Human immunodeficiency virus type 1 (HIV-1) gradually destroys the CD4$ ^{+} $ T cells leading to immune system dysfunction. HIV-1 can result in acquired immunodeficiency syndrome (AIDS) if antiretroviral drugs are not used. HIV/AIDS patients are more vulnerable to opportunistic infections or cancers. Human herpesvirus 8 (HHV-8) targets B cells and causes an AIDS-related cancer known as kaposi sarcoma (KS). Numerous investigations have demonstrated co-infection instances between HIV-1 and HHV-8. In this research, we investigated the co-dynamics of HIV-1 and HHV-8 in vivo using a system of delay differential equations (DDEs). The model explained the interactions between uninfected CD4$ ^{+} $ T cells, latently/actively HIV-1-infected CD4$ ^{+} $ T cells, free HIV-1 particles, uninfected B cells, latently/actively HHV-8-infected B cells, and free HHV-8 particles. Eight distributed-time delays were incorporated into the model to account for the delays that arose during the generation of both actively and latently infected cells, the activation of latent reservoirs, and the maturation of freshly discharged virions. By examining the nonnegativity and boundedness of the solutions, we demonstrated that the model was both mathematically and biologically well-posed. We calculated the model's equilibria and threshold numbers. We studied the global asymptotic stability of the model's equilibria by building appropriate Lyapunov functionals and applying the Lyapunov-LaSalle asymptotic stability theorem. Numerical simulations were used to display the results. For the basic reproduction numbers of HHV-8 single-infection ($ R_{1} $) and HIV-1 single-infection ($ R_{2} $), sensitivity analysis was carried out. Comparing HIV-1 or HHV-8 single infections with co-infections of HHV-8 and HIV-1 was shown. It's interesting to note that we detected larger amounts of HHV-8 and HIV-1 when they co-infect than when they are infected alone. This outcome aligned with several findings seen in the literature. The effect of antiviral drugs and time delays on the co-dynamics of HIV-1 and HHV-8 was investigated. We found that the delay parameter and drug effectiveness both contributed to a decrease in the basic reproduction numbers, $ R_{1} $ and $ R_{2} $. Less treatment efficacies will be needed to keep the system at the infection-free equilibrium and remove HIV-1 and HHV-8 from the body if a model with time delays is employed.
Citation: A. M. Elaiw, E. A. Almohaimeed, A. D. Hobiny. Stability of HHV-8 and HIV-1 co-infection model with latent reservoirs and multiple distributed delays[J]. AIMS Mathematics, 2024, 9(7): 19195-19239. doi: 10.3934/math.2024936
Human immunodeficiency virus type 1 (HIV-1) gradually destroys the CD4$ ^{+} $ T cells leading to immune system dysfunction. HIV-1 can result in acquired immunodeficiency syndrome (AIDS) if antiretroviral drugs are not used. HIV/AIDS patients are more vulnerable to opportunistic infections or cancers. Human herpesvirus 8 (HHV-8) targets B cells and causes an AIDS-related cancer known as kaposi sarcoma (KS). Numerous investigations have demonstrated co-infection instances between HIV-1 and HHV-8. In this research, we investigated the co-dynamics of HIV-1 and HHV-8 in vivo using a system of delay differential equations (DDEs). The model explained the interactions between uninfected CD4$ ^{+} $ T cells, latently/actively HIV-1-infected CD4$ ^{+} $ T cells, free HIV-1 particles, uninfected B cells, latently/actively HHV-8-infected B cells, and free HHV-8 particles. Eight distributed-time delays were incorporated into the model to account for the delays that arose during the generation of both actively and latently infected cells, the activation of latent reservoirs, and the maturation of freshly discharged virions. By examining the nonnegativity and boundedness of the solutions, we demonstrated that the model was both mathematically and biologically well-posed. We calculated the model's equilibria and threshold numbers. We studied the global asymptotic stability of the model's equilibria by building appropriate Lyapunov functionals and applying the Lyapunov-LaSalle asymptotic stability theorem. Numerical simulations were used to display the results. For the basic reproduction numbers of HHV-8 single-infection ($ R_{1} $) and HIV-1 single-infection ($ R_{2} $), sensitivity analysis was carried out. Comparing HIV-1 or HHV-8 single infections with co-infections of HHV-8 and HIV-1 was shown. It's interesting to note that we detected larger amounts of HHV-8 and HIV-1 when they co-infect than when they are infected alone. This outcome aligned with several findings seen in the literature. The effect of antiviral drugs and time delays on the co-dynamics of HIV-1 and HHV-8 was investigated. We found that the delay parameter and drug effectiveness both contributed to a decrease in the basic reproduction numbers, $ R_{1} $ and $ R_{2} $. Less treatment efficacies will be needed to keep the system at the infection-free equilibrium and remove HIV-1 and HHV-8 from the body if a model with time delays is employed.
