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June  2017, 14(3): 735-754. doi: 10.3934/mbe.2017041

Effect of the epidemiological heterogeneity on the outbreak outcomes

Alina Macacu and  Dominique J. Bicout ,

Biomathematics and Epidemiology, EPSP -TIMC, UMR 5525 CNRS, Grenoble Alpes University, VetAgro Sup Lyon, 1 avenue Bourgelat -69280 Marcy l'Etoile, France

* Corresponding author: Dominique J. Bicout

Received  May 08, 2016 Accepted  October 15, 2016 Published  December 2016

Multi-host pathogens infect and are transmitted by different kinds of hosts and, therefore, the host heterogeneity may have a great impact on the outbreak outcome of the system. This paper deals with the following problem: consider the system of interacting and mixed populations of hosts epidemiologically different, what would be the outbreak outcome for each host population composing the system as a result of mixing in comparison to the situation with zero mixing? To address this issue we have characterized the epidemic response function for a single-host population and defined a heterogeneity index measuring how host systems are epidemiologically different in terms of generation time, basic reproduction number $R_0$ and, therefore, epidemic response function. Based on the individual epidemiological characteristics of populations, with heterogeneities and mixing affinities, the response of subpopulations in a multi-host system is compared to that of a single-host system. The case of a two-host system, in which the infection transmission depends solely on the infection susceptibility of the receiver, is analyzed in detail. Three types of responses are observed: dilution, amplification or no effect, corresponding to lower, higher or equal attack rates, respectively, for a host population in an interacting multi-host system compared to the zero-mixing situation. We find that no effect is generally observed for zero heterogeneity. A dilution effect is always observed for all the host populations when their individual $R_{0,i} <1$. Whereas, when at least one of the individual $R_{0,i}>1$, then the hosts "$i$" with $R_{0,i}>R_{0,j}$ undergo a dilution effect while the hosts "$j$" undergo an amplification effect.

Keywords: Host heterogeneity, avian influenza, compartmental model, wildfowl, multi-host pathogens.
Mathematics Subject Classification: Primary: 92D30; Secondary: 92D25, 37N25.
Citation: Alina Macacu, Dominique J. Bicout. Effect of the epidemiological heterogeneity on the outbreak outcomes. Mathematical Biosciences & Engineering, 2017, 14 (3) : 735-754. doi: 10.3934/mbe.2017041
References:
[1]

F. R. Adler, The effects of averaging on the basic reproduction ratio, Mathematical Biosciences, 111 (1992), 89-98.  doi: 10.1016/0025-5564(92)90080-G.  Google Scholar

[2]

R. M. Anderson and R. M. May, Infectious Diseases of Humans/ Dynamics and Control, Oxford Science Publications, Oxford, 1991. Google Scholar

[3]

D. J. Bicout, Modélisation des Maladies Vectorielles, Habilitation á Diriger des Recherches -Université Joseph Fourier -Grenoble I, 2006. Google Scholar

[4]

J. D. Brown, D. E. Stallknecht and D. E. Swayne, Experimental infection of swans and geese with highly pathogenic avian influenza virus (H5N1) of asian lineage, Emerging Infectious Diseases, 14 (2008), 136-142.   Google Scholar

[5]

J. D. Brown, D. E. Stallknecht, J. R. Beck, D. L. Suarez and D. E. Swayne, Susceptibility of North american ducks and gulls to (H5N1) highly pathogenic avian influenza viruses, Emerging Infectious Diseases, 12 (2006), 1663-1670.   Google Scholar

[6]

H. Chen, Y. Li, Z. Li, J. Shi, K. Shinya, G. Deng, Q. Qi, G. Tian, S. Fan, H. Zhao, Y. Sun and Y. Kawaoka, Properties and Dissemination of H5N1 Viruses Isolated during an Influenza Outbreak in Migratory Waterfowl in Western China, Journal of Virology, 80 (2006), 5976-5983.   Google Scholar

[7]

H. Chen, G. J. D. Smith, S. Y. Zhang, K. Qin, J. Wang, K. S. Li, R. G. Webster, J. S. M. Peiris and Y. Guan, H5N1 virus outbreak in migratory waterfowl, Nature, 436 (2005), 191-192.   Google Scholar

[8]

M. de Jong, O. Diekmann and H. Heesterbeek, How does transmission of infection depend on population size, In Epidemic models: their structure and relation to data (eds. D. Mollison) Cambridge: Press Syndicate of the University of Cambridge, (1995), 84–94. Google Scholar

