### Dynamics of malaria-dengue fever and its optimal control

#### Abstract

The mosquito-borne infectious diseases like malaria and dengue are putative as important tropical infections and cause high morbidity and mortality around the world. In some cases, simultaneous coexistence of both the infections in one individual is seen which is very hard to distinguish as both diseases have almost similar symptoms. In this proposed article, dynamical system of non-linear differential equations is constructed with the help of mathematical modeling, which describe dynamics of the spread of these infectious diseases separately and concurrently. Basic reproduction number is evaluated to understand dynamical behaviour of the model. Local and global stability criteria have been deliberated rigorously. Control parameters are used to perceive effect of medication on these prevalent tropical diseases. Numerical simulations are used to observe effect of control parameters graphically.

#### Keywords

#### Full Text:

PDF#### References

Aldila, D., & Agustin, M. R. (2018, March). A Mathematical Model Of Dengue-Chikungunya Co-Infection In A Closed Population. In Journal of Physics: Conference Series (Vol. 974, No. 1, p. 012001). IOP Publishing.

Arino, J., McCluskey, C. C., & van den Driessche, P. (2003). Global results for an epidemic model with vaccination that exhibits backward bifurcation. SIAM Journal on Applied Mathematics, 64(1), 260-276.

Brauer, F., van den Driessche, P., & Wu, J. (2008). Mathematical Epidemiology. lecture notes in mathematics-springer verlag-, 1(1945), 1-2.

Carme, B., Matheus, S., Donutil, G., Raulin, O., Nacher, M., & Morvan, J. (2009). Concurrent dengue and malaria in Cayenne hospital, French Guiana. Emerging infectious diseases, 15(4), 668-671.

Castillo-Chavez, C., Blower, S., van den Driessche, P., Kirschner, D., & Yakubu, A. A. (Eds.). (2002). Mathematical approaches for emerging and reemerging infectious diseases: an introduction (Vol. 1). Springer Science & Business Media.

Charrel, R. N., Brouqui, P., Foucault, C., & De Lamballerie, X. (2005). Concurrent dengue and malaria. Emerging infectious diseases, 11(7), 1153-1154

Cox, J., Grillet, M. E., Ramos, O. M., Amador, M., & Barrera, R. (2007). Habitat segregation of dengue vectors along an urban environmental gradient. The American journal of tropical medicine and hygiene, 76(5), 820-826.

Deresinski, S. (2006). Concurrent Plasmodium vivax malaria and dengue.

Diekmann, O., Heesterbeek, J. A. P., & Metz, J. A. (1990). On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations. Journal of mathematical biology, 28(4), 365-382.

Epelboin, L., Hanf, M., Dussart, P., Ouar-Epelboin, S., Djossou, F., Nacher, M., & Carme, B. (2012). Is dengue and malaria co-infection more severe than single infections? A retrospective matched-pair study in French Guiana. Malaria journal, 11(1), 142-149.

Fleming, W. H., Rishel, R. W., Marchuk, G. I., Balakrishnan, A. V., Borovkov, A. A., Makarov, V. L., Rubinov, A.M., Liptser, R.S., Shiryayev, A.N., Krassovsky, N.N. & Subbotin, A. N. (1975). Applications of Mathematics. Deterministic and Stochastic Optimal Control.

Isea, R., & Lonngren, K. E. (2016). A preliminary mathematical model for the dynamic transmission of Dengue, Chikungunya and Zika. American Journal of Modern Physics and Application 2016; 3(2), 11-15.

Kaushik, R. M., Varma, A., Kaushik, R., & Gaur, K. J. (2007). Concurrent dengue and malaria due to Plasmodium falciparum and P. vivax. Transactions of the Royal Society of Tropical Medicine and Hygiene, 101(10), 1048-1050.

Khan, M. A., Islam, S., & Khan, S. A. (2014). Mathematical modeling towards the dynamical interaction of leptospirosis. Applied Mathematics & Information Sciences, 8(3), 1049-1056.

Kumari, N., & Sharma, S. (2018). Modeling the Dynamics of Infectious Disease Under the Influence of Environmental Pollution. International Journal of Applied and Computational Mathematics, 4, 1-24.

Li, M. Y., & Muldowney, J. S. (1996). A geometric approach to global-stability problems. SIAM Journal on Mathematical Analysis, 27(4), 1070-1083.

Mackey, T. K., & Liang, B. A. (2012). Lessons from SARS and H1N1/A: employing a WHO–WTO forum to promote optimal economic-public health pandemic response. Journal of public health policy, 33(1), 119-130.

Mahmoud, D. M., Hussein, H. M., El Gozamy, B. M. R., Thabet, H. S., Hassan, M. A., & Meselhey, R. A. A. (2019). Screening of Plasmodium parasite in vectors and humans in three villages in Aswan Governorate, Egypt. Journal of Parasitic Diseases, 43(1), 158-163.

McCluskey, C. C., & van den Driessche, P. (2004). Global analysis of two tuberculosis models. Journal of Dynamics and Differential Equations, 16(1), 139-166.

Obsomer, V., Defourny, P., & Coosemans, M. (2007). The Anopheles dirus complex: spatial distribution and environmental drivers. Malaria Journal, 6(1), 26-40.

Patel, S. K., Rajora, N., Kumar, S., Sahu, A., Kochar, S. K., Krishna, C. M., & Srivastava, S. (2019). Rapid discrimination of Malaria and Dengue Infected Patients Sera using Raman Spectroscopy. Analytical chemistry.

Rodenhuis-Zybert, I. A., Wilschut, J., & Smit, J. M. (2010). Dengue virus life cycle: viral and host factors modulating infectivity. Cellular and molecular life sciences, 67(16), 2773-2786.

Rupali, P. (2019). Introduction to Tropical Medicine. Infectious Disease Clinics, 33(1), 1-15.

Santana, V. D. S., Lavezzo, L. C., Mondini, A., Terzian, A. C. B., Bronzoni, R. V. D. M., Rossit, A. R. B., Machado, R.L.D., Rahal, P., Nogueira, M.C.L. & Nogueira, M. L. (2010). Concurrent dengue and malaria in the Amazon region. Revista da Sociedade Brasileira de Medicina Tropical, 43(5), 508-511.

Sharomi, O., Podder, C., Gumel, A., & Song, B. (2008). Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment. Mathematical Biosciences and Engineering, 5(1), 145-174.

Silva, C. J., & Torres, D. F. (2015). A TB-HIV/AIDS coinfection model and optimal control treatment. Discrete and Continuous Dynamical Systems, 35(9), 4639-4663.

Smith, R. A. (1986). Some applications of Hausdorff dimension inequalities for ordinary differential equations. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 104(3-4), 235-259.

Van den Driessche, P., & Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical biosciences, 180(1-2), 29-48.

Wangari, I. M., Davis, S., & Stone, L. (2016). Backward bifurcation in epidemic models: Problems arising with aggregated bifurcation parameters. Applied Mathematical Modelling, 40(2), 1669-1675.

Ward, D. I. (2006). A case of fatal Plasmodium falciparum malaria complicated by acute dengue fever in East Timor. The American journal of tropical medicine and hygiene, 75(1), 182-185.

Wiwanitkit, V. (2011). Concurrent malaria and dengue infection: a brief summary and comment. Asian Pacific journal of tropical biomedicine, 1(4), 326-327.

DOI: http://dx.doi.org/10.11121/ijocta.01.2020.00828

### Refbacks

- There are currently no refbacks.

Copyright (c) 2020 Nita H. Shah, Ankush H. Suthar, Ekta N. Jayswal

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