Minimum Vector Control Intensity to Get a Stable Fixed Point in a Mosquito Dynamic Model


  • F. H. Kawahama Universidade Federal de São Paulo
  • L. B. L. Santos
  • P. R. Cirilo
  • L. F. Souza
  • E. N. Macau



Modelling, Population Dynamic, Stability, Simulation


Vector-borne diseases are a cause of concern all around the world, especially in Brazil. In the past few years, the Brazilian health system faced recurrent epidemics such as Dengue and Malaria as well as new cases of Chikungunya, Zika and Yellow Fever. Vector control continues to be one of the most important counter measures against these types of diseases. Mathematical models are important tools for planning vector control strategies. In this work we present an approach in order to calculate what is the minimum vector control intensity to obtain stability in a simple population’s dynamics model of mosquitoes.We combined numerical simulations with analytic results. Transcritical bifurcations appear in our analysis considering different control’s parameters values for the eggs, larvae, pupae and adults mosquitoes populations. A discussion about combined strategies of vector control was also showed.


C. T. Codeço, D. A. Villela, and F. C. Coelho, “Estimating the effective reproduction number of dengue considering temperature-dependent generation

intervals,” Epidemics, vol. 25, pp. 101–111, 2018.

S. M. Raimundo, M. Amaku, and E. Massad, “Equilibrium analysis of a yellow

fever dynamical model with vaccination,” Computational and Mathematical

Methods in Medicine, vol. 2015, 2015.

B. Wahid, A. Ali, S. Rafique, and M. Idrees, “Global expansion of chikungunya virus: mapping the 64-year history,” International Journal of Infectious

Diseases, vol. 58, pp. 69 – 76, 2017.

R. Lowe, C. Barcellos, P. Brasil, O. G. Cruz, N. A. Honório, H. Kuper, and

M. S. Carvalho, “The zika virus epidemic in brazil: From discovery to future

implications,” Int J Environ Res Public Health, vol. 15(1):96, pp. 1–18, 2018.

L. Barsante, R. Cardoso, J. Luiz Acebal, C. Silva, and A. Eiras, “Modelo entomológico determinístico sob efeito da pluviosidade para o aedes aegypti e o aedes albopictus,” TEMA: Tendências em Matemática Aplicada e Computacional, vol. 19 n 2, pp. 289–303, 2018.

L. B. L. Santos, S. T. R. Pinho, R. F. S. Andrade, and et al, “Periodic forcing in a three-level cellular automata model for a vector-transmitted disease,”

Physical Review E, vol. 80, 2009.

H. M. Yang, J. L. Boldrini, A. C. Fassoni, and et al, “Fitting the incidence data from the city of campinas, brazil, based on dengue transmission modellings considering time-dependent entomological parameters,” PLOS One, 2016.

F. B. Agusto, S. Bewick, and W. F. Fagan, “Mathematical model of zika virus with vertical transmission,” Infectious Disease Modelling, vol. 2, pp. 244–267,2017.

L. Z. Goldani, “Yellow fever outbreak in brazil, 2017,” Brazilian Society of

Infectious Diseases, vol. 21, no. 2, pp. 123–124, 2017.

G. Phaijoo and D. Gurung, “Mathematical model of dengue disease transmission dynamics with control measures,” Journal of Advances in Mathematics

and Computer Science, vol. 23, pp. 1–12, 2017.

G. Adamu, M. Bawa, M. Jiya, and U. Chado, “A mathematical model for the

dynamics of zika virus via homotopy perturbation method,” Journal of Applied

Sciences and Environmental Management, vol. 21, p. 615, 2017.

R. M. Lana, M. M. Morais, T. F. M. d. Lima, T. G. d. S. Carneiro, L. M.

Stolerman, J. P. C. dos Santos, J. J. C. Cortés, A. E. Eiras, and C. T. Codeço,

“Assessment of a trap based aedes aegypti surveillance program using mathematical modeling,” PLOS One, vol. 13, pp. 1–16, 2018.

D. L. Kreider, R. G. Kuller, and D. R. Ostberg, Elementary Differential Equations. Edgard Blucher, 1972.

R. L. Devaney, An Introduction to Chaotic Dynamical System. Westview Press,

S. H. Strogatz, Nonlinear Dynamics and Chaos: with applications to physics,

biology, chemistry, and engineering. Addison-Wesley Publishing Company,




How to Cite

Kawahama, F. H., Santos, L. B. L., Cirilo, P. R., Souza, L. F., & Macau, E. N. (2023). Minimum Vector Control Intensity to Get a Stable Fixed Point in a Mosquito Dynamic Model. Trends in Computational and Applied Mathematics, 24(3), 521–533.



Original Article