DETERMINISTIC AND STOCHASTIC DENGUE EPIDEMIC MODEL: EXPLORING THE PROBABILITY OF EXTINCTION

  • Meksianis Z. Ndii Department of Mathematics, Faculty of Sciences and Engineering, University of Nusa Cendana
  • Yudi Ari Adi Department of Mathematics, Faculty of Science and Technology, Ahmad Dahlan University
  • Bertha S Djahi Department of Computer Sciences, Faculty of Sciences and Engineering, University of Nusa Cendana
DOI: https://doi.org/10.30598/barekengvol16iss2pp583-596
Keywords: Dengue, Probability of extinction, Modelling

Abstract

Dengue, a vector-borne disease, threatens the life of humans in tropical and subtropical regions. Hence, the dengue transmission dynamics need to be studied. An important aspect to be investigated is the probability of extinction. In this paper, deterministic and stochastic dengue epidemic models with two-age classes have been developed and analyzed, and the probability of extinction has been determined.  For the stochastic approach, we use the Continuous-Time Markov Chain model. The results show that vaccination of adult individuals leads to a lower number of adult infected individuals. Furthermore, the results showed that a higher number of initial infections causes a low probability of dengue extinction. Furthermore, factors contributing to an increase in the infection-related parameters have to be minimized to increase the potential reduction of dengue cases.

Downloads

Download data is not yet available.

References

S. Bhatt et al., “The effect of malaria control on Plasmodium falciparum in Africa between 2000 and 2015,” Nature, vol. 526, no. 7572, pp. 207–211, Oct. 2015, doi: 10.1038/nature15535.

N. L. Achee et al., “A Critical Assessment of Vector Control for Dengue Prevention,” PLOS Neglected Tropical Diseases, vol. 9, no. 5, p. e0003655, May 2015, doi: 10.1371/journal.pntd.0003655.

M. Z. Ndii, “Modelling the Use of Vaccine and Wolbachia on Dengue Transmission Dynamics,” Tropical Medicine and Infectious Disease, vol. 5, no. 2, 2020, doi: 10.3390/tropicalmed5020078.

Ferguson Neil M., Rodríguez-Barraquer Isabel, Dorigatti Ilaria, Mier-y-Teran-Romero Luis, Laydon Daniel J., and Cummings Derek A. T., “Benefits and risks of the Sanofi-Pasteur dengue vaccine: Modeling optimal deployment,” Science, vol. 353, no. 6303, pp. 1033–1036, Sep. 2016, doi: 10.1126/science.aaf9590.

M. Z. Ndii, A. R. Mage, J. J. Messakh, and B. S. Djahi, “Optimal vaccination strategy for dengue transmission in Kupang city, Indonesia,” Heliyon, vol. 6, no. 11, Nov. 2020, doi: 10.1016/j.heliyon.2020.e05345.

S. Sridhar et al., “Effect of Dengue Serostatus on Dengue Vaccine Safety and Efficacy,” N Engl J Med, vol. 379, no. 4, pp. 327–340, Jul. 2018, doi: 10.1056/NEJMoa1800820.

S. Biswal et al., “Efficacy of a Tetravalent Dengue Vaccine in Healthy Children and Adolescents,” N Engl J Med, vol. 381, no. 21, pp. 2009–2019, Nov. 2019, doi: 10.1056/NEJMoa1903869.

S. Biswal et al., “Efficacy of a tetravalent dengue vaccine in healthy children aged 4–16 years: a randomised, placebo-controlled, phase 3 trial,” The Lancet, vol. 395, no. 10234, pp. 1423–1433, May 2020, doi: 10.1016/S0140-6736(20)30414-1.

J. L. Arredondo-García et al., “Four-year safety follow-up of the tetravalent dengue vaccine efficacy randomized controlled trials in Asia and Latin America,” Clinical Microbiology and Infection, vol. 24, no. 7, pp. 755–763, Jul. 2018, doi: 10.1016/j.cmi.2018.01.018.

A. Bustamam, D. Aldila, and A. Yuwanda, “Understanding Dengue Control for Short- and Long-Term Intervention with a Mathematical Model Approach,” Journal of Applied Mathematics, vol. 2018, p. 9674138, Jan. 2018, doi: 10.1155/2018/9674138.

