On the role of vector modeling in a minimalistic epidemic model

Peter Rashkov; Ezio Venturino; Maira Aguiar; Nico Stollenwerk; Bob W. Kooi; Peter Rashkov; Ezio Venturino; Maira Aguiar; Nico Stollenwerk; Bob W. Kooi

doi:10.3934/mbe.2019215

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 4314-4338. doi: 10.3934/mbe.2019215

Previous Article Next Article

Research article Special Issues

On the role of vector modeling in a minimalistic epidemic model

1.
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, ul. Akademik Georgi Bonchev 8, 1113 Sofia, Bulgaria
2.
Dipartimento di Matematica “Giuseppe Peano”, Universit`a di Torino, via Carlo Alberto 10, 10123 Torino, Italy; Member of the INdAM research group GNCS
3.
Dipartimento di Matematica, Universit`a degli Studi di Trento, via Sommarive 14, 38123 Povo Trento, Italy
4.
Faculdade de Ciˆencias da Universidade de Lisboa, Campo Grande, C6-Piso 1, Gabinete C6.1.18, 1749-016 Lisboa, Portugal
5.
Faculty of Science, VU University, De Boelelaan 1085, 1081 HV Amsterdam, The Netherlands

Received: 19 December 2018 Accepted: 27 March 2019 Published: 16 May 2019

The motivation for the research reported in this paper comes from modeling the spread of vector-borne virus diseases. To study the role of the host versus vector dynamics and their interaction we use the susceptible-infected-removed (SIR) host model and the susceptible-infected (SI) vector model. When the vector dynamical processes occur at a faster scale than those in the host-epidemics dynamics, we can use a time-scale argument to reduce the dimension of the model. This is often implemented as a quasi steady-state assumption (qssa) where the slow varying variable is set at equilibrium and an ode equation is replaced by an algebraic equation. Singular perturbation theory will appear to be a useful tool to perform this derivation. An asymptotic expansion in the small parameter that represents the ratio of the two time scales for the dynamics of the host and vector is obtained using an invariant manifold equation. In the case of a susceptible-infected-susceptible (SIS) host model this algebraic equation is a hyperbolic relationship modeling a saturated incidence rate. This is similar to the Holling type Ⅱ functional response (Ecology) and the Michaelis-Menten kinetics (Biochemistry). We calculate the value for the force of infection leading to an endemic situation by performing a bifurcation analysis. The effect of seasonality is studied where the force of infection changes sinusoidally to model the annual fluctuations of the vector population. The resulting non-autonomous system is studied in the same way as the autonomous system using bifurcation analysis.
- asymptotic expansions,
- epidemic models,
- seasonally-forced models,
- quasi steady state assumption,
- geometrical singular perturbation,
- vector borne disease dynamics
Citation: Peter Rashkov, Ezio Venturino, Maira Aguiar, Nico Stollenwerk, Bob W. Kooi. On the role of vector modeling in a minimalistic epidemic model[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 4314-4338. doi: 10.3934/mbe.2019215

Related Papers:

Abstract

The motivation for the research reported in this paper comes from modeling the spread of vector-borne virus diseases. To study the role of the host versus vector dynamics and their interaction we use the susceptible-infected-removed (SIR) host model and the susceptible-infected (SI) vector model. When the vector dynamical processes occur at a faster scale than those in the host-epidemics dynamics, we can use a time-scale argument to reduce the dimension of the model. This is often implemented as a quasi steady-state assumption (qssa) where the slow varying variable is set at equilibrium and an ode equation is replaced by an algebraic equation. Singular perturbation theory will appear to be a useful tool to perform this derivation. An asymptotic expansion in the small parameter that represents the ratio of the two time scales for the dynamics of the host and vector is obtained using an invariant manifold equation. In the case of a susceptible-infected-susceptible (SIS) host model this algebraic equation is a hyperbolic relationship modeling a saturated incidence rate. This is similar to the Holling type Ⅱ functional response (Ecology) and the Michaelis-Menten kinetics (Biochemistry). We calculate the value for the force of infection leading to an endemic situation by performing a bifurcation analysis. The effect of seasonality is studied where the force of infection changes sinusoidally to model the annual fluctuations of the vector population. The resulting non-autonomous system is studied in the same way as the autonomous system using bifurcation analysis.

