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Analysis of an asymmetric two-strain dengue model. (English) Zbl 1310.92055
Summary: In this paper we analyse a two-strain compartmental dengue fever model that allows us to study the behaviour of a dengue fever epidemic. Dengue fever is the most common mosquito-borne viral disease of humans that in recent years has become a major international public health concern. The model is an extension of the classical compartmental susceptible-infected-recovered (SIR) model where the exchange between the compartments is described by ordinary differential equations (ODE). Two-strains of the virus exist so that a primary infection with one strain and secondary infection by the other strain can occur. There is life-long immunity to the primary infection strain, temporary cross-immunity and, after the secondary infection, life-long immunity to the secondary infection strains. Newborns are assumed susceptible. Antibody dependent enhancement (ADE) is a mechanism where the pre-existing antibodies to the previous dengue infection do not neutralize but rather enhance replication of the secondary strain. In the previously studied models, the two strains are identical with respect to their epidemiological functioning: that is, the epidemiological process parameters of the two strains were assumed equal. As a result, the mathematical model possesses a mathematical symmetry property. In this manuscript, we study a variant with epidemiological asymmetry between the strains: the force of infection rates differ while all other epidemiological parameters are equal. Comparison with the results for the epidemiologically symmetric model gives insight into its robustness. Numerical bifurcation analysis and simulation techniques including Lyapunov exponent calculation will be used to study the long-term dynamical behaviour of the model. For the single strain system, stable endemic equilibria exist and for the two-strain system, endemic equilibria, periodic solutions and also chaotic behaviour.

MSC:
92D30 Epidemiology
Software:
AUTO; Maple; MATCONT; Matlab
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[1] Adams, B.; Holmes, E.; Zhang, C.; Mammen, J. M.P.; Nimmannitya, S.; Kalayanarooj, S.; Boots, M., Cross-protective immunity can account for the alternating epidemic pattern of dengue virus serotypes circulating in Bangkok, Proc. Natl. Acad. Sci. USA, 103, 14234, (2006)
[2] Adams, B.; Sasaki, A., Antigenic distance and cross-immunity, invasibility and coexistence of pathogen strains in an epidemiological model with discrete antigenic space, Theor. Popul. Biol., 76, 157, (2009) · Zbl 1213.92055
[3] Aguiar, M.; Ballesteros, S.; Kooi, B. W.; Stollenwerk, N., The role of seasonality and import in a minimalistic multi-strain dengue model capturing differences between primary and secondary infections: complex dynamics and its implications for data analysis, J. Theor. Biol., 289, 181, (2011) · Zbl 1397.92613
[4] Aguiar, M.; Kooi, B. W.; Stollenwerk, N., Epidemiology of dengue fever: a model with temporary cross-immunity and possible secondary infection shows bifurcations and chaotic behaviour in wide parameter regions, Math. Model. Nat. Phenom., 3, 4, 48, (2008) · Zbl 1337.92126
[5] Aguiar, M.; Stollenwerk, N.; Kooi, B. W., Torus bifurcations, isolas and chaotic attractors in a simple dengue fever model with ade and temporary cross immunity, Int. J. Comput. Math., 86, 10/11, 1867, (2009) · Zbl 1173.92023
[6] Aguiar, M.; Stollenwerk, N.; Kooi, B. W., Scaling of stochasticity in dengue hemorrhagic fever epidemics, Math. Model. Nat. Phenom., 7, 1, (2012) · Zbl 1250.92025
[7] Andraud, M.; Hens, N.; Beutels, P., A simple periodic-forced model for dengue fitted to incidence data in Singapore, Math. Biosci., 244, 22, (2013) · Zbl 1279.92048
[8] Bianco, S.; Shaw, L. B.; Schwartz, I. B., Epidemics with multistrain interactions: the interplay between cross immunity and antibody-dependent enhancement, Chaos, 19, 043123, (2009)
[9] Billings, L.; Schwartz, I. B.; Shaw, L. B.; McCrary, M.; Burke, D. S.; Cummings, D. A.T., Instabilities in multiserotype disease models with antibody-dependent enhancement, J. Theor. Biol., 246, 18, (2007)
[10] Cummings, D. A.T.; Schwartz, I. B.; Billings, L.; Shaw, L. B.; Burke, D. S., Dynamic effects of antibody-dependent enhancement on the fitness of viruses, Proc. Natl. Acad. Sci. USA, 102, 42, 15259, (2005)
[11] Dhooge, A.; Govaerts, W.; Kuznetsov, Yu. A., Matcont: a MATLAB package for numerical bifurcation analysis of odes, ACM Trans. Math. Software, 29, 141, (2003) · Zbl 1070.65574
[12] E.J. Doedel, B. Oldeman, Auto 07p: continuation and bifurcation software for ordinary differential equations, Technical report, Concordia University, Montreal, Canada, 2009.
