Dynamical behavior of a vector-host epidemic model with demographic structure. (English) Zbl 1165.34382
Summary: We propose a model that tracks the dynamics of many diseases spread by vectors, such as malaria, dengue, or West Nile virus (all spread by mosquitoes). Our model incorporates demographic structure with variable population size which is described by nonlinear birth rate and linear death rate. The stability of the system is analyzed for the existence of the disease-free and endemic equilibria points. We find the basic reproduction number \(R_{0}\) in terms of measurable epidemiological and demographic parameters is the threshold condition that determines the dynamics of disease infection: if \(R_{0}<1\) the disease fades out, and for \(R_{0}>1\) the disease remains endemic. The threshold condition provides important guidelines for accessing control of the vector diseases, and implies that it is an efficient way to halt the spread of vector epidemic by reducing the carrying capacity of the environment for the vector and the host. Moreover, sufficient conditions are also obtained for the global stability of the unique endemic equilibrium \(E^{*}\).
References:
| [1] | Arino, J.; Mccluskey, C. C.; van Den Driessche, P., Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. Math., 64, 260-276 (2003) · Zbl 1034.92025 |
| [2] | Wang, W. D.; Ma, Z., Global dynamics of an epidemic model with time delay, Nonlinear Anal. RWA, 3, 365-373 (2002) · Zbl 0998.92038 |
| [3] | Wang, W. D.; Zhao, X. Q., An epidemic model in a patchy environment, Math. Biosci., 190, 97-112 (2004) · Zbl 1048.92030 |
| [4] | Feng, Z. L.; Velasco-Hernández, J. X., Competitive exclusion in a vector-host model for the dengue fever, J. Math. Biol., 35, 523-544 (1997) · Zbl 0878.92025 |
| [5] | Cruz-Pachecoa, G.; Estevab, L.; Montano-Hirosec, J. A.; Vargasd, C., Modelling the dynamics of West Nile Virus, Bull. Math. Biol., 67, 1157-1172 (2005) · Zbl 1334.92397 |
| [6] | Bowman, C.; Gumel, A. B.; Wu, J.; van den Driessche, P.; Zhu, H., A mathematical model for assessing control strategies against West Nile virus, Bull. Math. Biol., 67, 1107-1133 (2005) · Zbl 1334.92392 |
| [7] | J.F. Jiang, Z.P. Qiu, J. Wu, H. Zhu, Threshold conditions for West Nile virus outbreaks, Bull. Math. Biol. (submitted for publication); J.F. Jiang, Z.P. Qiu, J. Wu, H. Zhu, Threshold conditions for West Nile virus outbreaks, Bull. Math. Biol. (submitted for publication) · Zbl 1163.92036 |
| [8] | J.F. Jiang, Z.P. Qiu, The complete classification for dynamics in a nine dimensional West Nile virus model, SIAM J Appl. Math. (submitted for publication); J.F. Jiang, Z.P. Qiu, The complete classification for dynamics in a nine dimensional West Nile virus model, SIAM J Appl. Math. (submitted for publication) · Zbl 1184.92030 |
| [9] | Tumwiine, J.; Mugisha, J. Y.T.; Luboobi, L. S., A mathematical model for the dynamics of malaria in a human host and mosquito vector with temporary immunity, Appl. Math. Comput., 189, 1953-1965 (2007) · Zbl 1117.92039 |
| [10] | Wei, Hui-Ming; Li, Xue-Zhi; Martcheva, Maia, An epidemic model of a vector-borne disease with direct transmission and time delay, J. Math. Anal. Appl. (2008) · Zbl 1146.34059 |
| [11] | Anderson, R. M.; May, R. M., Co-evolution of host and parasites, Parasitology, 85, 411-426 (1982) |
| [12] | Brauer, F., Models for the spread of universally fatal diseases, J. Math. Biol., 28, 451-462 (1990) · Zbl 0718.92021 |
| [13] | Bremermann, H. J.; Thieme, H. R., A competitive exclusion principle for pathogen virulence, J. Math. Biol., 27, 179-190 (1989) · Zbl 0715.92027 |
| [14] | Pugliese, A., Population models for diseases with no recovery, J. Math. Biol., 28, 65-82 (1990) · Zbl 0727.92023 |
| [15] | Gao, L. Q.; Hethcote, H. W., Disease transmission models with density-dependent demographics, J. Math. Biol., 30, 717-731 (1992) · Zbl 0774.92018 |
| [16] | Cooke, K.; van den Driessche, P.; Zou, X., Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol., 39, 332-352 (1999) · Zbl 0945.92016 |
| [17] | Hethcote, H. W., The mathematics of infectious diseases, SIAM Rev., 42, 599-653 (2000) · Zbl 0993.92033 |
| [18] | van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180, 29-48 (2002) · Zbl 1015.92036 |
| [19] | Jiang, J. F., On the global stability of cooperative systems, Bull. London Math. Soc., 6, 455-458 (1994) · Zbl 0820.34030 |
| [20] | Thieme, H. R., Persistence under relaxed point-dissipativity with an application to an epidemic model, SIAM J. Math. Anal., 24, 407-435 (1993) · Zbl 0774.34030 |
| [21] | Smith, H. L., Monotone dynamical systems: An introduction to theory of competitive and cooperative systems, (Math. Surveys Monogr., vol. 41 (1995), AMS: AMS Providence, RI) · Zbl 0821.34003 |
| [22] | Castillo-Chavez, C.; Thieme, H. R.; Arino, O.; Kimmel, M., Asymptotically autonomous epidemic models, Mathematical Population Dynamics: Analysis of Heterogeneity. Mathematical Population Dynamics: Analysis of Heterogeneity, Theory of Epidemics. Wuetz 1, 33-50 (1995) |
| [23] | Smith, R. A., Some applications of Hausdorff dimension inequalities for ordinary differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 104, 235-259 (1986) · Zbl 0622.34040 |
| [24] | Li, M. Y.; Muldowney, J., On Bendixson’s criterion, J. Differential Equations, 106, 27-39 (1994) · Zbl 0786.34033 |
| [25] | Li, M. Y.; Muldowney, J., A geometric approach to global-stability problems, SIAM J. Math. Anal., 27, 1070-1083 (1996) · Zbl 0873.34041 |
| [26] | Martin, R. H., Logarithmic norms and projections applied to linear differential systems, J. Math. Anal. Appl., 45, 432-454 (1974) · Zbl 0293.34018 |
| [27] | Muldowney, J. S., Dichotomies and asymptotic behavior for linear differential systems, Trans. Amer. Math. Soc., 283, 465-484 (1984) · Zbl 0559.34049 |
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.