| [1] | UNAIDS: Global HIV & AIDS statistics-Fact sheet, 2023. Available from: https://www.unaids.org/en/resources/fact-sheet. |
| [2] |
M. Bilal, M. A. Z. Raja, I. Ahmad, R. Khan, M. Shoaib, Dynamical analysis of nonlinear combined drug therapy model for HIV infection: Bayesian regularization technique intelligent networks, Biomed. Signal Proces. Control, 88 (2024), 105629. https://doi.org/10.1016/j.bspc.2023.105629 doi: 10.1016/j.bspc.2023.105629
|
| [3] |
T. F. Schulz, Y. Chang, P. S. Moore, Kaposi's sarcoma's associated herpesvirus (human herpesvirus 8), J. Gen. Virol., 79 (1998), 1573–1591. https://doi.org/10.1128/9781555818289.ch3 doi: 10.1128/9781555818289.ch3
|
| [4] |
Y. Chang, E. Cesarman, M. S. Pessin, F. Lee, J. Culpepper, D. M. Knowles, et al., Identification of herpesvirus-like DNA sequences in AIDS-sssociated kaposi's sarcoma, Science, 266 (1994), 1865–1869. https://doi.org/10.1126/science.7997879 doi: 10.1126/science.7997879
|
| [5] |
C. Boshoff, R. A. Weiss, Kaposi's sarcoma-associated herpesvirus, Adv. Cancer Res., 75 (1998), 57–87. https://doi.org/10.1016/S0065-230X(08)60739-3 doi: 10.1016/S0065-230X(08)60739-3
|
| [6] |
M. J. Cannon, A. S. Laney, P. E. Pellett, Human herpesvirus 8: current issues, Clin. Infect. Dis., 37 (2003), 82–87. https://doi.org/10.1086/375230 doi: 10.1086/375230
|
| [7] |
S. J. Dollery, R. J. Santiago-Crespo, D. Chatterjee, E. A. Berger, Glycoprotein K8. 1A of Kaposi's sarcoma-associated herpesvirus is a critical B cell tropism determinant independent of its heparan sulfate binding activity, J. Virol., 93 (2019), e01876-18. https://doi.org/10.1128/JVI.01876-18 doi: 10.1128/JVI.01876-18
|
| [8] |
S. L. Swain, K. K. McKinstry, T. M. Strutt, Expanding roles for CD4$^{+}$T cells in immunity to viruses, Nat. Rev. Immunol., 12 (2012), 136–148. https://doi.org/10.1038/nri3152 doi: 10.1038/nri3152
|
| [9] |
E. Rohner, N. Wyss, Z. Heg, Z. Faralli, S. M. Mbulaiteye, U. Novak, et al., HIV and human herpesvirus 8 co-infection across the globe: systematic review and meta-analysis, Int. J. Cancer, 138 (2016), 45–54. https://doi.org/10.1002/ijc.29687 doi: 10.1002/ijc.29687
|
| [10] |
G. A. Malonga, A. Jary, V. Leducq, D. Moudiongui Mboungou Malanda, A. L. M. Boumba, E. Chicaud, et al., Seroprevalence and molecular diversity of Human Herpesvirus 8 among people living with HIV in Brazzaville, Congo. Sci. Rep., 11 (2021), 17442. https://doi.org/10.1038/s41598-021-97070-4 doi: 10.1038/s41598-021-97070-4
|
| [11] |
L. C. Pierrotti, A. Etzel, L. M. Sumita, P. E. Braga, J. Eluf-Neto, V. A. U. F. de Souza, et al., Human herpesvirus 8 (HHV-8) infection in HIV/AIDS patients from Santos, Brazil: seroprevalence and associated factors, Sex. Transm. Dis., 32 (2005), 57–63. https://doi.org/10.1097/01.olq.0000148300.33428.6e doi: 10.1097/01.olq.0000148300.33428.6e
|
| [12] |
M. Masiá, C. Robledano, V. Ortiz de la Tabla, P. Antequera, B. Lumbreras, I. Hernández, et al., Coinfection with human herpesvirus 8 is associated with persistent inflammation and immune activation in virologically suppressed HIV-infected patients, PLoS One, 9 (2014), e105442. https://doi.org/10.1371/journal.pone.0105442 doi: 10.1371/journal.pone.0105442
|
| [13] |