[13] Ferguson, N.; Anderson, R.; Gupta, S., The effect of antibody-dependent enhancement on the transmission dynamics and persistence of multiple-strain pathogens, Proc. Natl. Acad. Sci. USA, 96, 9, 790, (1999)
[14] Gibbons, R.; Kalanarooj, S.; Jarman, R.; Nisalak, A.; Vaughn, D.; Endy, T.; Mammen, M.; Srikiatkhachorn, A., Analysis of repeat hospital admissions for dengue to estimate the frequency of third or fourth dengue infections resulting in admissions and dengue hemorrhagic fever and serotype sequences, Am. J. Trop. Med. Hyg., 77, 5, 910, (2007)
[15] Gubler, D. J., Epidemic dengue/dengue hemorrhagic fever as a public health, social and economic problem in the 21st century, Trends Microbiol., 10, 100, (2002)
[16] Guckenheimer, J.; Holmes, P., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Applied Mathematical Sciences, vol. 42, (1985), Springer-Verlag New York
[17] Hu, K.; Thoens, C.; Bianco, S.; Edlund, S.; Davis, M.; Douglas, J.; Kaufman, J. H., The effect of antibody-dependent enhancement, cross immunity and vector population on the dynamics of dengue fever, J. Theor. Biol., 319, 62, (2013) · Zbl 1406.92577
[18] Johansson, M. A.; Hombach, J.; Cummings, D. A.T., Models of the impact of dengue vaccines: a review of current research and potential approaches, Vaccine, 29, 5860, (2011)
[19] Kawaguchi, I.; Sasaki, A.; Boots, M., Why are dengue virus serotypes so distantly related? enhancement and limiting serotype similarity between dengue virus strains, Proc. R. Soc. Lond. Ser. B, 270, 2241, (2003)
[20] Kooi, B. W.; Aguiar, M.; Stollenwerk, N., Bifurcation analysis of a family of multi-strain epidemiology models, J. Comput. Appl. Math., 252, 148, (2013) · Zbl 1288.92022
[21] Kuznetsov, Yu. A., Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, vol. 112, (2004), Springer-Verlag New York · Zbl 1082.37002
[22] Maple, Maple software, Maplesoft, Waterloo, Ontario, Canada, 2008.
[23] Matlab, Matlab package, The MathWorks, Natick, Massachusetts, USA, 2008.
[24] Mier-y-Teran-Romero, L.; Schwartz, I. B.; Cummings, D. A.T., Breaking the symmetry: immune enhancement increases persistence of dengue viruses in the presence of asymmetric transmission rates, J. Theor. Biol., 332, 203, (2013) · Zbl 1330.92067
[25] Matheus, S.; Deparis, X.; Labeau, B.; Lelarge, J.; Morvan, J.; Dussart, P., Discrimination between primary and secondary dengue virus infection by an immunoglobulin G avidity test using a single acute-phase serum sample, J. Clin. Microbiol., 43, 6, 2793, (2005)
[26] Nagao, Y.; Koelle, K., Decreases in dengue transmission may act to increase the incidence of dengue hemorrhagic fever, Proc. Natl. Acad. Sci. USA, 105, 2238, (2008)
[27] Schwartz, I. B.; Shaw, L. B.; Cummings, D. A.T.; Billings, L.; McCrary, L.; Burke, D. S., Chaotic de-synchronization of multi-strain diseases, Phys. Rev. E, 72, 066201, (2005)
[28] Seydel, R., Practical bifurcation and stability analysis - from equilibrium to chaos, (1994), Springer-Verlag New York · Zbl 0806.34028
[29] Stollenwerk, N.; Aguiar, M.; Ballesteros, S.; Boto, J.; Kooi, B. W.; Mateus, L., Dynamic noise, chaos and parameter estimation in population biology, Interf. Focus, 2, 2, 156, (2012)
[30] Wearing, H.; Rohani, P., Ecological and immunological determinants of dengue epidemics, Proc. Natl. Acad. Sci. USA, 103, 11802, (2006)
[31] WHO, Dengue and dengue hemorrhagic fever, fact sheet 117, Technical report, World Health Organization, 2009.
[32] Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos, Texts in Applied Mathematics, vol. 2, (1990), Springer-Verlag New York · Zbl 0701.58001
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