[9]

M. C. M. de Jong, O. Diekmann and J. A. P. Heesterbeek, The computation of R0 for discrete-time epidemic models with dynamic heterogeneity, Mathematical Biosciences, 119 (1994), 97-114.   Google Scholar

[10]

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, Journal of Mathematical Biology, 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar

[11]

O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models, Journal of the Royal Society Interface, 7 (2010), 873-885.  doi: 10.1098/rsif.2009.0386.  Google Scholar

[12]

A. P. Dobson, Population dynamics of pathogens with multiple host species, Am. Nat., 164 (2004), S64-S78.  doi: 10.1086/424681.  Google Scholar

[13]

D. Doctrinal, S. Ruette, J. Hars, M. Artois and D. J. Bicout, Spatial and temporal analysis of the highly pathogenic avian influenza (H5N1) outbreak in the Dombes Area, France in 2006, Wildfowl, 2 (2009), 202-214.   Google Scholar

[14]

J. Dushoff and S. Levin, The effects of population heterogeneity on disease invasion, Mathematical Biosciences, 128 (1995), 25-40.  doi: 10.1016/0025-5564(94)00065-8.  Google Scholar

[15]

P. L. Flint, Applying the scientific method when assessing the influence of migratory birds on the dispersal of H5N1, Virology Journal, 4 (2007), 132 (1-3).   Google Scholar

[16]

L. Gall-Reculé, F. X. Briand, A. Schmitz, O. Guionie, P. Massin and V. Jestin, Double introduction of highly pathogenic H5N1 avian influenza virus into France in early 2006, Avian Pathology, 37 (2008), 15-23.   Google Scholar

[17]

M. Gauthier-Clerc, C. Lebarbenchon and F. Thomas, Recent expansion of highly pathogenic avian influenza H5N1: a critical review, Ibis, 149 (2007), 202-214.   Google Scholar

[18]

V. Guberti and S. H. Newman, Guidelines on Wild Bird Surveillance for Highly Pathogenic Avian Influenza H5N1 Virus, Journal of Wildlife Diseases, 43 (2007), S29-S34.   Google Scholar

[19]

J. Hars, S. Ruette, M. Benmergui, C. Fouque, J. Y. Fournier, A. Legouge, M. Cherbonnel, D. Baroux, C. Dupuy and V. Jestin, The epidemiology of the highly pathogenic H5N1 avian influenza in Mute Swan (Cygnus olor) and other Anatidae in the Dombes region (France), 2006, J Wildlife Dis, 44 (2008), 811-823.   Google Scholar

[20]

J. Hars, S. Ruette, M. Benmergui, C. Fouque, J. Y. Fournier, A. Legouge, M. Cherbonnel, D. Baroux, C. Dupuy and V. Jestin, Rôle Epidémiologique du Cygne Tuberculé et des Autres Anatidés Dans L'épisode D'influenza Aviaire H5N1 HP Dans la Dombes en 2006, ONCFS Rapport Scientifique, 2006. Google Scholar

[21]

J. A. P. Heesterbeek, Abrief history of R0 and a recipe for its calculation, Acta Biotheoretica, 50 (2002), 189-204.   Google Scholar

[22]

D. Kalthoff, A. Breithaupt, J. P. Teifke, A. Globig, T. Harder, T. C. Mettenleiter and M. Beer, Highly pathogenic avian influenza virus (H5N1) in experimentally infected adult mute swans, Emerging Infectious Diseases, 14 (2008), 1267-1270.   Google Scholar

[23]

J. Keawcharoen, D. van Riel, G. van Amerongen, T. Bestebroer, W. E. Beyer, R. van Lavieren, A. D. M. E. Osterhaus, R. A. M. Fouchier and T. Kuiken, Wild ducks as long-distance vectors of highly pathogenic avian influenza virus (H5N1), Emerging Infectious Diseases, 14 (2008), 600-607.   Google Scholar

[24]

F. Keesing, R. D. Holt and R. S. Ostfeld, Effects of species diversity on disease risk, Ecology letters, 9 (2006), 485-498.  doi: 10.1111/j.1461-0248.2006.00885.x.  Google Scholar

[25]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. Lond. A, 115 (1927), 700-721.   Google Scholar

[26]

