093/imammb/18.1.41. [28] L. M. Rueda, K. J. Patel, R. C. Axtell and R. E. Stinner, Temperature-dependent development and survival rates of Culex quinquefasciatus and Aedes aegypti (Diptera: Culicidae), J. Medical Entomology, 27 (1990), 892-898.  doi: 10.1093/jmedent/27.5.892. [29] A. C. T. Saulo, A. E. Bermudez and A. M. Loaiza, Controlling Aedes aegypti mosquitoes by using ovitraps: A mathematical model, Appl. Math. Sci., 11 (2017), 1123-1131.  doi: 10.12988/ams.2017.712. [30] Y. Song and S. Yuan, Bifurcation analysis in a predator prey system with time delay, Nonlinear Anal. Real World Appl., 7 (2006), 265-284.  doi: 10.1016/j.nonrwa.2005.03.002. [31] G. Vacus, Expansion G$\acute{e}$ographique d'Aedes Albopictus. Quel Risque de Maladies $\acute{E}$mergentes en France M$\acute{e}$tropolitaine?, Ph.D. thesis, Institut National de M$\acute{e}$decine Agricole, 2012. [32] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [33] R. K. Walsh, C. L. Aguilar, L. Facchinelli and L. Valerio, Assessing the impact of direct and delayed density dependence in natural larval populations of Aedes aeygpti, Amer. J. Tropical Medicine and Hygiene, 89 (2013), 68-77. [34] World Health Organization, Dengue Haemorrhagic Fever: Diagnosis, Treatment, Prevention and Control, Geneva, Switzerland, 1997. [35] World Health Organization, Dengue guidelines for diagnosis, treatment, prevention and control, Geneva, Switzerland, 2009. [36] X. Zhou, Y. Wu, Y. Li and X. Yao, Stability and Hopf bifurcation analysis on a two-neuron network with discrete and distributed delays, Chaos Solitons Fractals, 40 (2009), 1493-1505.  doi: 10.1016/j.chaos.2007.09.034.
Schematic representation describing the interaction between females ($F$), eggs ($E$) and pupae ($P$) of Ae. aegypti population
Surfaces representing the effect of parameters (left $\gamma$ and $\alpha$) and (right $\sigma$ and $\alpha$) on the variations of the population reproduction number $R$
Surfaces representing the effect of parameters(left $\beta$ and $\alpha$) and (right $\mu_{F}$ and $\alpha$) on the variations of the population reproduction number $R$
Surfaces representing the effect of parameters (left $\mu_{E}$ and $\alpha$) and (right $\mu_{P}$ and $\alpha$) on the variations of the population reproduction number $R$
Surfaces representing the effect of parameters (left $\mu_{P}$ and $\mu_{F}$) and (right $\mu_{P}$ and $\beta$) on the variations of the population reproduction number $R$
Surface representing the effect of parameters $\mu_{P}$ and $\gamma$ on the variations of the population reproduction number $R$
Stability of $E_0$ for $\tau = 0$ in $(t,FEP)$ plane (left) and in $(F,E,P)$ space (right) and non existence of $E_1$ for $\mu_F = 5$ and $R = 0.269$
Instability of $E_0$ and stability of $E_1$ for $\tau = 0$ in $(t,FEP)$ plane (left) and in $(F,E,P)$ space (right) with $R = 8.43$
Instability of $E_0$ and stability of $E_1$ for $\tau = 5$ in $(t,FEP)$ plane (left) and in $(F,E,P)$ space (right)
Mikhailov hodograph indicating the stability of $E_1$ for $\tau = 5$ (left) and instability of $E_1$ for $\tau = 50$ (right). Note that, The steady state $E_1$ is stable if the curve crosses imaginary axis above zero (at all crossing points)
Periodic solution bifurcated from the steady state $E_1$ for $\tau = \tau_0 = 10.35$ in $(t,FEP)$ plane (left) and in $(F,E,P)$ space (right)
Periodic solution bifurcated from the steady state $E_1$ for $\tau = \tau_0+\epsilon = 10.85$ with $\epsilon = 0.5$ in $(t,FEP)$ plane (left) and in $(F,E,P)$ space (right)
Chaotic solution bifurcated from the steady state $E_1$ for $\tau = \tau_0+\epsilon = 11.15$ with $\epsilon = 0.8$ in $(t,FEP)$ plane (left) and in $(F,E,P)$ space (right)
Existence of pair purely imaginary roots (left). The branches of bifurcation (diagram of bifurcation), branch $2$ (blue line) is the branch of the periodic orbits that arise from the Hopf point, branch $3$ (red line) is the branch of period doubling bifurcation point in branch $2$ and branch $4$ (green line) is the branch of period doubling bifurcation point in branch $3$ (right). Note that, the branch $1$ is the steady state $E_1$ with amplitude $0$
Parameters estimation
 parameter value reference $\gamma$ $0.90$ Assumed $\alpha$ $0 .6$ Assumed $\sigma$ $4$ [29] $\beta$ $0.4$ [29] $\mu_F$ $0.16$ Assumed $\mu_E$ $0.15$ Assumed $\mu_P$ $0.01$ Assumed $k$ $500$ Assumed
 parameter value reference $\gamma$ $0.90$ Assumed $\alpha$ $0 .6$ Assumed $\sigma$ $4$ [29] $\beta$ $0.4$ [29] $\mu_F$ $0.16$ Assumed $\mu_E$ $0.15$ Assumed $\mu_P$ $0.01$ Assumed $k$ $500$ Assumed
The sensitivity indices of the population reproduction number $R$
 Parameter Sensitivity index Index at parameters value $\gamma$ $\frac{\mu_{E}}{\gamma+\mu_{E}}$ $+0.1428$ $\alpha$ $\frac{\mu_{P}}{\alpha+\mu_{P}}$ $+0.366$ $\sigma$ $+1$ $+1$ $\beta$ $+1$ $+1$ $\mu_F$ $-1$ $-1$ $\mu_E$ $-\frac{\mu_{E}}{\gamma+\mu_{E}}$ $-0.1428$ $\mu_P$ $-\frac{\mu_{P}}{\gamma+\mu_{P}}$ $-0.336$
 Parameter Sensitivity index Index at parameters value $\gamma$ $\frac{\mu_{E}}{\gamma+\mu_{E}}$ $+0.1428$ $\alpha$ $\frac{\mu_{P}}{\alpha+\mu_{P}}$ $+0.366$ $\sigma$ $+1$ $+1$ $\beta$ $+1$ $+1$ $\mu_F$ $-1$ $-1$ $\mu_E$ $-\frac{\mu_{E}}{\gamma+\mu_{E}}$ $-0.1428$ $\mu_P$ $-\frac{\mu_{P}}{\gamma+\mu_{P}}$ $-0.336$
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