Threshold dynamics of a vector-borne disease model with spatial structure and vector-bias. (English) Zbl 1431.35214
Summary: In this paper, we formulate and investigate a vector-borne disease model with vector-bias in a bounded spatial domain. The main result is a threshold dynamics characterized in terms of the basic reproduction number \(\operatorname{Re}_0\) or equivalently in terms of the principal eigenvalue of the linearized system. Roughly speaking, if \(\operatorname{Re}_0 < 1\) then the semi-trivial equilibrium is globally asymptotically stable and the disease dies out; if \(\operatorname{Re}_0 > 1\) then the disease persists.
MSC:
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 92D30 | Epidemiology |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B35 | Stability in context of PDEs |
| 35P99 | Spectral theory and eigenvalue problems for partial differential equations |
Keywords:
spatial heterogeneity; global asymptotic stability; basic reproduction number; threshold dynamicsReferences:
| [1] | Cai, Y.; Ding, Z.; Yang, B.; Peng, Z.; Wang, W., Transmission dynamics of zika virus with spatial structure-a case study in rio de janeiro, Brazil, Physica A, 514, 729-740 (2019) |
| [2] | Cai, Y.; Wang, K.; Wang, W., Global transmission dynamics of a zika virus model, Appl. Math. Lett., 92, 190-195 (2019) · Zbl 1414.35232 |
| [3] | Gao, D.; Lou, Y.; He, D., Prevention and control of zika as a mosquito-borne and sexually transmitted disease: a mathematical modeling analysis, Sci. Rep., 6, 28070 (2016) |
| [4] | Magal, P.; Webb, G.; Wu, Y., On a vector-host epidemic model with spatial structure, Nonlinearity, 31, 5589-5614 (2018) · Zbl 1408.35199 |
| [5] | Tang, B.; Xiao, Y.; Wu, J., Implication of vaccination against dengue for zika outbreak, Sci. Rep., 6, 35623 (2016) |
| [6] | Wang, L.; Zhao, H., Dynamics analysis of a zika-dengue co-infection model with dengue vaccine and antibody-dependent enhancement, Physica A, 522, 248-273 (2019) · Zbl 07561951 |
| [7] | Wang, X.; Zhao, X.-Q., A periodic vector-bias malaria model with incubation period, SIAM J. Appl. Math., 77, 181-201 (2017) · Zbl 1401.35188 |
| [8] | Fitzgibbon, W. E.; Morgan, J. J.; Webb, G. F., An outbreak vector-host epidemic model with spatial structure: the 2015-2016 zika outbreak in rio de janeiro, Theor. Biol. Med. Model., 14, 7 (2017) |
| [9] | Villela, D. A.M.; Bastos, L. S.; de Carvalho, L. M., Zika in rio de janeiro: assessment of basic reproduction number and comparison with dengue outbreaks, Epidemiol. Inf., 145, 1649-1657 (2016) |
| [10] | Lacroix, R.; Mukabana, W. R.; Gouagna, L. C.; Koella, J. C., Malaria infection increases attractiveness of humans to mosquitoes, PLoS Biol., 3, Article e298 pp. (2005) |
| [11] | Daniel, T. L.; Kingsolver, J. G., The feeding strategy and the mechanics of blood sucking in insects, J. Theoret. Biol., 105, 661-677 (1983) |
| [12] | Bai, Z.; Peng, R.; Zhao, X.-Q., A reaction-diffusion malaria model with seasonality and incubation period, J. Math. Biol., 77, 201-228 (2018) · Zbl 1390.35145 |
| [13] | Cantrell, R. S.; Cosner, C., Spatial Ecology Via Reaction-Diffusion Equations (2003), John Wiley & Sons, Ltd: John Wiley & Sons, Ltd Chichester · Zbl 1059.92051 |
| [14] | Thieme, H. R., Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70, 188-211 (2009) · Zbl 1191.47089 |
| [15] | Wang, W.; Zhao, X.-Q., Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11, 1652-1673 (2012) · Zbl 1259.35120 |
| [16] | Smith, H. L.; Zhao, X.-Q., Robust persistence for semidynamical systems, Nonlinear Anal., 47, 6169-6179 (2001) · Zbl 1042.37504 |
| [17] | Magal, P.; Zhao, X.-Q., Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37, 251-275 (2005) · Zbl 1128.37016 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
