Estimation of time-space-varying parameters in dengue epidemic models

Karunia Putra Wijaya


There are nowadays a huge load of publications about dengue epidemic models, which mostly employ deterministic differential equations. The analytical properties of deterministic models are always of particular interest by many experts, but their validity – if they can indeed track some empirical data – is an increasing demand by many practitioners. In this view, the data can tell to which figure the solutions yielded from the models should be; they drift all the involving parameters towards the most appropriate values. By prior understanding of the population dynamics, some parameters with inherently constant values can be estimated forthwith; some others can sensibly be guessed. However, solutions from such models using sets of constant parameters most likely exhibit, if not smoothness, at least noise-free behavior; whereas the data appear very random in nature. Therefore, some parameters cannot be constant as the solutions to seemingly appear in a high correlation with the data. We were aware of impracticality to solve a deterministic model many times that exhaust all trials of the parameters, or to run its stochastic version with Monte Carlo strategy that also appeals for a high number of solving processes. We were also aware that those aforementioned non-constant parameters can potentially have particular relationships with several extrinsic factors, such as meteorology and socioeconomics of the human population. We then study an estimation of time-space-varying parameters within the framework of variational calculus and investigate how some parameters are related to some extrinsic factors. Here, a metric between the aggregated solution of the model and the empirical data serves as the objective function, where all the involving state variables are kept satisfying the physical constraint described by the model. Numerical results for some examples with real data are shown and discussed in details.


Dengue epidemics; seasonal-spatial model; parameter estimation; variational calculus.

Full Text:



S. L. Sharara and S. S. Kanj. War and infectious diseases: Challenges of the Syrian civil war. PLoS Pathogens 10(11) (2014) e1004438–4.

Global strategy for dengue prevention and control, 2012–2020. (World Health Organization, Geneva, Switzerland, 2012).

Comprehensive guidelines for prevention and control of dengue and dengue haemorrhagic fever. Revised and expanded edition. (SEARO Technical Publication Series No. 60, World Health Organization South-East Asia office, 2011).

S. Bhatt, P. W. Gething, O. J. Brady, J. P. Messina, A. W. Farlow, C. L. Moyes, et al. The global distribution and burden of dengue. Nature 496(7446) (2013) 504–507.

J. D. Stanaway, D. S. Shepard, E. A. Undurraga, Y. A. Halasa, L. E. Coffeng, O. J. Brady, S. I. Hay, N. Bedi, I. M. Bensenor, C. A. Castaeda-Orjuela, T-W. Chuang, K. B. Gibney, Z. A. Memish, A. Rafay, K. N. Ukwaja, N. Yonemoto, C. J. L. Murray. The global burden of dengue: an analysis from the Global Burden of Disease Study 2013. The Lancet Infectious Diseases 16(6)

(2016) 712–723.

W. O. Kermack, A. G. McKendrick. A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London, Series A, Containing papers of a mathematical and physical character 115 (1927) 700–721.

J. D. Murray. Mathematical Biology 3rd Ed. (Springer Science & Business Media, 1989).

E. M. Lotfi, M. Maziane, K. Hattaf and N. Yousfi. Partial differential equations of an epidemic model with spatial diffusion. International Journal of Partial Differential Equations 2014 186437–6 (2014).

A. Okubo and S. A. Levin. Diffusion and Ecological Pproblems: Modern Perspectives 2nd Ed. (Springer, NewYork, 2001).

M. Mimura and M. Kawasaki. Spatial segregation in competitive interaction-diffusion equations. Journal of Mathematical Biology 9 (1980) 49–64.

M. Mimura and M. Yamaguti. Pattern formation in interacting and diffusive systems in population biology. Advances in Biophysics 15 (1982) 19–65.

G. Galiano, M. L. Garz´on and A J¨ungel. Analysis and numerical solution of a nonlinear cross-diffusion system arising in population dynamics. Revista de la Real Academia de Ciencias Exactas, F´ısicas y Naturales Serie A Mathem´aticas 95 (2001) 281–295.

M. Bendahmane and M. Langlais. A reaction-diffusion system with cross-diffusion modeling the spread of an epidemic disease. Journal of Evolution Equations 10 (2010) 883–904.

D. J. Gubler and M. Meltzer. The impact of dengue/dengue hemorrhagic fever on the developing world. Advances in Virus Research 53 (1999) 35–70.

