Study of A Delayed SIVA Within-Host Model of Dengue Virus Transmission
DOI:
https://doi.org/10.5614/cbms.2022.5.2.1Keywords:
Immune response; Time delay; Hopf BifurcationAbstract
During the process of immune response to the infection caused by dengue virus, antibodies are generated by plasma cells which are produced by B-cells. In some cases, it is observed that there is a delay in the production of plasma cells from B-cells which causes a delay in the immune response. We propose a SIVA within-host model of the virus transmission with delayed immune response to articulate the dynamics of the cell and virus population. The stability analysis of different equilibrium states is also studied. The basic reproduction number (BRN) of the model is computed using next generation matrix (NGM) method. The local stability analysis is discussed using the method of linearisation. The stability conditions of the equilibrium states are validated using the Lienard - Chipart criterion. Hopf bifurcation analysis is carried out as the system has time lag in the immune response. Three equilibrium states, namely, virus free equilibrium state, endemic equilibrium state with and without immune response, have been observed. It has been found that the virus free equilibrium state is locally asymptotically stable if BRN is less than or equal to 1. Additionally, the conditions for the stability of the endemic equilibrium points are derived and elaborated. Numerical simulations for different values of time delay parameter ? are presented and illustrated using graphs. A Hopf bifurcation is observed if the delay parameter ? crosses a threshold value and then the system becomes unstable with periodic solution. To determine the relative importance of the model parameters to the virus transmission and prevalence, sensitivity analysis of the parameters is illustrated using graphs. Due to the time lag in the immune response, an increase in the virus growth is observed in large quantity. As a result, the infection spreads more quickly within the host.
References
Gubler, D.J., Dengue and dengue hemorrhagic fever, Clin Microbiol Rev., 11(3), pp. 480-496, 1998.
Dengue and severe dengue, WHO, https://www.who.int/news-room/fact-sheets/detail/dengue-and-severe-dengue, accessed June, 2020.
Brady, O.J., Gething, P.W., Bhatt, S., Messina, J.P., Brownstein, J.S., Hoen, A.G., Moyes, C.L., Farlow, A.W., Scott, T.W. and Hay, S.I., Refining the global spatial limits of dengue virus transmission by evidence-based consensus, PLOS Neglected Tropical Diseases, 6(8), p. e1760, 2012.
Bhatt, S., Gething, P.W., Brady, O.J., Messina, J.P., Farlow, A.W., Moyes, C.L., Drake, J.M., Brownstein, J.S., Hoen, A.G., Sankoh, O., Myers, M.F., George, D.B., Jaenisch, T., William, G.R. Wint, Simmons, C.P., Scott, T.W., Farrar, J.J. and Hay, S., The global distribution and burden of dengue, Nature, 496(7446), pp. 504-507, 2013.
Global warming would foster spread of dengue fever into some temperate regions, http://www.sciencedaily.com/releases/1998/03/980310081157.htm/, Science Daily, 1998.
Scott, T.W. and Morrison, A.C., Vector dynamics and transmission of dengue virus: implications for dengue surveillance and prevention strategies: vector dynamics and dengue prevention, Curr. Topics Microbiol. Immunol., 338, pp. 115-128, 2010.
Normile D., Tropical medicine. Surprising new dengue virus throws a spanner in disease control efforts, Science, 342(6147), p. 415, 2013.
Dwivedi, V.D., Tripathi, I.P., Tripathi, R.C., Bharadwaj, S. and Mishra, S.K., Genomics, proteomics and evolution of Dengue virus, Briefings in Functional Genomics, 16(4), pp. 217-227, 2017.
Kalayanarooj, S., Clinical Manifestations and Management of Dengue/DHF/DSS, Trop. Med. Health., 39(4), pp. 83-87, 2011.
Jindadamrongwech, S., Thepparit, C. and Smith, D.R., Identification of GRP 78 (BiP) as a liver cell expressed receptor element for dengue virus serotype 2, Arch Virol, 149(5), pp. 915-927, 2004.
Wu, S.J., Grouard-Vogel, G., Sun, W., Mascola, J.R., Brachtel, E., Putvatana, R., Louder, M.K., Filgueira, L., Marovich, M.A., Wong, H.K., Blauvelt, A., Murphy, G.S., Robb, M.L., Innes, B.L., Birx, D.L., Hayes, C.G. and Frankel, S.S., Human skin Langerhans cells are targets of dengue virus infection, Nat. Med., 6(7), pp. 816-820, 2000.