A. Abidemi and N. A. B. Aziz, “Optimal control strategies for dengue fever spread in Johor, Malaysia,” Computer Methods and Programs in Biomedicine, vol. 196, p. 105585, Nov. 2020, doi: 10.1016/j.cmpb.2020.105585.

A. Abidemi and N. A. B. Aziz, “Analysis of deterministic models for dengue disease transmission dynamics with vaccination perspective in Johor, Malaysia,” International Journal of Applied and Computational Mathematics, vol. 8, no. 1, p. 45, Feb. 2022, doi: 10.1007/s40819-022-01250-3.

E. Soewono and G. Lahodny, “On the effect of postponing pregnancy in a Zika transmission model,” Advances in Difference Equations, vol. 2021, no. 1, p. 140, Feb. 2021, doi: 10.1186/s13662-021-03308-w.

H. Fahlena, R. Kusdiantara, N. Nuraini, and E. Soewono, “Dynamical analysis of two-pathogen coinfection in influenza and other respiratory diseases,” Chaos, Solitons & Fractals, vol. 155, p. 111727, Feb. 2022, doi: 10.1016/j.chaos.2021.111727.

Fatmawati and M. A. Khan, “The dynamics of dengue infection through fractal-fractional operator with real statistical data,” Alexandria Engineering Journal, vol. 60, no. 1, pp. 321–336, Feb. 2021, doi: 10.1016/j.aej.2020.08.018.

C. J. Tay et al., “Dengue epidemiological characteristic in Kuala Lumpur and Selangor, Malaysia,” Mathematics and Computers in Simulation, vol. 194, pp. 489–504, Apr. 2022, doi: 10.1016/j.matcom.2021.12.006.

M. Z. Ndii, N. Anggriani, J. J. Messakh, and B. S. Djahi, “Estimating the reproduction number and designing the integrated strategies against dengue,” Results in Physics, vol. 27, p. 104473, Aug. 2021, doi: 10.1016/j.rinp.2021.104473.

C. Champagne and B. Cazelles, “Comparison of stochastic and deterministic frameworks in dengue modelling,” Mathematical Biosciences, vol. 310, pp. 1–12, Apr. 2019, doi: 10.1016/j.mbs.2019.01.010.

W.-J. Feng, L.-M. Cai, and K. Liu, “Dynamics of a dengue epidemic model with class-age structure,” International Journal of Biomathematics, vol. 10, no. 08, p. 1750109, 2017, doi: 10.1142/S1793524517501091.

S. B. Maier, E. Massad, M. Amaku, M. N. Burattini, and D. Greenhalgh, “The Optimal Age of Vaccination Against Dengue with an Age-Dependent Biting Rate with Application to Brazil,” Bulletin of Mathematical Biology, vol. 82, no. 1, p. 12, Jan. 2020, doi: 10.1007/s11538-019-00690-1.

S. B. Maier, X. Huang, E. Massad, M. Amaku, M. N. Burattini, and D. Greenhalgh, “Analysis of the optimal vaccination age for dengue in Brazil with a tetravalent dengue vaccine,” Mathematical Biosciences, vol. 294, pp. 15–32, 2017, doi: https://doi.org/10.1016/j.mbs.2017.09.004.

A. K. Supriatna, E. Soewono, and S. A. van Gils, “A two-age-classes dengue transmission model,” Mathematical Biosciences, vol. 216, no. 1, pp. 114–121, 2008, doi: https://doi.org/10.1016/j.mbs.2008.08.011.

N. Ganegoda, T. Götz, and K. Putra Wijaya, “An age-dependent model for dengue transmission: Analysis and comparison to field data,” Applied Mathematics and Computation, vol. 388, p. 125538, Jan. 2021, doi: 10.1016/j.amc.2020.125538.

O. Diekmann, J. A. P. Heesterbeek, and M. G. Roberts, “The construction of next-generation matrices for compartmental epidemic models,” Journal of The Royal Society Interface, vol. 7, no. 47, pp. 873–885, Jun. 2010, doi: 10.1098/rsif.2009.0386.