References

[1]	N. Ferguson, R. Anderson and S. Gupta, The effect of antibody-dependent enhancement on the transmission dynamics and persistence of multiple-strain pathogens, Proc. Natl. Acad. Sci. USA, 96 (1999), 790–794.
[2]	L. Billings, I. B. Schwartz, L. B. Shaw, et al., Instabilities in multiserotype disease models with antibody-dependent enhancement, J. Theor. Biol., 246 (2007), 18–27.
[3]	M. Aguiar, S. Ballesteros, B. W. Kooi, et al., 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 (2011), 181–196.
[4]	B. W. Kooi, M. Aguiar and N. Stollenwerk, Bifurcation analysis of a family of multi-strain epidemiology models, J. Comput. Appl. Math., 252 (2013), 148–158.
[5]	T.-T. Zheng and L.-F. Nie, Modelling the transmission dynamics of two-strain Dengue in the presence awareness and vector control, J. Theor. Biol., 443 (2018), 82–91.
[6]	Z. Feng and J. X. Velasco-Hernandez, Competitive exclusion in a vector-host model for the dengue fever, J. Math. Biol., 35 (1997), 523–544.
[7]	M. Zhu and Y. Xu, A time-periodic dengue fever model in a heterogeneous environment, Math. Comput. Simulat., 155 (2019), 115–129.
[8]	B. Althouse, J. Lessler, A. Sall, et al., Synchrony of sylvatic dengue isolations: A multi-host, multi-vector SIR model of dengue virus transmission in Senegal, PLoS Negl. Trop. Dis., 6 (2012), e1928.
[9]	F. Rocha, M. Aguiar, M. Souza, et al., Time-scale separation and centre manifold analysis describing vector-borne disease dynamics, Int. J. Comput. Math., 90 (2013), 2105–2125.
[10]	A. Segel and M. Slemrod, The Quasi-Steady-State Assumption: A case study in perturbation, SIAM Rev., 31 (1989), 446–477.
[11]	F. Rocha, L. Mateus, U. Skwara, et al., Understanding dengue fever dynamics: a study of seasonality in vector-borne disease models, Int. J. Comput. Math., 93 (2016), 1405–1422.
[12]	T. G¨otz, K. P. Wijaya and E. Venturino, Introducing seasonality in an SIR-UV epidemic model: an application to dengue, in 18th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2018 (ed. J. V. Aguiar), Wiley Online Library, 2018, Online ISSN:2577-7408.
[13]	Y. Nagao and K. Koelle, Decreases in dengue transmission may act to increase the incidence of dengue hemorrhagic fever, Proc. Natl. Acad. Sci. USA, 105 (2008), 2238–2243.
[14]	C. K. R. T. Jones, Geometric singular perturbation theory, in Dynamical Systems: Lectures Given at the 2nd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Montecatini Terme, Italy, June 13–22, 1994 (ed. R. Johnson), Springer, Berlin, Heidelberg, 1995, 44–118.
[15]	G. Hek, Geometric singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347–386.
[16]	B. W. Kooi and J.-C. Poggiale, Modelling, singular perturbation and bifurcation analyses of bitrophic food chains, Math. Biosci., 301 (2018), 93–110.
[17]	C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Entomol., 91 (1959), 385–398.
[18]	M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, Am. Nat., 97 (1963), 209–223.
[19]	N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana U. Math. J., 21 (1971), 193–226.
[20]	N. Fenichel, Geometric singular perturbation theory, J. Differ. Equations, 31 (1979), 53–98.
[21]	A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Math. Med. Biol., 21 (2004), 75–83.
[22]	M. Kot, Elements of Mathematical Ecology, Cambridge University Press, Cambridge, 2001.
[23]	O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365–382.
[24]	O. Diekmann and J. A. P. Heesterbeek, Mathematical epidemiology of infectious diseases, Wiley series in mathematical and computional biology, Wiley, Chichester, 2000.
[25]	V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43–61.
[26]	K. Dietz, Overall population patterns in the transmission cycle of infectious disease agents, in Population biology of infectious diseases (eds. R. M. Anderson and R. M. May), Springer, Berlin, Heidelberg, 1982, 87–102.
[27]	J. Mena-Lorca and H. W. Hethcote, Dynamic models of infectious diseases as regulators of population size, J. Math. Biol., 30 (1992), 693–716.
[28]	B. W. Kooi, G. A. K. van Voorn and K. P. Das, Stabilization and complex dynamics in a predatorprey model with predator suffering from an infectious disease, Ecol. Complex., 8 (2011), 113–122.
[29]	F. Irwin and H. N. Wright, Some properties of polynomial curves, Ann. Math. Second Series, 19 (1917), 152–158.
[30]	L. Edelstein-Keshet, Mathematical Models in Biology, SIAM, Philadelphia, 2005.
[31]	E. J. Doedel and B. Oldeman, AUTO 07p: Continuation and Bifurcation software for ordinary differential equations, Technical report, Concordia University, Montreal, Canada, 2009.
[32]	A.-M. Helt and E. Harris, S-phase-dependent enhancement of dengue virus 2 replication in mosquito cells, but not in human cells, J. Virology, 79 (2005), 13218–13230.
[33]	J. Hily, A. García, A. Moreno, M. Plaza, M.Wilkinson, A. Fereres, A. Fraile and F. García-Arenal, The relationship between host lifespan and pathogen reservoir potential: An analysis in the system Arabidopsis thaliana-Cucumber mosaic virus, PLoS Pathog., 10 (2014), e1004492.
[34]	N. M. Nguyen, D. Thi Hue Kien, T. V. Tuan, et al., Host and viral features of human dengue cases shape the population of infected and infectious Aedes aegypti mosquitoes, Proc. Natl. Acad. Sci. USA, 110 (2013), 9072–9077.
[35]	J. Yu, Modeling mosquito population suppression based on delay differential equations, SIAM J. Appl. Math., 78 (2018), 3168–3187.
[36]	M. R. W. de Valdez, D. Nimmo, J. Betz, et al., Genetic elimination of dengue vector mosquitoes, Proc. Natl. Acad. Sci. USA, 108 (2011), 4772–4775.

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)