D. Watanabe, S. Iida, K. Hirota, T. Ueji, T. Matsumura, Y. Nishida, et al., Evaluation of human herpesvirus-8 viremia and antibody positivity in patients with HIV infection with human herpesvirus-8-related diseases, J. Med. Virol., 95 (2023), e29324. https://doi.org/10.1002/jmv.29324 doi: 10.1002/jmv.29324
|
| [14] |
H. Lambarey, M. J. Blumenthal, A. Chetram, W. Joyimbana, L. Jennings, C. Orrell, et al., Reactivation of Kaposi's sarcoma-associated herpesvirus (KSHV) by SARS-CoV-2 in non-hospitalised HIV-infected patients, Ebiomedicine, 100 (2024), 1–13. https://doi.org/10.1016/j.ebiom.2024.104986 doi: 10.1016/j.ebiom.2024.104986
|
| [15] |
M. A. Nowak, C. R. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74–79. https://doi.org/10.1126/science.272.5258.74 doi: 10.1126/science.272.5258.74
|
| [16] | M. A. Nowak, R. M. May, Virus dynamics: mathematical principles of immunology and virology, New York: Oxford University Press, 2000. https://doi.org/10.1093/oso/9780198504184.001.0001 |
| [17] |
T. Inoue, T. Kajiwara, T. Sasaki, Global stability of models of humoral immunity against multiple viral strains, J. Biol. Dynam., 4 (2010), 282–295. https://doi.org/10.1080/17513750903180275 doi: 10.1080/17513750903180275
|
| [18] |
M. Dhar, S. Samaddar, P. Bhattacharya, Modeling the effect of non-cytolytic immune response on viral infection dynamics in the presence of humoral immunity, Nonlinear Dyn., 98 (2019), 637–655. https://doi.org/10.1007/s11071-019-05219-8 doi: 10.1007/s11071-019-05219-8
|
| [19] |
S. Wang, D. Zou, Global stability of in host viral models with humoral immunity and intracellular delays, Appl. Math. Model., 36 (2012), 1313–1322. https://doi.org/10.1016/j.apm.2011.07.086 doi: 10.1016/j.apm.2011.07.086
|
| [20] |
J. Xu, Y. Zhou, Y. Li, Y. Yang, Global dynamics of a intracellular infection model with delays and humoral immunity, Math. Methods Appl. Sci., 39 (2016), 5427–5435. https://doi.org/10.1002/mma.3927 doi: 10.1002/mma.3927
|
| [21] |
S. Tang, Z. Teng, H. Miao, Global dynamics of a reaction-diffusion virus infection model with humoral immunity and nonlinear incidence, Comput. Math. Appl., 78 (2019), 786–806. https://doi.org/10.1016/j.camwa.2019.03.004 doi: 10.1016/j.camwa.2019.03.004
|
| [22] |
T. Kajiwara, T. Sasaki, Y. Otani, Global stability for an age-structured multistrain virus dynamics model with humoral immunity, J. Appl. Math. Comput., 62 (2020), 239–279. https://doi.org/10.1007/s12190-019-01283-w doi: 10.1007/s12190-019-01283-w
|
| [23] |
J. Lin, R. Xu, X. Tian, Threshold dynamics of an HIV-1 virus model with both virus-to-cell and cell-to-cell transmissions, intracellular delay, and humoral immunity, Appl. Math. Comput., 315 (2017), 516–530. https://doi.org/10.1016/j.amc.2017.08.004 doi: 10.1016/j.amc.2017.08.004
|
| [24] |
Y. Luo, L. Zhang, T. Zheng, Z. Teng, Analysis of a diffusive virus infection model with humoral immunity, cell-to-cell transmission and nonlinear incidence, Phys. A, 535 (2019), 122415. https://doi.org/10.1016/j.physa.2019.122415 doi: 10.1016/j.physa.2019.122415
|
| [25] |
H. Miao, R. Liu, M. Jiao, Global dynamics of a delayed latent virus model with both virus-to-cell and cell-to-cell transmissions and humoral immunity, J. Inequal. Appl., 2021 (2021), 156. https://doi.org/10.1186/s13660-021-02691-y doi: 10.1186/s13660-021-02691-y
|
| [26] |
