# American Institute of Mathematical Sciences

2010, 7(1): 195-211. doi: 10.3934/mbe.2010.7.195

## Models for the spread and persistence of hantavirus infection in rodents with direct and indirect transmission

 1 Louisiana State University in Shreveport, Department of Mathematics, Shreveport, LA 71115, United States 2 Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, United States 3 Université Victor Segalen Bordeaux 2, IMB UMR CNRS 5251 & INRIA Bordeaux Sud Ouest projet Anubis, case 36, UFR Sciences et Modelisation, 3 ter place de la Victoire, 33076 Bordeaux Cedex, France

Received  March 2009 Revised  July 2009 Published  January 2010

Hantavirus, a zoonotic disease carried by wild rodents, is spread among rodents via direct contact and indirectly via infected rodent excreta in the soil. Spillover to humans is primarily via the indirect route through inhalation of aerosolized viral particles. Rodent-hantavirus models that include direct and indirect transmission and periodically varying demographic and epidemiological parameters are studied in this investigation. Two models are analyzed, a nonautonomous system of differential equations with time-periodic coefficients and an autonomous system, where the coefficients are taken to be the time-average. In the nonautonomous system, births, deaths, transmission rates and viral decay rates are assumed to be periodic. For both models, the basic reproduction numbers are calculated. The models are applied to two rodent populations, reservoirs for a New World and for an Old World hantavirus. The numerical examples show that periodically varying demographic and epidemiological parameters may substantially increase the basic reproduction number. Also, large variations in the viral decay rate in the environment coupled with an outbreak in rodent populations may lead to spillover infection in humans.
Citation: Curtis L. Wesley, Linda J. S. Allen, Michel Langlais. Models for the spread and persistence of hantavirus infection in rodents with direct and indirect transmission. Mathematical Biosciences & Engineering, 2010, 7 (1) : 195-211. doi: 10.3934/mbe.2010.7.195
 [1] Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37 [2] Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166 [3] Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595 [4] Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455 [5] Zhengxin Zhou. On the Poincaré mapping and periodic solutions of nonautonomous differential systems. Communications on Pure & Applied Analysis, 2007, 6 (2) : 541-547. doi: 10.3934/cpaa.2007.6.541 [6] Tom Burr, Gerardo Chowell. The reproduction number $R_t$ in structured and nonstructured populations. Mathematical Biosciences & Engineering, 2009, 6 (2) : 239-259. doi: 10.3934/mbe.2009.6.239 [7] Paolo Gidoni, Alessandro Margheri. Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 585-606. doi: 10.3934/dcds.2019024 [8] Ling Xue, Caterina Scoglio. Network-level reproduction number and extinction threshold for vector-borne diseases. Mathematical Biosciences & Engineering, 2015, 12 (3) : 565-584. doi: 10.3934/mbe.2015.12.565 [9] Gerardo Chowell, Catherine E. Ammon, Nicolas W. Hengartner, James M. Hyman. Estimating the reproduction number from the initial phase of the Spanish flu pandemic waves in Geneva, Switzerland. Mathematical Biosciences & Engineering, 2007, 4 (3) : 457-470. doi: 10.3934/mbe.2007.4.457 [10] Rui Xu, M.A.J. Chaplain, F.A. Davidson. Periodic solutions of a discrete nonautonomous Lotka-Volterra predator-prey model with time delays. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 823-831. doi: 10.3934/dcdsb.2004.4.823 [11] Meng Fan, Qian Wang. Periodic solutions of a class of nonautonomous discrete time semi-ratio-dependent predator-prey systems. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 563-574. doi: 10.3934/dcdsb.2004.4.563 [12] Zhijun Liu, Weidong Wang. Persistence and periodic solutions of a nonautonomous predator-prey diffusion with Holling III functional response and continuous delay. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 653-662. doi: 10.3934/dcdsb.2004.4.653 [13] Christian Pötzsche. Nonautonomous continuation of bounded solutions. Communications on Pure & Applied Analysis, 2011, 10 (3) : 937-961. doi: 10.3934/cpaa.2011.10.937 [14] Tommaso Leonori, Ireneo Peral, Ana Primo, Fernando Soria. Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6031-6068. doi: 10.3934/dcds.2015.35.6031 [15] Arno Berger. Counting uniformly attracting solutions of nonautonomous differential equations. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 15-25. doi: 10.3934/dcdss.2008.1.15 [16] João Ferreira Alves, Michal Málek. Zeta functions and topological entropy of periodic nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 465-482. doi: 10.3934/dcds.2013.33.465 [17] Russell Johnson, Francesca Mantellini. A nonautonomous transcritical bifurcation problem with an application to quasi-periodic bubbles. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 209-224. doi: 10.3934/dcds.2003.9.209 [18] D. Motreanu, V. V. Motreanu, Nikolaos S. Papageorgiou. Nonautonomous resonant periodic systems with indefinite linear part and a nonsmooth potential. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1401-1414. doi: 10.3934/cpaa.2011.10.1401 [19] Luís Silva. Periodic attractors of nonautonomous flat-topped tent systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1867-1874. doi: 10.3934/dcdsb.2018243 [20] Jian-Jun Xu, Junichiro Shimizu. Asymptotic theory for disc-like crystal growth (I) --- Basic state solutions. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1091-1116. doi: 10.3934/dcdsb.2004.4.1091

2018 Impact Factor: 1.313