Stochastic and random models of Malaria Disease with vertical transmission

Abstract

Malaria is an infectious disease which affects both humans and animals. In this study, the existing mathematical model of malaria disease with vertical transmission is analyzed in random enviroment. Random effect terms are added to the parameters of the deterministic model to form a system of random differential equations. Similarly, stochastic noise is added to the deterministic system to obtain a stochastic model. Finally, the results from the deterministic, random and stochastic model are compared to comment on the random behavior of the disease.

Keywords

• Malaria,stochastic differential equation,random differential equation,Milstein method

Kaynakça

1. Aron, J.L. (1989). Mathematical modelling of immunity to malaria. Math. Biosci., 90, 385-396.
2. Anderson, R.M. & May, R.M. (1991). Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford.
3. Bailey, N.T.J. (1988). The Biomathematics of Malaria. In: Malaria: Principles and Practice of Malariology, Oxford Uni. Press, London.
4. Bekiryazici, Z., Merdan, M., Kesemen, T. & Najmuldeen, M. (2016). Mathematical Modeling of Dengue Disease under Random Effects. Mathematical Sciences and Applications E-Notes, 4(2), 58-70.
5. Caraballo, H. (2014). Emergency department management of mosquito-borne illness: Malaria, dengue, and west nile virus. Emergency Medicine Practice, 16(5), 1-23.
6. Chiyaka, C., Garira, W. & Dube, S. (2007). Transmission model of endemic human malaria in a partially immune population. Mathematical and Computer Modelling, 46, 806-822.
7. Feller, W. (1971). An Introduction to Probability Theory and Its Applications, vol. 2, John Wiley & Sons, New York.
8. Gaudart, J., Touré, O., Dessay, N., Dicko, A.I., Ranque, S., Forest, L., Demongeot, J. & Doumbo, O.K. (2009). Modelling malaria incidence with environmental dependency in a locality of Sudanese savannah area, Mali, Malaria Journal, 8(61).

9. Gerardin, J., Ouédraogo, A.L., McCarthy, K.A., Eckhoff, P.A. & Wenger, E.A. (2009). Characterization of the infectious reservoir of malaria with an agent-based model calibrated to age-stratified parasite densities and infectiousness, Malaria Journal, 14(231).
10. Gurarie, D. & McKenzie, F.E. (2007). A stochastic model of immune-modulated malaria infection and disease in children, Math Biosci, 210(2), 576-597.
11. Merdan, M. & Khaniyev, T. (2008). On the Behavior of Solutions under the Influence of Stochastic Effect of Avian-Human Influenza Epidemic Model, International Journal of Biotechnology and Biochemistry, 4(1), 75-100.
12. Ngwa, G.A. & Shu, W.S. (2000). A Mathematical Model for Endemic Malaria with Variable Human and Mosquito Population, Mathematical and Computer Modelling, 32, 747-763.
13. Nirwani, N., Badshah, V.D. & Khandewal, R. (2015). A Mathematical Model of Malaria Disease with Vertical Transmission. Journal of Mathematics Research, 7(3), 159-164.
14. Smith, T.A. (2008). Estimation of heterogeneity in malaria transmission by stochastic modelling of apparent deviations from mass action kinetics. Malaria Journal, 7(12).
15. World Health Organization (2015). Fact Sheet: World Malaria Report 2015.
16. Yang, H.M. (2000). Malaria transmission model for different levels of acquired immunity and temperature dependent parameters (vector). J Public Health, 34, 223-231.