O. M. Chimbola, E. M. Lungu, B. Szomolay, Optimal control application to a Kaposi's sarcoma treatment model, Int. J. Biomath., 15 (2022), 2150081. https://doi.org/10.1142/S1793524521500819 doi: 10.1142/S1793524521500819
|
| [27] |
A. M. Elaiw, A. D. Al Agha, S. A. Azoz, E. Ramadan, Global analysis of within-host SARS-CoV-2/HIV coinfection model with latency, Eur. Phys. J. Plus, 137 (2022), 174. https://doi.org/10.1140/epjp/s13360-022-02387-2 doi: 10.1140/epjp/s13360-022-02387-2
|
| [28] |
A. M. Elaiw, N. H. AlShamrani, Analysis of a within-host HIV/HTLV-I co-infection model with immunity, Virus Res., 295 (2021), 198204. https://doi.org/10.1016/j.virusres.2020.198204 doi: 10.1016/j.virusres.2020.198204
|
| [29] |
R. Birger, R. Kouyos, J. Dushoff, B. Grenfell, Modeling the effect of HIV coinfection on clearance and sustained virologic response during treatment for hepatitis C virus, Epidemics, 12 (2015), 1–10. https://doi.org/10.1016/j.epidem.2015.04.001 doi: 10.1016/j.epidem.2015.04.001
|
| [30] |
H. Nampala, L. S. Luboobi, J. Y. T. Mugisha, C. Obua, M. Jablonska-Sabuka, Modelling hepatotoxicity and antiretroviral therapeutic effect in HIV/HBV coinfection, Math. Biosci., 302 (2018), 67–79. https://doi.org/10.1016/j.mbs.2018.05.012 doi: 10.1016/j.mbs.2018.05.012
|
| [31] | F. Nani, M. Jin, Dynamics of HIV-1 associated Kaposi sarcoma during HAART therapy, Math and Computer Science Working Papers, 2011. |
| [32] |
F. Nani, M. Jin, Analysis of dynamics of HIV-1 associated Kaposi sarcoma during HAART and ACI, Brit. J. Math. Comput. Sci., 19 (2016), 1–22. https://doi.org/10.9734/BJMCS/2016/20358 doi: 10.9734/BJMCS/2016/20358
|
| [33] |
R. F. Kaondera-Shava, E. Lungu, B. Szomolay, A novel mathematical model of AIDS-associated Kaposi's sarcoma: analysis and optimal control, Biosystems, 200 (2021), 104318. https://doi.org/10.1016/j.biosystems.2020.104318 doi: 10.1016/j.biosystems.2020.104318
|
| [34] |
B. Szomolay, E. M. Lungu, A mathematical model for the treatment of AIDS-related Kaposi's sarcoma, J. Biol. Syst., 22 (2014), 495–522. https://doi.org/10.1142/S0218339014500247 doi: 10.1142/S0218339014500247
|
| [35] |
O. M. Chimbola, E. M. Lungu, B. Szomolay, Effect of innate and adaptive immune mechanisms on treatment regimens in an AIDS-related Kaposi's Sarcoma model, J. Biol. Dyn., 15 (2021), 213–249. https://doi.org/10.1080/17513758.2021.1912420 doi: 10.1080/17513758.2021.1912420
|
| [36] |
G. Huang, Y. Takeuchi, W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010), 2693–2708. https://doi.org/10.1137/090780821 doi: 10.1137/090780821
|
| [37] |
A. Alshorman, X. Wang, M. J. Meyer, L. Rong, Analysis of HIV models with two time delays, J. Biol. Dyn., 2 (2017), 40–64. https://doi.org/10.1080/17513758.2016.1148202 doi: 10.1080/17513758.2016.1148202
|
| [38] |
H. Liu, J. F. Zhang, Dynamics of two time delays differential equation model to HIV latent infection, Phys. A, 514 (2019), 384–395. https://doi.org/10.1016/j.physa.2018.09.087 doi: 10.1016/j.physa.2018.09.087
|
| [39] |
Y. Wang, M. Lu, J. Liu, Global stability of a delayed virus model with latent infection and Beddington-DeAngelis infection function, Appl. Math. Lett., 107 (2020), 106463. https://doi.org/10.1016/j.aml.2020.106463 doi: 10.1016/j.aml.2020.106463
|
| [40] |
