## THERMAL SCIENCE

International Scientific Journal

### Thermal Science - Online First

online first only
### Local stability of dengue model using the fractional order system with different memory effect on the host and vector population

**ABSTRACT**

In this study, we formulate a fractional order dengue model by considering different order dynamics on human and mosquito population. The order of the differential equation is associated with the index of memory. Both human and mosquito carry a different value of order α to showcase the different memory effect implies to each of them in the transmission process. Local stability of the equilibria is obtained based on the threshold parameter related to the basic reproduction number, denoted by R0. Finally, numerical simulations of the model are conducted to study the dynamical behaviour of the system.

**KEYWORDS**

PAPER SUBMITTED: 2018-11-22

PAPER REVISED: 2018-12-26

PAPER ACCEPTED: 2019-01-07

PUBLISHED ONLINE: 2019-03-09

- De Los Reyes V, A. A., Escaner IV, J. M. L., Dengue in the Philippines: model and analysis of parameters affecting transmission, Journal of Biological Dynamics, 12 (2018), 1, pp. 894-912
- Gumel, A. B., Causes of backward bifurcations in some epidemiological models, Journal of Mathematical Analysis and Applications, 395 (2012), pp. 355-365
- Sharp, T., Centers for disease control and prevention (CDC), www.cdc.gov/dengue
- Derouich, M., Boutayeb, A., Dengue fever: mathematical modelling and computer simulation, Applied Mathematics and Computation, 177 (2006), 2, pp. 528-544
- Pinho, S. T. R., et al., Modelling the dynamics of dengue real epidemics, Philosophical Transactions of The Royal Society, 368 (2010), pp. 5679-5693
- Esteva, L., Vargas, C., Analysis of dengue transmission model, Mathematical Biosciences, 15 (1998), 2, pp. 131-151
- Yang, H. M., Ferreira, C. P., Assessing the effects of vector control on dengue transmission, Applied Mathematics and Computation, 198 (2008), pp. 401-413
- Garba, S. M., et al., Backward bifurcations in dengue transmission dynamics, Mathematical Biosciences, 215 (2008), pp. 11-25
- Esteva, L., Yang, H. M., Assessing the effects of temperature and dengue virus load on dengue transmission, Journal of Biological Systems, 23 (2015), 4, pp. 527-554
- Phaijoo, G. R., Gurung, D. B., Mathematical model of dengue disease transmission dynamics with control measures, Journal of Advances in Mathematics and Computer Science, 23 (2017), 2, pp. 1-12
- Rodrigues, H. S., et al., Dengue disease, basic reproduction number and control, International Journal of Computer Mathematics, 89 (2012), 3, pp. 334-346
- Side, S., Noorani, M. S. M., SEIR model for transmission of dengue fever in Selangor Malaysia, International Journal of Modern Physics: Conference Series, 9 (2012), pp. 380-389
- Side, S., Noorani, M. S. M., A SIR model for spread of dengue fever disease (simulation for South Sulawesi, Indonesia and Selangor, Malaysia), World Journal of Modelling and Simulation, 9 (2013), 2, pp. 96-105
- McCall, P. J., Kelly, D. W., Learning and memory in disease vectors, Trends. Parasitol, 18 (2002), 10, pp. 4229-43
- Takken, W., Verhulst, N. O., Host preferences of blood-feeding mosquitoes, Annu. Rev. Entomol., 58 (2013), pp. 433-453
- Chilaka, N., et al., Visual and olfactory associated learning in malaria vector Anopheles gambiae sensu stricto, Malaria Journal, 11 (2012), 11
- Hanert, E., et al., Front dynamics in fractional-order epidemic model, Journal Theor. Biol., 279 (2011), 1, pp. 9-16
- Agarwal, R. P., et al., Existence of solutions for integro-differential equations of fractional order with nonlocal three-point fractional boundary conditions, Advances in Difference Equations, 123 (2013)
- Du, M. et al., Measuring memory with the order of fractional derivative, Sci Rep., 3 (2013)
- Pooseh, S. et al., Fractional derivatives in dengue epidemics, AIP Conf. Proc. 1389 (2011), 739
- Diethelm, K., A fractional calculus based model for the simulation of an outbreak of dengue fever, Nonlinear Dynamics, 71 (2013), 4, pp. 613-619
- Sardar, T., et al., A generic model for a single strain mosquito-transmitted disease with memory on the host and the vector, Mathematical Biosciences, 263 (2015), pp. 18-36
- Sardar, T., et al., A mathematical model of dengue transmission with memory, Communications in Nonlinear Science and Numerical Simulation, 22 (2015), 1-3, pp. 511-525
- Hamdan, N. I., Kilicman, A., A fractional order SIR epidemic model for dengue transmission, Chaos, Solitons and Fractals, 114 (2018), pp. 55-62
- Machado, J. T., et al., Recent history of fractional calculus, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 3, pp. 1140-1153
- Matignon, D., Stability results for fractional differential equations with applications to control processing, In Proceedings of the Computational Engineering in Systems Application, 2 (1996), pp. 963-968
- Rostamy, D., Mottaghi, E., Stability analysis of a fractional-order epidemics model with multiple equilibriums, Advances in Difference Equations, 170 (2016)
- El-Shahed, M., Alsaedi, A., The fractional SIRC model and Influenza A., Mathematical Problems in Engineering, (2011)
- Garrappa, R., Trapezoidal methods for fractional differential equations: theoretical and computational aspects. Math Comput Simul, 11 (2015), pp. 96-112
- ***, data.gov.my, www.data.gov.my
- Li, C. P., Zhang, F. R., A survey on the stability of fractional differential equations, The European Physical Journal Special Topics, 193 (2011), pp. 27-47
- Kelly, D. W., Why are some people bitten more than others?, J. Med. Entomol., 17 (2001),12, pp. 578-581
- Kelly, D. W., Thompson, C., Epidemiology and optimal foraging: modelling the ideal free distribution of insect vectors, Parasitology, 120 (2000), pp. 319-327
- Chen, L., et al., Fractional order models for system identification of thermal dynamics of buildings, Energy and Buildings, 133 (2016), pp. 381-388
- Ouhsaine, L., et al., A general fractional-order heat transfer model for Photovoltaic/Thermal hybrid systems and its observer design, Energy Procedia, 139 (2017), pp. 49-54