Kliks, S.C., Nisalak, A., Brandt, W.E., Wahl, L. and Burke, D.S., Antibody-dependent enhancement of dengue virus growth in human monocytes as a risk factor for dengue haemorrhagic fever, Am. J. Trop. Med. Hyg., 40, pp. 444-451, 1989.
Murphy, B.R. and Whitehead, S.S., Immune response to dengue virus and prospects for a vaccine, Annu. Rev. Immunol., 29, pp. 587-619, 2011.
Janeway, C.A., Travers, P., Walport, M. and Shlomchik, M.J., Immunobiology: the immune system in health and disease, 5th edition, Garland Science, New York, 2001.
Chaturvedi, U.C., Tandon, P., Mathur, A. and Kumar, A., Host defence mechanisms against dengue virus infection of mice, J. Gen. Virol., 39, pp. 293-302, 1978.
Halstead, S.B., Pathogenesis of dengue: challenges to molecular biology, Science, 239(4839), pp. 476-481, 1988.
Gujarati, T.P. and Ambika, G., Virus antibody dynamics in primary and secondary dengue infections, Journal of mathematical biology, 69(6), pp. 1773-1800, 2014.
Rodrigo, G. and Bernhard, R., Dengue virus infection: current concepts in immune mechanisms and lessons from murine models, Immunology, 141(2), pp. 143-156, 2014.
Malavige, G.N., Fernando, S., Fernando, D.J. and Seneviratne, S.L., Dengue viral infections, Postgrad. Med. Journal, 80(948), pp. 588-601, 2004.
Sabin, A.B., Research on dengue during World War II, American J. Trop. Med. Hyg., 1(1), pp. 30-50, 1952.
Reich, N.G., Shrestha, S., King, A.A., Rohani, P., Lessler, J., Kalayanarooj, S., Yoon I., Gibbons, R.V., Burke, D.S. and Cummings, D.A., Interactions between serotypes of dengue highlight epidemiological impact of cross-immunity, R. Soc. Interface, 10(86), pp. 1-9, 2013.
Sangkawibha, N., Rojanasuphot, S., Ahandrik, S., Viriyapongse, S., Jatanasen, S., Salttul, V., Phanthumachinda, B. and Halstead, S.B., Risk factors in dengue shock syndrome: a prospective epidemiologic study in Rayong, Thailand. I. The 1980 outbreak, American Journal of Epidemiology, 120(5), pp. 653-669, 1984.
John, A.S. and Rathore, A.P.S., Adaptive immune responses to primary and secondary dengue virus infections, Nat Rev Immunol, 19(4), pp. 218-230, 2019.
Esteva, L. and Vargas C., Analysis of a dengue disease transmission model, Math. Biosci., 150(2), pp. 131-151, 1998.
Esteva L. and Vargas, C., A model for dengue disease with variable human population, J. Math. Biol., 38(3), pp. 220-240, 1999.
Fischer, D.B. and Halstead, S.B., Observations related to pathogenesis of dengue hemorrhagic fever. V. Examination of agspecific sequential infection rates using a mathematical model, Yale. J. Biol. Med., 42(5), pp. 329-349, 1970.
Nowak, M.A. and May, R.M., Virus dynamics: mathematical principles of immunology and virology, Oxford University Press Inc., New York, 2000.
Murase, A., Sasaki, T. and Kajiwara, T., Stability analysis of pathogen-immune interaction dynamics, J. Math. Biol., 51(3), pp. 247-267, 2005.
Garba, S.M. and Gumel, A.B., Effect of cross-immunity on the transmission dynamics of two strains of dengue, International Journal of Computer Mathematics, 87(10), pp. 2361-2384, 2010.
Nuraini, N., Soewono, E. and Sidarto, K.A., A mathematical model of dengue internal transmission process, J. Indonesia Math. Soc.(MIHMI), 13(1), pp. 123-132, 2007.
Ansari, H. and Hesaaraki, M., A With-In Host Dengue Infection Model with Immune Response and Beddington-DeAngelis Incidence Rate, Applied Mathematics, 3, pp. 177-184, 2012.
Dibrov, B.F., Livshits, M.A. and Volkenstein, M.V., Mathematical Model of Immune Processes, J. theor. Biol., 65(4), pp. 609-631, 1977.
Dibrov, B.F., Livshits, M.A. and Volkenstein, M.V., Mathematical Model of Immune Processes. II. Kinetlc Features of Antigen-Antibody Interrelations, J. theor. Biol, 69(1), pp. 23-39, 1977.
Fowler, A.C., Approximate Solution of a Model of Biological Immune Responses Incorporating Delay, J. Math. Biology, 13(1), pp. 23-45, 1981.
Gourley, S.A., Kuang, Y., and Nagy, J.D., Dynamics of a delay differential equation model of hepatitis B virus infection, Journal of Biological Dynamics, 2(2), pp. 140-153, 2008.
Li, M.Y. and Shua, H., Global Dynamics of an In-host Viral Model with Intracellular Delay, Bulletin of Mathematical Biology, 72(6), pp. 1492-1505, 2010.
Huang, G., Beretta, E. and Takeuchi, Y., Global Stabilty for Epidemic Model with Constant Latency and Infectious Periods, Mathematical Biosciences and Engineering, 9(2), pp. 297-312, 2012.
Yang, Y., Zou, L. and Ruan, S., Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions, Mathematical Biosciences, 270, pp. 183-191, 2015.
Wang, S. and Zou, D., Global stability of in-host viral models with humoral immunity and intracellular delays, Applied Mathematical Modelling, 36(3), pp. 1313-1322, 2012.
Nuraini, N., Tasman, H., Soewono, E. and Sidarto, K.A., A with-in host Dengue infection model with immune response, Math. Comput. Model, 49(5-6), pp. 1148-1155, 2008.
Kanumoori, D.S.S.M., Bhanu Prakash, D., Vamsi, D.K.K. and Carani, B. Sanjeevi, A study of within-host dynamics of Dengue infection incorporating both humoral and cellular response with a time delay for production of antibodies, Comput. Math. Biophys., 9(1), pp. 66-80, 2021.
Camargo, F. de A., Adimy, M., Esteva, L., Metayer, C. and Ferreira, C.P., Modeling the relationship between antibody-dependent enhancement and disease severity in secondary dengue infection, Bull. Math. Biol., 83(8), pp. 1-28, 2021.
Sebayang, A.A., Fahlena, H., Anam, V., Knopoff, D., Stollenwerk, N., Aguiar, M. and Soewono, E., Modeling Dengue Immune Responses Mediated by Antibodies: A Qualitative Study, Biology, 10(9), pp. 1-18, 2021.
Christine, A.K., Adam, D.W. and Timothy, P.E., Mobilization and activation of the innate immune response to dengue virus, Front. Cell Infect. Microbiol., 10, pp. 1-16, 2020.
Kurane, I. and Takasaki, T., Dengue fever and dengue haemorrhagic fever: challenges of controlling an enemy still at large, Rev. Med. Virol., 11(5), pp. 301-311, 2001.
Gantmacher, F., The Theory of Matrices, American Mathematical Society., 2, pp. 221-225, 2000.
Lienard and Chipart Criterion, https://en.wikipedia.org/wiki/Lienard-Chipart criterion.
Diekmann, O., Heesterbeek, J.A.P. and Metz, J.A.J., On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28(4), pp. 365-382, 1990.
Driessche, P.V. and Watmough, J., Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180(1-2), pp. 29-48, 2002.
Dieudonne, J., Foundations of Modern Analysis, Academic Press, New York, 1960.
Mishra, A. and Gakkhar, S., A micro-epidemic model for primary dengue infection, Communications in Nonlinear Science and Numerical Simulation, 47, pp. 426-437, 2017.
Freedman, H.L. and Sree Hari Rao, V., The trade-off between mutual interference and time lags in predator-prey systems, Bulletin of Mathematical Biology, 45(6), pp. 991-1004, 1983.
Jack Hale, Theory of functional differential equation, Springer Verlag, 1977.
Srivastava, P. K. and Chandra, Peeyush, Hopf bifurcation and periodic solutions in a dynamical model for HIV and immune response, Differential Equations and Dynamical Systems, 16(1), pp. 77-100, 2008.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Communication in Biomathematical Sciences
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