X. Zhou, L. Zhang, T. Zheng, H. Li, Z. Teng, Global stability for a delayed HIV reactivation model with latent infection and Beddington-DeAngelis incidence, Appl. Math. Lett., 117 (2021), 107047. https://doi.org/10.1016/j.aml.2021.107047 doi: 10.1016/j.aml.2021.107047
|
| [41] |
J. Xu, G. Huang, Global stability and bifurcation analysis of a virus infection model with nonlinear incidence and multiple delays, Fractal Fract., 7 (2023), 583. https://doi.org/10.3390/fractalfract7080583 doi: 10.3390/fractalfract7080583
|
| [42] |
Y. Yang, Y. Dong, Y. Takeuchi, Global dynamics of a latent HIV infection model with general incidence function and multiple delays, Discrete Cont. Dyn. Syst.-Ser. B, 24 (2019), 783–800. https://doi.org/10.3934/dcdsb.2018207 doi: 10.3934/dcdsb.2018207
|
| [43] |
Y. Wang, M. Lu, D. Jiang, Viral dynamics of a latent HIV infection model with Beddington-DeAngelis incidence function, B-cell immune response and multiple delays, Math. Biosci. Eng., 18 (2021), 274–299. https://doi.org/10.3934/mbe.2021014 doi: 10.3934/mbe.2021014
|
| [44] |
M. Y. Li, H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections, SIAM J. Appl. Math., 70 (2010), 2434–2448. https://doi.org/10.1137/090779322 doi: 10.1137/090779322
|
| [45] | J. K. Hale, S. M. Verduyn Lunel, Introduction to functional differential equations, New York: Springer-Verlag, 1993. https://doi.org/10.1007/978-1-4612-4342-7 |
| [46] | Y. Kuang, Delay differential equations: with applications in population dynamics, Academic press, 1993. |
| [47] |
A. S. Perelson, D. E. Kirschner, R. De Boer, Dynamics of HIV-1 infection of CD4+ T cells, Math. Biosci., 114 (1993), 81–125. https://doi.org/10.1016/0025-5564(93)90043-A doi: 10.1016/0025-5564(93)90043-A
|
| [48] |
R. V. Culshaw, S. Ruan, A delay-differential equation model of HIV infection of $\mathrm{CD} 4^{+}$T-cells, Math. Biosci., 165 (2000), 27–39. https://doi.org/10.1016/s0025-5564(00)00006-7 doi: 10.1016/s0025-5564(00)00006-7
|
| [49] |
M. Hadjiandreou, R. Conejeros, V. S. Vassiliadis, Towards a long-term model construction for the dynamic simulation of HIV-1 infection, Math. Biosci. Eng., 4 (2007), 489–504. https://doi.org/10.3934/mbe.2007.4.489 doi: 10.3934/mbe.2007.4.489
|
| [50] |
E. A. Hernandez-Vargas, R. H. Middleton, Modeling the three stages in HIV infection, J. Theor. Biol., 320 (2013), 33–40. https://doi.org/10.1016/j.jtbi.2012.11.028 doi: 10.1016/j.jtbi.2012.11.028
|
| [51] |
S. K. Sahani, Yashi, Effects of eclipse phase and delay on the dynamics of HIV-1 infection, J. Biol. Syst., 26 (2018), 421–454. https://doi.org/10.1142/S0218339018500195 doi: 10.1142/S0218339018500195
|
| [52] |
S. Pankavich, The effects of latent infection on the dynamics of HIV-1, Differ. Equat. Dyn. Syst., 24 (2016), 281–303. https://doi.org/10.1007/s12591-014-0234-6 doi: 10.1007/s12591-014-0234-6
|
| [53] |
A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879–883. https://doi.org/10.1016/j.bulm.2004.02.001 doi: 10.1016/j.bulm.2004.02.001
|
| [54] | E. A. Barbashin, Introduction to the theory of stability, Wolters-Noordhoff, Groningen, 1970. |
| [55] | J. P. La Salle, The stability of dynamical systems, SIAM, Philadelphia, 1976. |
| [56] |
A. M. Lyapunov, The general problem of the stability of motion, Taylor Francis, 55 (1992), 531–534. https://doi.org/10.1080/00207179208934253 doi: 10.1080/00207179208934253
