An example of a compartmental model of malaria transmission with treatment and environmental sanitation, which can be studied algebraically, is presented. Exact methods from algebra, geometry and computer algebra have been used to determine all the equilibria of the system of ordinary differential equations representing the model and to study their stability and bifurcations.

epidemiological models, malaria, sanitation effect, stability of equilibria, bifurcation, Gröbner basis.

Received: December 30, 2020; Accepted: January 16, 2021; Published: February 15, 2021

How to cite this article: Adamou Otto, Morou Amidou and Zakari Yaou Moussa, An algebraic approach for a malaria transmission model with environmental sanitation, Advances in Differential Equations and Control Processes 24(1) (2021), 67-78. DOI: 10.17654/DE024010067

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] W. W. Adams and P. Loustaunau, An introduction to Gröbner bases, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 3, 1994, 289 pp.
[2] S. Basu, R. Pollack and M. F. Roy, Algorithms in real algebraic geometry, Algorithms and Computation in Mathematics, Springer-Verlag, Berlin, 2nd ed., Vol. 10, 2006, 662 pp.
[3] T. Becker and V. Weispfenning, Gröbner Bases: A Computational Approach to Commutative Algebra (in cooperation with Heinz Kredel), Graduate Texts in Mathematics, Springer-Verlag, New York, Vol. 141, 2011, 576 pp.
[4] L. Billings, A. Fiorillo and I. B. Schwartz, Vaccinations in disease models with antibody-dependent enhancement, Math. Biosci. 211(2) (2008), 265-281.
[5] C. W. Brown, M. El Kahoui, D. Novotni and A. Weber, Algorithmic methods for investigating equilibria in epidemic modeling, J. Symbolic Comput. 41(11) (2006), 1157-1173.
[6] D. Cox, J. Little and D. O’Shea, Ideals, Varieties, and algorithms, 3rd ed., Undergraduate Texts in Mathematics, Springer, New York, Vol. 301, 2007.
[7] M. El Kahoui and A. Otto, Stability of disease free equilibria in epidemiological models, Math. Comput. Sci. 2 (2009), 517-533.
[8] N. Ferguson, R. Anderson and S. Gupta, The effect of antibody-dependant enhancement on the transmission dynamics and persistence of multiple strain pathogens, Proc. Natl. Acad. Sci. USA 96 (1999), 790-794.
[9] H. W. Hethcote, The mathematic of infectious disease, SIAM Rev. 42(4) (2000), 599-653.
[10] R. Ross, The Prevention of Malaria, 2nd ed., John Murray, London, 1911.
[11] 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.
[12] S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, Texts in Applied Mathematics, 2nd ed., Springer-Verlag, New York, Vol. 864, Series, Vol. 2, 2003, 844 pp.