H. Kida, R. Yanagawa and Y. Matsuoka, Duck influenza lacking evidence of disease signs and immune response, Infect. Immun, 30 (1980), 547-553.   Google Scholar

[27]

A. M. Kilpatrick, A. A. Chmura, D.W. Gibbons, R. C. Fleischer, P. P. Marra and P. Daszak, Predicting the global spread of H5N1 avian influenza, Proc Natl Acad Sci USA, 103 (2006), 19368-19373.   Google Scholar

[28]

J. Liu, H. Xiao, F. Lei, Q. Zhu, K. Qin, X. -w Zhang, X. -l. Zhang, D. Zhao, G. Wang, Y. Feng, J. Ma, W. Liu, J. Wang and G. F. Gao, Highly pathogenic H5N1 influenza virus infection in migratory birds, Science, 309 (2005), 1206. Google Scholar

[29]

H. Nishiura, B. Hoye, M. Klaassen, S. Bauer and H. Heesterbeek, How to find natural reservoir hosts from endemic prevalence in a multi-host population: A case study of influenza in waterfowl, Epidemics, 1 (2009), 118-128.  doi: 10.1016/j.epidem.2009.04.002.  Google Scholar

[30]

B. Olsen, V. J. Munster, A. Wallensten, J. Waldenström, A. D. M. E. Osterhaus and R. A. M. Fouchier, Global Patterns of Influenza A Virus in Wild Birds, Science, 312 (2006), 384-388.  doi: 10.1126/science.1122438.  Google Scholar

[31]

M. René and D. J. Bicout, Influenza aviaire: Modélisation du risque d'infection des oiseaux á partir d'étangs contaminés, Epidémiologie et santé animale, 51 (2007), 95-109.   Google Scholar

[32]

A. Satelli, S. Tarantola and K. P.-S. Chan, Quantitative model-independent method for global sensitivity analysis of model output, Technometrics, 41 (1999), 39-56.   Google Scholar

[33]

M. E. J. Woolhouse, L. H. Taylor and D. T. Haydon, Population biology of multi-host pathogens, Science, 292 (2001), 1109-1112.   Google Scholar

[34]

G. Zhang, D. Shoham, S. Davydof, J. D. Castello, S. O. Rogers and D. Gilichinsky, Evidence of influenza A virus RNA in Siberian lake ice, Journal of Virology, 80 (2006), 12229-12235.   Google Scholar

show all references

References:
[1]

F. R. Adler, The effects of averaging on the basic reproduction ratio, Mathematical Biosciences, 111 (1992), 89-98.  doi: 10.1016/0025-5564(92)90080-G.  Google Scholar

[2]

R. M. Anderson and R. M. May, Infectious Diseases of Humans/ Dynamics and Control, Oxford Science Publications, Oxford, 1991. Google Scholar

[3]

D. J. Bicout, Modélisation des Maladies Vectorielles, Habilitation á Diriger des Recherches -Université Joseph Fourier -Grenoble I, 2006. Google Scholar

[4]

J. D. Brown, D. E. Stallknecht and D. E. Swayne, Experimental infection of swans and geese with highly pathogenic avian influenza virus (H5N1) of asian lineage, Emerging Infectious Diseases, 14 (2008), 136-142.   Google Scholar

[5]

J. D. Brown, D. E. Stallknecht, J. R. Beck, D. L. Suarez and D. E. Swayne, Susceptibility of North american ducks and gulls to (H5N1) highly pathogenic avian influenza viruses, Emerging Infectious Diseases, 12 (2006), 1663-1670.   Google Scholar

[6]

H. Chen, Y. Li, Z. Li, J. Shi, K. Shinya, G. Deng, Q. Qi, G. Tian, S. Fan, H. Zhao, Y. Sun and Y. Kawaoka, Properties and Dissemination of H5N1 Viruses Isolated during an Influenza Outbreak in Migratory Waterfowl in Western China, Journal of Virology, 80 (2006), 5976-5983.   Google Scholar

[7]

H. Chen, G. J. D. Smith, S. Y. Zhang, K. Qin, J. Wang, K. S. Li, R. G. Webster, J. S. M. Peiris and Y. Guan, H5N1 virus outbreak in migratory waterfowl, Nature, 436 (2005), 191-192.   Google Scholar

[8]

M. de Jong, O. Diekmann and H. Heesterbeek, How does transmission of infection depend on population size, In Epidemic models: their structure and relation to data (eds. D. Mollison) Cambridge: Press Syndicate of the University of Cambridge, (1995), 84–94. Google Scholar

