GLOBAL RESULTS FOR AN HIV/AIDS MODEL WITH MULTIPLE SUSCEPTIBLE CLASSES AND NONLINEAR INCIDENCE

Wei Yang

doi:10.11948/20190199

2020 Volume 10 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2020 > 10(1): 335-349

Next Article Previous Article

Wei Yang. GLOBAL RESULTS FOR AN HIV/AIDS MODEL WITH MULTIPLE SUSCEPTIBLE CLASSES AND NONLINEAR INCIDENCE[J]. Journal of Applied Analysis & Computation, 2020, 10(1): 335-349. doi: 10.11948/20190199

Citation:

Wei Yang. GLOBAL RESULTS FOR AN HIV/AIDS MODEL WITH MULTIPLE SUSCEPTIBLE CLASSES AND NONLINEAR INCIDENCE[J]. Journal of Applied Analysis & Computation, 2020, 10(1): 335-349. doi: 10.11948/20190199

GLOBAL RESULTS FOR AN HIV/AIDS MODEL WITH MULTIPLE SUSCEPTIBLE CLASSES AND NONLINEAR INCIDENCE

Wei Yang^,

Academy for Engineering and Technology, Fudan University, Shanghai, 200433, China

Corresponding author: Email address: yangwei@fudan.edu.cn(W. Yang)
Fund Project: The author was supported by National Science Foundation of Shanghai (No.18ZR1404300) and National Natural Science Foundation of China (No.11401109)

Abstract

Abstract

In this paper, an HIV/AIDS epidemic model is proposed in which there are two susceptible classes. Two types of general nonlinear incidence functions are employed to depict the scenarios of infection among cautious and incautious individuals. Qualitative analyses are performed, in terms of the basic reproduction number $ \mathcal{R}_0 $, to gain the global dynamics of the model: the disease-free equilibrium is of global asymptotic stability when $ \mathcal{R}_0\leq 1 $; a unique endemic equilibrium exists and globally asymptotically stable $ \mathcal{R}_0> 1 $. The introduction of cautious susceptible and the resulting multiple transmission functions has positive effect on HIV/AIDS prevalence. Numerical simulations are carried out to illustrate and extend the obtained analytical results.
- HIV/AIDS /
- cautious susceptible /
- general nonlinear incidence /
- basic reproduction number
MSC: 34K20, 92D30

FullText(HTML)