M. Aguiar, S. Ballesteros, B. W. Kooi, and N. Stollenwerk. The role of seasonality and import in a minimalistic multi-strain dengue model capturing differences between primary and secondary infections: complex dynamics and its implications for data analysis. Journal of Theoretical Biology 289 (2011) 181–196.

F. Rocha, M. Aguiar, M. Souza, and Nico Stollenwerk. Time-scale separation and centre manifold analysis describing vector-borne disease dynamics. International Journal of Computer Mathematics 90(10) (2013) 2105–2125.

T. G¨otz, N. Altmeier, W. Bock, R. Rockenfeller, Sutimin, K. P. Wijaya. Modeling dengue data from Semarang, Indonesia. Ecological Complexity 30 (2017) 57–62.

H. G. Bock. Recent advances in parameter identification techniques for ordinary differential equations, in P. Deuflhard and E. Hairer (Eds.). Numerical treatment of inverse problems in differential and integral equations. (Birkh¨auser, 1983) 95–121.

K. P. Wijaya, T. G¨otz and E. Soewono. Advances in mosquito dynamics modeling. Mathematical Methods in the Applied Sciences 39(16) (2016) 4750–4763.

E. Zeidler. Nonlinear Functional Analysis and its Applications, Vol. II/A. (Springer, New York, 1990).

J. Wloka. Partial Differential Equations. (Cambridge University Press, 1987).

E. Hille and R. S. Phillips. Functional Analysis and Semigroups. (American Mathematical Society Colloquial Publication, Providence, RI, 1957).

F. Tr¨oltzsch. Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Vol. 112. (American Mathematical Society, Providence, RI, 2010).

L. C. Evans. Partial differential equations, 2nd Ed. (American Mathematical Society, Providence, RI, 2010).

R. Dautray and J. -L. Lions. Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5: Evolution Problems. I, With the collaboration of Michel Artola, Michel Cessenat and H´el´ene Lanchon, Translated from the French by Alan Craig. (Springer-Verlag, Berlin, 1992).

D. Wachsmuth. The regularity of the positive part of functions in L2(I;H1())H1(I;H1()) with applications to parabolic equations. Commentationes Mathematicae Universitatis Carolinae 57(3) (2016) 327–332.

D. Henry. Geometric Theory of Semilinear Parabolic Equations . (Springer-Verlag, New York, 1981).

M. H. Protter and H. F. Weinberger. Maximum Principles in Differential Equations, 2nd ed. (Springer-Verlag, Berlin, 1984).

G. F. Raggett. An efficient gradient technique for the solution of optimal control problems. Computer Methods in Applied Mechanics and Engineering 12 (1977) 315–322.

T. G¨otz and S. S. N. Perera. Optimal control of melt-spinning processes. Journal of Engineering Mathematics 67(3) (2010) 153–163.

K. P. Wijaya, T. G¨otz and E. Soewono. An optimal control model of mosquito reduction management in a dengue endemic region. International Journal of Biomathematics 7(5) (2014) 1450056–22.

The Health Office of the City of Semarang via Last accessed December 11, 2016, 19:38.

D. He and D. J. D. Earn.Epidemiological effects of seasonal oscillations in birth rates. Theoretical Population Biology 72 (2007) 274–291.

S. M. Henson and J. M. Cushing. The effect of periodic habitat fluctuations on a nonlinear insect population model. Journal of Mathematical Biology 36 (1997) 201–226..

J. M. Ireland, B. D. Mestel and R. A. Norman. The effect of seasonal host birth rates on disease persistence. Mathematical Biosciences 206 (2007) 31–45.

F. Sauvage, M. Langlais and D. Pontier. Predicting the emergence of human hantavirus disease using a combination of viral dynamics and rodent demographic patterns. Epidemiology & Infection 135 (2007) 46–56.

L. J. S. Allen and A. M. Burgin. Comparison of deterministic and stochastic SIS and SIRS models in discrete time. Mathematical Biosciences 163(1) (2000) 1–33.

R. K. McCormack and L. J. S. Allen. Multi-patch deterministic and stochastic models for wildlife diseases. Journal of Biological Dynamics 1(1) (2006) 63–85.

J. D’Errico. Surface Fitting using Gridfit, Last accessed December 11, 2016, 20:11.



  • There are currently no refbacks.

View My Stats

Creative Commons License

This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.

This journal published by: Indonesian Bio-Mathematical Society, Pusat Pemodelan Matematika dan Simulasi, Jalan Ganesa No. 10 Bandung 40116