|
| [57] |
G. Rappocciolo, H. R. Hensler, M. Jais, T. A. Reinhart, A. Pegu, F. J. Jenkins, et al., Human herpesvirus 8 infects and replicates in primary cultures of activated B lymphocytes through DC-SIGN, J. Virol., 82 (2008), 4793–4806. https://doi.org/10.1128/jvi.01587-07 doi: 10.1128/jvi.01587-07
|
| [58] |
M. Renardy, C. Hult, S. Evans, J. J. Linderman, D. E. Kirschner, Global sensitivity analysis of biological multiscale models, Curr. Opin. Biomed. Eng., 11 (2019), 109–116. https://doi.org/10.1016/j.cobme.2019.09.012 doi: 10.1016/j.cobme.2019.09.012
|
| [59] |
Z. Zi, Sensitivity analysis approaches applied to systems biology models, IET Syst. Biol., 5 (2011), 336–346. https://doi.org/10.1049/iet-syb.2011.0015 doi: 10.1049/iet-syb.2011.0015
|
| [60] |
S. Marino, I. B. Hogue, C. J. Ray, D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol., 254 (2008), 178–196. https://doi.org/10.1016/j.jtbi.2008.04.011 doi: 10.1016/j.jtbi.2008.04.011
|
| [61] |
V. M. S. de Morais, E. L. S. de Lima, G. G. D. O. M. Cahú, T. R. R. Lopes, J. P. Gonçales, M. T. C. Muniz, et al., MBL2 gene polymorphisms in HHV-8 infection in people living with HIV/AIDS, Retrovirology, 15 (2018), 1–9. https://doi.org/10.1186/s12977-018-0456-8 doi: 10.1186/s12977-018-0456-8
|
| [62] |
R. Minami, M. Yamamoto, S. Takahama, H. Ando, T. Miyamura, E. Suematsu, Human herpesvirus 8 DNA load in the leukocytes correlates with the platelet counts in HIV type 1-infected individuals, AIDS Res. Hum. Retrov., 25 (2009), 1–8. https://doi.org/10.1089/aid.2007.0260 doi: 10.1089/aid.2007.0260
|
| [63] |
D. Oktafiani, N. L. A. Megasari, E. Fitriana, N. Nasronudin, M. I. Lusida, S. Soetjipto, Detection of human herpesvirus-8 antigen in HIV-infected patients in East Java, Indonesia, Afr. J. Infect. Dis., 12 (2018), 43–46. https://doi.org/10.21010/ajid.v12i2.7 doi: 10.21010/ajid.v12i2.7
|
| [64] |
R. Tedeschi, M. Enbom, E. Bidoli, A. Linde, P. De Paoli, J. Dillner, Viral load of human herpesvirus 8 in peripheral blood of human immunodeficiency virus-infected patients with Kaposi's sarcoma, J. Clin. Microbiol., 39 (2001), 4269–4273. https://doi.org/10.1128/jcm.39.12.4269-4273.2001 doi: 10.1128/jcm.39.12.4269-4273.2001
|
| [65] |
M. Mercader, B. Taddeo, J. R. Panella, B. Chandran, B. J. Nickoloff, K. E. Foreman, Induction of HHV-8 lytic cycle replication by inflammatory cytokines produced by HIV-1-infected T cells, Am. J. Pathol., 156 (2000), 1961–1971. https://doi.org/10.1016/S0002-9440(10)65069-9 doi: 10.1016/S0002-9440(10)65069-9
|
| [66] |
N. Bellomo, D. Burini, N. Outada, Multiscale models of Covid-19 with mutations and variants, Netw. Heterog. Media, 17 (2022), 293–310. https://doi.org/10.3934/nhm.2022008 doi: 10.3934/nhm.2022008
|
| [67] |
D. Burini, D. A. Knopoff, Epidemics and society-A multiscale vision from the small world to the globally interconnected world, Math. Models Methods Appl. Sci., 34 (2024), 1564–1596. https://doi.org/10.1142/S0218202524500295 doi: 10.1142/S0218202524500295
|
Figures(9) / Tables(5)
A. M. Elaiw, E. A. Almohaimeed, A. D. Hobiny. Stability of HHV-8 and HIV-1 co-infection model with latent reservoirs and multiple distributed delays[J]. AIMS Mathematics, 2024, 9(7): 19195-19239. doi: 10.3934/math.2024936
DownLoad: