Musharif AHMED, Saad ZAFAR, Muhammad Aamer SALEEM, Muhammad ZUBAIR, Ijaz Mansoor QURESHI

Between-host HIV model: stability analysis and solution using memetic computing

HIV poses a great threat to humanity for two major reasons. First it attacks the immunity system of thebody and second, it is epidemic in nature. Mathematical models of HIV have been instrumental in understanding andcontrolling the infection. In this paper, we solve the between host epidemic model of HIV, described by nonlinear coupleddifferential equations, by using memetic computing. Under this model, the sexually active population is divided intofour classes and we investigate the transfer of individuals from one class to another. The solution consists of Bernsteinpolynomials whose parameters have been optimized by using differential evolution as global and sequential quadraticprogramming as local optimizer. Our second contribution is the stability analysis of this model. The disease-freeequilibrium is stable while endemic equilibrium is unstable within the practical range of the values of parameters.

PDF

___

[1] Bushnaq S, Khan S A, Shah K, Zaman G. Mathematical analysis of HIV/AIDS infection model with Caputo-Fabrizio fractional derivative. Cogent Mathematics & Statistics 2018; 5 (1): 1432521. doi:10.1080/23311835.2018.1432521
[2] Akpa O M, Oyejola B A. Modeling the Transmission Dynamics of HIV/AIDS epidemics: an introduction and a review. The Journal of Infection in Developing Countries 2010; 4 (10): 597-608. doi:10.3855/jidc.542
[3] Ma Z, Li J. Dynamical Modeling and Analysis of Epidemics 1st ed. Singapore: World Scientific Publishing Co. Inc., 2009.
[4] Angelis DD, Day NE, Gill ON. Acquired immune deficiency syndrome projections in England and Wales: Interplay of methodology and data. Journal of the Royal Statistical Society: Series A (Statistics in Society) 2002; 161 (2): 167-176. doi:10.1111/1467-985X.00096
[5] Chin J, Lwanga S K. Estimation and projection of adult AIDS cases: a simple epidemiological model. Bulletin of the World Health Organization 1991; 69 (4): 399-406. PMC2393240
[6] Bortz DM, Nelson PW. Model selection and mixed-effects modeling of HIV infection dynamics. Bulletin of Mathematical Biology 2006; 68 (8): 2005-2025. doi:10.1007/s11538-006-9084-x
[7] Nowak MA, Bangham CR. Population dynamics of immune responses to persistent viruses. Science 1996; 272 (5258): 74-79. doi:10.1126/science.272.5258.74
[8] Perelson A, Nelson P. Mathematical analysis of HIV-1 dynamics in vivo. SIAM review 1999; 41 (1): 3-44. doi:10.1137/S0036144598335107
[9] Perelson AS, Neumann AU, Markowitz M, Leonard JM, Ho DD. HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science 1996; 271 (5255): 1582-1586. doi:10.1126/science.271.5255.1582
[10] Haberman S. Actuarial review of models for describing and predicting the spread of HIV infection and AIDS. Journal of the Institute of Actuaries 1990; 117 (2): 319-405. doi:10.1017/S0020268100043122
[11] Isham V. Mathematical modelling of the transmission dynamics of HIV infection and AIDS: a review. Mathematical and Computer Modelling 1989; 12 (9): 1187. doi:10.1016/0895-7177(89)90269-0
[12] Cassels S, Clark SJ, Morris M. Mathematical models for HIV transmission dynamics: tools for social and behavioral science research. Journal of Acquired Immune Deficiency Syndromes 2008; 47 Suppl 1: S34-39. doi:10.1097/QAI.0b013e3181605da3
[13] Ghoreishi M, Ismail AIB, Alomari AK. Application of the homotopy analysis method for solving a model for HIV infection of CD4+ t-cells. Mathematical and Computer Modelling 2011; 54 (11): 3007-3015. doi:10.1016/j.mcm.2011.07.029
[14] Yüzbaşı Ş. A numerical approach to solve the model for HIV infection of CD4+t cells. Applied Mathematical Modelling 2012; 36 (12): 5876-5890. doi:10.1016/j.apm.2011.12.021
[15] Parand K, Yousefi H, Fotouhifar M, Delkhosh M, Hosseinzadeh M. Shifted boubaker lagrangian approach for solving biological systems. International Journal of Biomathematics 2018; 11 (3): 1850039. doi:10.1142/S1793524518500390
[16] Ongun MY. The laplace adomian decomposition method for solving a model for HIV infection of CD4+t cells. Mathematical and Computer Modelling 2011; 53 (5): 597-603. doi:10.1016/j.mcm.2010.09.009
[17] Venkatesh SG, Balachandar SR, Ayyaswamy SK, Balasubramanian K. A new approach for solving a model for HIV infection of CD4+t-cells arising in mathematical chemistry using wavelets. Journal of Mathematical Chemistry 2016; 54 (5): 1072-1082. doi:10.1007/s10910-016-0604-0
[18] Ashyralyev A, Hincal E, Kaymakamzade B. Numerical solutions of the system of partial differential equations for observing epidemic models. In: AIP Conference Proceedings; Mersin, Turkey; 2018, 1997 (1): 020050. doi:10.1063/1.5049044
[19] Basak US, Datta BK, Ghose PK. Mathematical analysis of an HIV/AIDS epidemic model. American Journal of Mathematics and Statistics 2015; 5 (5): 253-258. doi:10.5923/j.ajms.20150505.05
[20] Akinduko OB, Peter OJ. Semi analytic method for solving HIV/AIDS epidemic model. International Journal of Modern Biology and Medicine 2018; 9 (1): 1-8.
[21] Pratibha R, Divya J, Saxena PV. Approximate analytical solution with stability analysis of HIV/AIDS model. Cogent Mathematics 2016; 3 (1): 1206692. doi:10.1080/23311835.2016.1206692
[22] Malik SA, Qureshi IM, Amir M, Malik AN. Nature inspired computational approach to solve the model for HIV infection of CD4 T Cells. Research Journal of Recent Sciences 2014; 3 (6): 67-76.
[23] Kurdi M. An improved island model memetic algorithm with a new cooperation phase for multi-objective job shop scheduling problem. Computers & Industrial Engineering 2017; 111: 183-201. doi:10.1016/j.cie.2017.07.021
[24] Jaddi NS, Salwani A. Hybrid of genetic algorithm and great deluge algorithm for rough set attribute reduction. Turkish Journal of Electrical Engineering & Computer Sciences 2013; 21 (6): 1737-1750. doi:10.3906/elk-1202-113
[25] Zahoor RMA, Khan JA, Qureshi IM. Evolutionary computation technique for solving Riccati differential equation of arbitrary order. World Academy of Science, Engineering and Technology 2009; 3 (10): 303-309.
[26] Junaid A, Raja M, Qureshi IM. Evolutionary computing approach for the solution of initial value problems in ordinary differential equations. World Academy of Science, Engineering and Technology 2009; 3 (7): 578-5581
[27] Dorratoltaj N, Nikin-Beers R, Ciupe SM, Eubank SG, Abbas KM. Multi-scale immunoepidemiological modeling of within-host and between-host HIV dynamics: systematic review of mathematical models. PeerJ 2017; 5:e3877. doi:10.7717/peerj.3877
[28] Cuadros DF, García-Ramos G. Variable effect of co-infection on the HIV infectivity: Within-host dynamics and epidemiological significance. Theoretical Biology and Medical Modelling 2012; 9 (1): 9. doi:10.1186/1742-4682-9-9
[29] Shen M, Xiao Y, Rong L. Global stability of an infection-age structured HIV-1 model linking within-host and between-host dynamics. Mathematical Biosciences 2015; 263: 37-50. doi:10.1016/j.mbs.2015.02.003
[30] Yeghiazarian L, Cumberland WG, Yang OO. A stochastic multi-scale model of HIV-1 transmission for decisionmaking: Application to a MSM population. PloS One 2013; 8 (11): e70578. doi:10.1371/journal.pone.0070578
[31] Castillo-Chavez C, Song B. Dynamical models of tuberculosis and their applications. Mathematical Biosciences and Engineering 2004; 1 (2): 361-404. doi:10.3934/mbe.2004.1.361
[32] Al-Sheikh S, Musali F, Alsolami M. Stability analysis of an HIV/AIDS epidemic model with screening. International Mathematical Forum 2011; 6:3251-3273.
[33] Storn R, Price K. Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization 1997; 11 (4): 341-359. doi:10.1023/A:1008202821328
[34] Coelho L, Mariani VC, Ferreira da Luz MV, Leite LV. Novel gamma differential evolution approach for multiobjective transformer design optimization. IEEE Transactions on Magnetics 2013; 49 (5): 2121-2124. doi:10.1109/TMAG.2013.2243134
[35] dos Santos Amorim EP, Xavier CR, Campos RS, dos Santos RW. Comparison between genetic algorithms and differential evolution for solving the history matching problem. In: Computational Science and Its Applications 2012, Lecture Notes in Computer Science, 7333:635-648. doi:/10.1007/978-3-642-31125-3_48
[36] Engelbrecht AP. Computational Intelligence: An Introduction 2nd ed. West Sussex, England: John Wiley & Sons, 2007
[37] Lorentz GG. Bernstein Polynomials, USA: American Mathematical Society, 2012.
[38] Hertz A, Kobler D. A framework for the description of evolutionary algorithms. European Journal of Operational Research 2000; 126 (1): 1-12. doi:10.1016/S0377-2217(99)00435-X
[39] Houck CR, Joines JA, Kay MG. Comparison of genetic algorithms, random restart and two-opt switching for solving large location-allocation problems. Computers & Operations Research 1996; 23 (6): 587-596. doi:10.1016/0305- 0548(95)00063-1
[40] Houck CR, Joines LA, Kay MG, Wilson JR. Empirical investigation of the benefits of partial lamarckianism. Evolutionary Computation 1997; 5 (1): 31-60. doi:10.1162/evco.1997.5.1.31