C. Castillo-Chavez, S. Blower, P. van den Driessche, D. Kirschner, and A. A. Yakubu, Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, no. v. 1. Springer, 2002. [Online]. Available: https://books.google.co.id/books?id=pR4CqiTSTMwC

L. J. S. Allen, “A primer on stochastic epidemic models: Formulation, numerical simulation, and analysis,” Infectious Disease Modelling, vol. 2, no. 2, pp. 128–142, May 2017, doi: 10.1016/j.idm.2017.03.001.

L. J. S. Allen and G. E. Lahodny, “Extinction thresholds in deterministic and stochastic epidemic models,” null, vol. 6, no. 2, pp. 590–611, Mar. 2012, doi: 10.1080/17513758.2012.665502.

L. J. S. Allen and P. van den Driessche, “Relations between deterministic and stochastic thresholds for disease extinction in continuous- and discrete-time infectious disease models,” Mathematical Biosciences, vol. 243, no. 1, pp. 99–108, May 2013, doi: 10.1016/j.mbs.2013.02.006.

M. Al-Zoughool et al., “Using a stochastic continuous-time Markov chain model to examine alternative timing and duration of the COVID-19 lockdown in Kuwait: what can be done now?,” Archives of Public Health, vol. 80, no. 1, p. 22, Jan. 2022, doi: 10.1186/s13690-021-00778-y.

G. E. Lahodny, R. Gautam, and R. Ivanek, “Estimating the probability of an extinction or major outbreak for an environmentally transmitted infectious disease,” null, vol. 9, no. sup1, pp. 128–155, Jun. 2015, doi: 10.1080/17513758.2014.954763.

S. Marino, I. B. Hogue, C. J. Ray, and D. E. Kirschner, “A methodology for performing global uncertainty and sensitivity analysis in systems biology,” Journal of Theoretical Biology, vol. 254, no. 1, pp. 178–196, Sep. 2008, doi: 10.1016/j.jtbi.2008.04.011.

J. Wu, R. Dhingra, M. Gambhir, and J. V. Remais, “Sensitivity analysis of infectious disease models: methods, advances and their application,” Journal of The Royal Society Interface, vol. 10, no. 86, p. 20121018, Sep. 2013, doi: 10.1098/rsif.2012.1018.

G. Chowell et al., “Estimation of the reproduction number of dengue fever from spatial epidemic data,” Mathematical Biosciences, vol. 208, no. 2, pp. 571–589, Aug. 2007, doi: 10.1016/j.mbs.2006.11.011.

M. A. Khan and Fatmawati, “Dengue infection modeling and its optimal control analysis in East Java, Indonesia,” Heliyon, vol. 7, no. 1, Jan. 2021, doi: 10.1016/j.heliyon.2021.e06023.

L. S. Sepulveda and O. Vasilieva, “Optimal control approach to dengue reduction and prevention in Cali, Colombia,” Mathematical Methods in the Applied Sciences, vol. 39, no. 18, pp. 5475–5496, Dec. 2016, doi: 10.1002/mma.3932.

Published
2022-06-01
How to Cite
[1]
M. Ndii, Y. Adi, and B. Djahi, “DETERMINISTIC AND STOCHASTIC DENGUE EPIDEMIC MODEL: EXPLORING THE PROBABILITY OF EXTINCTION”, BAREKENG: J. Math. & App., vol. 16, no. 2, pp. 583-596, Jun. 2022.
Issue
Vol 16 No 2 (2022): BAREKENG: Jurnal Ilmu Matematika dan Terapan
Section
Articles

Authors who publish with this Journal agree to the following terms:

  1. Author retain copyright and grant the journal right of first publication with the work simultaneously licensed under a creative commons attribution license that allow others to share the work within an acknowledgement of the work’s authorship and initial publication of this journal.
  2. Authors are able to enter into separate, additional contractual arrangement for the non-exclusive distribution of the journal’s published version of the work (e.g. acknowledgement of its initial publication in this journal).
  3. Authors are permitted and encouraged to post their work online (e.g. in institutional repositories or on their websites) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published works.