[9]

M. C. M. de Jong, O. Diekmann and J. A. P. Heesterbeek, The computation of R0 for discrete-time epidemic models with dynamic heterogeneity, Mathematical Biosciences, 119 (1994), 97-114.   Google Scholar

[10]

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, Journal of Mathematical Biology, 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar

[11]

O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models, Journal of the Royal Society Interface, 7 (2010), 873-885.  doi: 10.1098/rsif.2009.0386.  Google Scholar

[12]

A. P. Dobson, Population dynamics of pathogens with multiple host species, Am. Nat., 164 (2004), S64-S78.  doi: 10.1086/424681.  Google Scholar

[13]

D. Doctrinal, S. Ruette, J. Hars, M. Artois and D. J. Bicout, Spatial and temporal analysis of the highly pathogenic avian influenza (H5N1) outbreak in the Dombes Area, France in 2006, Wildfowl, 2 (2009), 202-214.   Google Scholar

[14]

J. Dushoff and S. Levin, The effects of population heterogeneity on disease invasion, Mathematical Biosciences, 128 (1995), 25-40.  doi: 10.1016/0025-5564(94)00065-8.  Google Scholar

[15]

P. L. Flint, Applying the scientific method when assessing the influence of migratory birds on the dispersal of H5N1, Virology Journal, 4 (2007), 132 (1-3).   Google Scholar

[16]

L. Gall-Reculé, F. X. Briand, A. Schmitz, O. Guionie, P. Massin and V. Jestin, Double introduction of highly pathogenic H5N1 avian influenza virus into France in early 2006, Avian Pathology, 37 (2008), 15-23.   Google Scholar

[17]

M. Gauthier-Clerc, C. Lebarbenchon and F. Thomas, Recent expansion of highly pathogenic avian influenza H5N1: a critical review, Ibis, 149 (2007), 202-214.   Google Scholar

[18]

V. Guberti and S. H. Newman, Guidelines on Wild Bird Surveillance for Highly Pathogenic Avian Influenza H5N1 Virus, Journal of Wildlife Diseases, 43 (2007), S29-S34.   Google Scholar

[19]

J. Hars, S. Ruette, M. Benmergui, C. Fouque, J. Y. Fournier, A. Legouge, M. Cherbonnel, D. Baroux, C. Dupuy and V. Jestin, The epidemiology of the highly pathogenic H5N1 avian influenza in Mute Swan (Cygnus olor) and other Anatidae in the Dombes region (France), 2006, J Wildlife Dis, 44 (2008), 811-823.   Google Scholar

[20]

J. Hars, S. Ruette, M. Benmergui, C. Fouque, J. Y. Fournier, A. Legouge, M. Cherbonnel, D. Baroux, C. Dupuy and V. Jestin, Rôle Epidémiologique du Cygne Tuberculé et des Autres Anatidés Dans L'épisode D'influenza Aviaire H5N1 HP Dans la Dombes en 2006, ONCFS Rapport Scientifique, 2006. Google Scholar

[21]

J. A. P. Heesterbeek, Abrief history of R0 and a recipe for its calculation, Acta Biotheoretica, 50 (2002), 189-204.   Google Scholar

[22]

D. Kalthoff, A. Breithaupt, J. P. Teifke, A. Globig, T. Harder, T. C. Mettenleiter and M. Beer, Highly pathogenic avian influenza virus (H5N1) in experimentally infected adult mute swans, Emerging Infectious Diseases, 14 (2008), 1267-1270.   Google Scholar

[23]

J. Keawcharoen, D. van Riel, G. van Amerongen, T. Bestebroer, W. E. Beyer, R. van Lavieren, A. D. M. E. Osterhaus, R. A. M. Fouchier and T. Kuiken, Wild ducks as long-distance vectors of highly pathogenic avian influenza virus (H5N1), Emerging Infectious Diseases, 14 (2008), 600-607.   Google Scholar

[24]

F. Keesing, R. D. Holt and R. S. Ostfeld, Effects of species diversity on disease risk, Ecology letters, 9 (2006), 485-498.  doi: 10.1111/j.1461-0248.2006.00885.x.  Google Scholar

[25]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. Lond. A, 115 (1927), 700-721.   Google Scholar

[26]

