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A novel approach to modelling the spatial spread of airborne diseases: an epidemic model with indirect transmission. (English) Zbl 1467.92186
Summary: We formulated and analyzed a class of coupled partial and ordinary differential equation (PDE-ODE) model to study the spread of airborne diseases. Our model describes human populations with patches and the movement of pathogens in the air with linear diffusion. The diffusing pathogens are coupled to the SIR dynamics of each population patch using an integro-differential equation. Susceptible individuals become infected at some rate whenever they are in contact with pathogens (indirect transmission), and the spread of infection in each patch depends on the density of pathogens around the patch. In the limit where the pathogens are diffusing fast, a matched asymptotic analysis is used to reduce the coupled PDE-ODE model into a nonlinear system of ODEs, which is then used to compute the basic reproduction number and final size relation for different scenarios. Numerical simulations of the reduced system of ODEs and the full PDE-ODE model are consistent, and they predict a decrease in the spread of infection as the diffusion rate of pathogens increases. Furthermore, we studied the effect of patch location on the spread of infections for the case of two population patches. Our model predicts higher infections when the patches are closer to each other.
##### MSC:
 92D30 Epidemiology 34C60 Qualitative investigation and simulation of ordinary differential equation models 35Q92 PDEs in connection with biology, chemistry and other natural sciences
##### Software:
ode113; Ode15s; FlexPDE; Matlab; ode23s; ode45; ode23
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##### References:
 [1] C. J. Noakes, C. B. Beggs, P. A. Sleigh, K. G. Kerr, Modelling the transmission of airborne infections in enclosed spaces, Epidemiol. Infect., 134 (2006), 1082-1091. [2] C. B. Beggs, The airborne transmission of infection in hospital [3] C. M. Issarow, N. Mulder, R. Wood, Modelling the risk of airborne infectious disease using exhaled air. J. Theor. Biol., 372 (2015), 100-106. · Zbl 1342.92242 [4] Z. Xu, D. Chen, An SIS epidemic model with diffusion, Appl. Math. Ser. B, 32 (2017), 127-146. · Zbl 1389.35064 [5] J. Ge, K. Kim, Z. Lin, H. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differ. Equ., 259 (2015), 5486-5509. · Zbl 1341.35171 [6] M. Liu, Y. Xiao, Modeling and analysis of epidemic diffusion with population migration, J. Appl.Math., 2013 (2003), 583648. · Zbl 1397.92651 [7] N. Ziyadi, S. Boulite, M. L. Hbid, S. Touzeau, Mathematical analysis of a PDE epidemiological model applied to scrapie transmission, Commun. Pure Appl. Anal., 7 (2008), 659. · Zbl 1144.35436 [8] J. Gou, M. J. Ward, An asymptotic analysis of a 2-D model of dynamically active compartments coupled by bulk diffusion, J. Nonlinear Sci., 26 (2016), 979-1029. · Zbl 1439.92024 [9] F. Brauer, A new epidemic model with indirect transmission, J. Biol. Dyn., 11 (2017), 285-293. · Zbl 1447.92400 [10] J. F. David, Epidemic models with heterogeneous mixing and indirect transmission, J. Biol. Dyn., 12 (2018), 375-399. · Zbl 1447.92412 [11] M. J. Ward, J. B. Keller, Strong localized perturbations of eigenvalue problems, SIAM J. Appl.Math., 53 (1993), 770-798. · Zbl 0778.35081 [12] PDE solutions Inc, FlexPDE 6, 2019. [13] F. Brauer, C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, 40, Springer, 2001. · Zbl 0967.92015 [14] O. Diekmann, J. A. P. Heesterbeek, J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations, J.Math. Biol., 28 (1990), 365-382. · Zbl 0726.92018 [15] P. V. den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. · Zbl 1015.92036 [16] D. Bichara, Y. Kang, C. Castillo-Chavez, R. Horan, C. Perrings, SIS and SIR epidemic models under virtual dispersal, Bull. Math. Biol., 77 (2015), 2004-2034. · Zbl 1339.92078 [17] F. Brauer, Epidemic models with heterogeneous mixing and treatment, Bull. Math. Biol., 70 (2008), 1869-1885. · Zbl 1147.92033 [18] J. Arino, F. Brauer, P. Van Den Driessche, J. Watmough, J. Wu, A final size relation for epidemic models, Math. Biosci. Eng., 4 (2007), 159-175. · Zbl 1123.92030 [19] F. Brauer, Age-of-infection and the final size relation, Math. Biosci. Eng., 5 (2008), 681-690. · Zbl 1166.92321 [20] F. Brauer, The final size of a serious epidemic, Bull. Math. Biol., 81 (2019), 869-877. · Zbl 1415.92172 [21] F. Brauer, A final size relation for epidemic models of vector-transmitted diseases, Infect. Dis.Model., 2 (2017), 12-20. [22] F. Brauer, C. Castillo-Chaavez, Mathematical models for communicable diseases, volume 84. SIAM, 2012. · Zbl 1353.92001 [23] F. Brauer, C. Castillo-Chavez, Z. Feng, Mathematical models in epidemiology, 2018. [24] L. F. Shampine, M. W. Reichelt, The Matlab ODE suite, SIAM J. Sci. Comput., 18 (1997), 1-22. · Zbl 0868.65040 [25] L. Zhang, Z.-C. Wang, Y. Zhang, Dynamics of a reaction-diffusion waterborne pathogen model with direct and indirect transmission, Comput. Math. Appl., 72 (2016), 202-215. · Zbl 1443.92189 [26] T. Kolokolnikov, M. S. Titcombe, M. J. Ward, Optimizing the fundamental Neumann eigenvalue for the Laplacian in a domain with small traps, Europ. J. Appl. Math., 16 (2005), 161-200. · Zbl 1090.35070 [27] S. Chinviriyasit, W. Chinviriyasit, Numerical modelling of an SIR epidemic model with diffusion, Appl. Math. Comput., 216 (2010), 395-409. · Zbl 1184.92028 [28] H. Huang, M. Wang, The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Cont. Dyn-B, 20 (2015), 2039-3050. [29] K. Ik Kim, Z. Lin, Asymptotic behavior of an SEI epidemic model with diffusion, Math. Comput.Model., 47 (2008), 1314-1322. · Zbl 1145.35400 [30] E. M. Lotfi, M. Maziane, K. Hattaf, N. Yousfi, Partial differential equations of an epidemic model with spatial diffusion, Int. J. Part. Differ. Eq., 2014 (2014), 186437. · Zbl 1300.92103 [31] F. A. Milner, R. Zhao, Analysis of an SIR model with directed spatial diffusion, Math. Popul.Stud., 15 (2008), 160-181. · Zbl 1151.92322
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