References

[1]	V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Mathematical Biosciences, 1978, 42(1), 43–61. Google Scholar
[2]	H. de Arazoza and R. Lounes, A nonlinear model for a sexually transmitted disease with contact tracing, Mathematical Medicine and Biology: A Journal of the IMA, 2002, 19(3), 221–234. doi: 10.1093/imammb/19.3.221 CrossRef Google Scholar
[3]	O. Diekmann, J. Heesterbeek and J. Metz, 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, 1990, 28(4), 365–382. Google Scholar
[4]	P. Georgescu and Y. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal, SIAM Journal on Applied Mathematics, 2007, 67(2), 337–353. Google Scholar
[5]	H. Hethcote, M. Lewis and P. van den Driessche, An epidemiological model with a delay and a nonlinear incidence rate, Journal of Mathematical Biology, 1989, 27(1), 49–64. doi: 10.1007/BF00276080 CrossRef Google Scholar
[6]	H. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear incidence, Journal of Mathematical Biology, 1991, 29(3), 271–287. doi: 10.1007/BF00160539 CrossRef Google Scholar
[7]	W. Huang, M. Han and K. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Mathematical Biosciences and Engineering, 2010, 7(1), 51–66. Google Scholar
[8]	H. Huo and L. Feng, Global stability for an HIV/AIDS epidemic model with different latent stages and treatment, Applied Mathematical Modelling, 2013, 37(3), 1480–1489. Google Scholar
[9]	J. Jia and G. Qin, Stability analysis of HIV/AIDS epidemic model with nonlinear incidence and treatment, Advances in Difference Equations, 2017, 2017(1), 136. Google Scholar
[10]	W. Kermack and A. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London. Series A, 1927, 115(772), 700–721. doi: 10.1098/rspa.1927.0118 CrossRef Google Scholar
[11]	A. Korobeinikov and P. Maini, Non-linear incidence and stability of infectious disease models, Mathematical Medicine and Biology: A Journal of the IMA, 2005, 22(2), 113–128. doi: 10.1093/imammb/dqi001 CrossRef Google Scholar
[12]	J. Levy, Pathogenesis of human immunodeficiency virus infection, Microbiological Reviews, 1994, 46(2), 113. Google Scholar
[13]	J. Li, Y. Yang, Y. Xiao and S. Liu, A class of Lyapunov functions and the global stability of some epidemic models with nonlinear incidence, Journal of Applied Analysis and Computation, 2016, 6(1), 38–46. Google Scholar
[14]	A. Liapunov, Stability of motion, Elsevier, 2016. Google Scholar
[15]	W. Liu, S. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, Journal of Mathematical Biology, 1986, 23(2), 187–204. doi: 10.1007/BF00276956 CrossRef Google Scholar
[16]	Z. Ma and J. Li, Dynamical modeling and analysis of epidemics, International Association of Geodesy Symposia, 2009, 106(B11), xiv+498. Google Scholar
[17]	R. Naresh, A. Tripathi and S. Omar, Modelling the spread of AIDS epidemic with vertical transmission, Applied Mathematics and Computation, 2006, 178(2), 262–272. Google Scholar
[18]	S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, Journal of Differential Equations, 2003, 188(1), 135–163. doi: 10.1016/S0022-0396(02)00089-X CrossRef Google Scholar
[19]	C. Silva and D. Torres, Global stability for a HIV/AIDS model, arXiv preprint arXiv: 1704.05806, 2017. Google Scholar
[20]	L. Simpson and G. A., Mathematical assessment of the role of pre-exposure prophylaxis on HIV transmission dynamics, Applied Mathematics and Computation, 2017, 293, 168–193. Google Scholar
[21]	B. Sounvoravong and S. Guo, Dynamics of a diffusive SIR epidemic model with time delay, Journal of Nonlinear Modeling and Analysis, 2019, 1(3), 319–334. Google Scholar
[22]	G. Statistics, HIV BASICS: Overview: Data & Trend. HIV gov., 2018. https://www.hiv.gov/hiv-basics/overview/data-and-trends/global-statistics. Google Scholar
[23]	C. Sun, J. Arino and S. Portet, Intermediate filament dynamics: Disassembly regulation, International Journal of Biomathematics, 2017, 10(1), 1750015 (22 pages). doi: 10.1142/S1793524517500152 CrossRef Google Scholar
[24]	C. Sun, Y. Hsieh and G. P., A model for HIV transmission with two interacting high-risk groups, Nonlinear Analysis: Real World Applications, 2018, 40, 170–184. Google Scholar
[25]	P. van den Driessche and J. Watmough, A simple SIS epidemic model with a backward bifurcation, Journal of Mathematical Biology, 2000, 40(6), 525–540. doi: 10.1007/s002850000032 CrossRef Google Scholar
[26]	P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 2002, 180(1), 29–48 Google Scholar
[27]	R. Weiss, How does HIV cause AIDS?, Science, 1993, 260(5112), 1273–1279. doi: 10.1126/science.8493571 CrossRef Google Scholar
[28]	W. Yang, Z. Shu, J. Lam and C. Sun, Global dynamics of an HIV model incorporating senior male clients, Applied Mathematics and Computation, 2017, 311, 203–216. doi: 10.1016/j.amc.2017.05.030 CrossRef Google Scholar
[29]	Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear Analysis: Real World Applications, 2010, 11(2), 995–1004. doi: 10.1016/j.nonrwa.2009.01.040 CrossRef Google Scholar
[30]	T. Yusuf and F. Benyah, Optimal strategy for controlling the spread of HIV/AIDS disease: a case study of South Africa, Journal of Biological Dynamics, 2012, 6(2), 475–494. doi: 10.1080/17513758.2011.628700 CrossRef Google Scholar