H. Kida, R. Yanagawa and Y. Matsuoka, Duck influenza lacking evidence of disease signs and immune response, Infect. Immun, 30 (1980), 547-553.   Google Scholar

[27]

A. M. Kilpatrick, A. A. Chmura, D.W. Gibbons, R. C. Fleischer, P. P. Marra and P. Daszak, Predicting the global spread of H5N1 avian influenza, Proc Natl Acad Sci USA, 103 (2006), 19368-19373.   Google Scholar

[28]

J. Liu, H. Xiao, F. Lei, Q. Zhu, K. Qin, X. -w Zhang, X. -l. Zhang, D. Zhao, G. Wang, Y. Feng, J. Ma, W. Liu, J. Wang and G. F. Gao, Highly pathogenic H5N1 influenza virus infection in migratory birds, Science, 309 (2005), 1206. Google Scholar

[29]

H. Nishiura, B. Hoye, M. Klaassen, S. Bauer and H. Heesterbeek, How to find natural reservoir hosts from endemic prevalence in a multi-host population: A case study of influenza in waterfowl, Epidemics, 1 (2009), 118-128.  doi: 10.1016/j.epidem.2009.04.002.  Google Scholar

[30]

B. Olsen, V. J. Munster, A. Wallensten, J. Waldenström, A. D. M. E. Osterhaus and R. A. M. Fouchier, Global Patterns of Influenza A Virus in Wild Birds, Science, 312 (2006), 384-388.  doi: 10.1126/science.1122438.  Google Scholar

[31]

M. René and D. J. Bicout, Influenza aviaire: Modélisation du risque d'infection des oiseaux á partir d'étangs contaminés, Epidémiologie et santé animale, 51 (2007), 95-109.   Google Scholar

[32]

A. Satelli, S. Tarantola and K. P.-S. Chan, Quantitative model-independent method for global sensitivity analysis of model output, Technometrics, 41 (1999), 39-56.   Google Scholar

[33]

M. E. J. Woolhouse, L. H. Taylor and D. T. Haydon, Population biology of multi-host pathogens, Science, 292 (2001), 1109-1112.   Google Scholar

[34]

G. Zhang, D. Shoham, S. Davydof, J. D. Castello, S. O. Rogers and D. Gilichinsky, Evidence of influenza A virus RNA in Siberian lake ice, Journal of Virology, 80 (2006), 12229-12235.   Google Scholar

Figure 1.  Cumulative distribution function (cdf) for the attack rate (left panel) and the reduced extinction time (right panel), for x = 0 and R0 = 0.5 (dashed line), 1 (dotted line), and 2 (solid line).
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Figure 2.  Distributions of attack rate a (left) and of reduced extinction time τ (right) for R0 = 2 and x = 0
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Figure 3.  Mean attack rate A (dashed lines) and mean reduced extinction time T (solid lines) as a function of R0 for x = 0 (left panel) and x = 0.9 (right panel)
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Figure 4.  Probability ω(Ith) of an outbreak occurrence as a function of R0 for x = 0 and different values of the threshold Ith. Star markers represent the mean attack rate A
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Figure 5.  Reduced one-dimensional heterogeneity, Hh/y, as a function of z, for different values of y. The values y = 0.25, 1 and 2.3 correspond to f1 = 1 − f2 = 0.8, 0.5 and 0.3, respectively, filled circles to z = h2/h1 = 0.53 [with (h1, h2) = (1.5, 0.8)], and filled diamonds to z = 0.1, 0.53, 0.625, and 1.
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Figure 6.  Sensitivity analysis on $\mathcal{R}_0$ using extended FAST method. For each parameter, the light area represents the main effect and the gray area the interaction effect between parameters
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Figure 7.  Contour diagrams in the space {R0, 1, R0, 2} showing level curves of ${\mathcal R}_0$ = 0.5, 1, …, 3.5 (quoted numbers) for different Φ and f1 and for a total population size N = 5000
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Figure 8.  Cumulative distribution function (cdf) for the attack rate for host 1 (dashed line), host 2 (dotted line) and the total population (solid line) in a two-host system. The initial conditions are I1(0) = 1 and I2(0) = 0, with parameters x1 = 0, x2 = 0.9, f1 = 1 − f2 = 0.8, R0, 1 = 1.5 and R0, 2 = 0.8, corresponding to HR = 0.043 [filled circle (y, z) = (0.25, 0.53) in Fig. 5], and Φ = 0.5 for a global ${\mathcal R}_0$ = 1.4
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Figure 9.  Effects of the heterogeneity and mixing on the outbreak outcome

