### Mathematical Biosciences and Engineering

2019, Issue 6: 6708-6727. doi: 10.3934/mbe.2019335
Research article Special Issues

# Basic reproduction number of SEIRS model on regular lattice

• Received: 01 March 2019 Accepted: 16 July 2019 Published: 25 July 2019
• In this paper we give a basic reproduction number $\mathscr{R}_0$ of an SEIRS model on regular lattice using next-generation matrix approach. Our result is straightforward but differs from the basic reproduction numbers for various fundamental epidemic models on the regular lattice which have been shown so far. Sometimes it is caused by the difference of derivation methods for $\mathscr{R}_0$ although their threshold of infectious rates for epidemic outbreak remain the same. Then we compare our $\mathscr{R}_0$ to the ones by these previous studies from several epidemic points of view.

Citation: Kazunori Sato. Basic reproduction number of SEIRS model on regular lattice[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 6708-6727. doi: 10.3934/mbe.2019335

### Related Papers:

• In this paper we give a basic reproduction number $\mathscr{R}_0$ of an SEIRS model on regular lattice using next-generation matrix approach. Our result is straightforward but differs from the basic reproduction numbers for various fundamental epidemic models on the regular lattice which have been shown so far. Sometimes it is caused by the difference of derivation methods for $\mathscr{R}_0$ although their threshold of infectious rates for epidemic outbreak remain the same. Then we compare our $\mathscr{R}_0$ to the ones by these previous studies from several epidemic points of view.

 [1] O. Diekmann, H. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princeton University Press, Princeton and Oxford, 2013. [2] 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. [3] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. [4] C. Castillo-Chavez, Z. Feng and W. Huang, On the computation of $\mathcal{R}_0$ and its role on global stability, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, Springer, New York (2002), 229–250. [5] M. Martcheva, An Introduction to Mathematical Epidemiology, Springer: New York, 2015, 16. [6] K. Saeki and A. Sasaki, The role of spatial heterogeneity in the evolution of local and global infections of viruses, PLoS Comput. Biol., 14 (2018), e1005952. [7] W. O. Kermack and A. G. Mc.Kendrick, A contributions to the mathematical theory of epidemics, Proc. R. Soc. A, 115 (1991), 700–721. [8] M. J. Keeling, The effects of local spatial structure on epidemiological invasions, P. Roy. Soc. Lond. B Bio., 266 (1999), 859–867. [9] H. Matsuda, N. Ogita, A. Sasaki, et al., Statistical mechanics of population: The lattice Lotoka-Volterra model, Prog. Theor. Phys., 88 (1992), 1035–1049. [10] C. T. Bauch, The spread of infectious diseases in spatially structured populations: An invasory pair approximation, Math. Biosci., 198 (2005), 217–237. [11] N. Ringa and C. T. Bauch, Dynamics and control of foot-and-mouth disease in endemic countries: A pair approximation model, J. Theor. Biol., 357 (2014), 150–159. [12] H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer, New York, 2017. [13] I. Z. Kiss, J. C. Miller and P. L. Simon, Mathematics of Epidemics on Networks, Cham: Springer (2017) 598. [14] P. Trapman, Reproduction numbers for epidemics on networks using pair approximation, Math. Biosci., 210 (2007), 464–489.
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