The reduced equivalent reproduction number, η1, for host 1, and global reproductive number ${\mathcal R}_0$ (from Eq.(10)) as a function of assortative mixing Φ for various values of heterogeneity, HR, and R0, 2. The initial conditions are I1(0) = 1 and I2(0) = 0, with parameters x1 = x2 = 0, and f1 = 1−f2 = 0.5. Values of HR correspond to filled diamonds along the line y = 1 in Fig. 5 with HR = 0 (R0, 1 = R0, 2 = 0.8 for z = 1), 0.05 (R0, 1 = 0.8; R0, 2 = 0.5 for z = 0.625), 0.092 (R0, 1 = 0.8; R0, 2 = 1.5 for z = 1.88) and 0.67 (R0, 1 = 2; R0, 2 = 0.2 for z = 0.1).

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Figure 10.  Impact of the initial conditions on effects of the heterogeneity and mixing on the outbreak outcome

Reduced equivalent reproduction numbers ηi (i = 1, 2) and global reproductive number ${\mathcal R}_0$ (from Eq.(10)) as a function of the assortative mixing Φ for the two hosts for different heterogeneity. The initial conditions are I1(0) = 1 and I2(0) = 0 (filed symbols for η1 and open symbols for η2), and I1(0) = 0 and I2(0) = 1 (open symbols for η1 and filled symbols for η2) with the parameters x1 = 0, x2 = 0.9, R0, 1 = 1.5 and R0, 2 = 0.8. Values of HR correspond to filled circles at z = R0, 2/R0, 1 = 0.53 in Fig. 5 with HR = 0.043 (f1 = 1− f2 = 0.8 for y = 0.25), 0.093 (f1 = 1− f2 = 0.5 for y = 1) and 0.101 (f1 = 1 − f2 = 0.3 for y = 2.3)

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Figure 11.  Left panel: Probability of minor epidemics as a function of R0. Triangle markers represent data from stochastic simulations and solid line through the data Eq.(18) for x = 0. Right panel: Mean attack rate as a function of R0 for x = 0, comparison of simulations (solid line) and the formula in Eq.(16) (dashed line)
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Table 1.  Synthetic summary of stochastic simulations for constructing the phase diagram of the outbreak response at individual host level as a function of the combined effects of mixing (Φ ≠ 0) and heterogeneity. Dilution, no effect and amplification responses correspond to ηi < 1, = 1 and > 1, respectively, where ηi in Eq. (12) is the ratio of the equivalent to the bare basic reproduction number. These observations are symmetric with respect to inversion of host 1 and 2, and for each host i the effect on the outbreak response increases when fi (fj) decreases (increases), and conversely
heterogeneity outbreak response
host 1 host 2
$\begin{array}{l} \bullet \;{H_R} > 0\;\;\;*{R_{0,1}}＆ {R_{0,2}} < 1\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;*{R_{0,1}} < {R_{0,2}}\;{\rm{with}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{at}}\;{\rm{least}}\;{\rm{one}}\;{R_{0,i}} > 1 \end{array} $ dilution dilution
amplification dilution
$\begin{array}{l} \bullet \;{H_R} = 0\;\;\;*{x_1} = {x_2}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;*{x_1} < {x_2}\;{\rm{and}}\;\left\{ \begin{array}{l} {R_{0,i}} < 1\\ {R_{0,i}} > 1 \end{array} \right. \end{array}$ no effect no effect
dilution dilution
no effect amplification
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heterogeneity outbreak response
host 1 host 2
$\begin{array}{l} \bullet \;{H_R} > 0\;\;\;*{R_{0,1}}＆ {R_{0,2}} < 1\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;*{R_{0,1}} < {R_{0,2}}\;{\rm{with}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{at}}\;{\rm{least}}\;{\rm{one}}\;{R_{0,i}} > 1 \end{array} $ dilution dilution
amplification dilution
$\begin{array}{l} \bullet \;{H_R} = 0\;\;\;*{x_1} = {x_2}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;*{x_1} < {x_2}\;{\rm{and}}\;\left\{ \begin{array}{l} {R_{0,i}} < 1\\ {R_{0,i}} > 1 \end{array} \right. \end{array}$ no effect no effect
dilution dilution
no effect amplification
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