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2015, 12(3): 565-584. doi: 10.3934/mbe.2015.12.565

## Network-level reproduction number and extinction threshold for vector-borne diseases

 1 Kansas State Epicenter, Department of Electrical & Computer Engineering, Kansas State University, Manhattan, KS 66506, United States, United States

Received  June 2014 Revised  January 2015 Published  January 2015

The basic reproduction number of deterministic models is an essential quantity to predict whether an epidemic will spread or not. Thresholds for disease extinction contribute crucial knowledge of disease control, elimination, and mitigation of infectious diseases. Relationships between basic reproduction numbers of two deterministic network-based ordinary differential equation vector-host models, and extinction thresholds of corresponding stochastic continuous-time Markov chain models are derived under some assumptions. Numerical simulation results for malaria and Rift Valley fever transmission on heterogeneous networks are in agreement with analytical results without any assumptions, reinforcing that the relationships may always exist and proposing a mathematical problem for proving existence of the relationships in general. Moreover, numerical simulations show that the basic reproduction number does not monotonically increase or decrease with the extinction threshold. Consistent trends of extinction probability observed through numerical simulations provide novel insights into mitigation strategies to increase the disease extinction probability. Research findings may improve understandings of thresholds for disease persistence in order to control vector-borne diseases.
Citation: 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
##### References:
 [1] L. J. Allen and G. E. Lahodny Jr, Extinction thresholds in deterministic and stochastic epidemic models, Journal of Biological Dynamics, 6 (2012), 590-611. doi: 10.1080/17513758.2012.665502. [2] L. J. Allen and P. van den Driessche, Relations between deterministic and stochastic thresholds for disease extinction in continuous- and discrete-time infectious disease models, Mathematical Biolosciences, 243 (2013), 99-108. doi: 10.1016/j.mbs.2013.02.006. [3] J. Arino and P. van den Driessche, The basic reproduction number in a multi-city compartmental epidemic model, Lecture Notes in Control and Information Sciences, 294 (2003), 135-142. doi: 10.1007/978-3-540-44928-7_19. [4] M. Bates, The natural history of mosquitoes, American Journal of Public Health, 39 (1949), p1592. [5] D. Bisanzio, L. Bertolotti, L. Tomassone, G. Amore, C. Ragagli, A. Mannelli, M. Giacobini and P. Provero, Modeling the spread of vector-borne diseases on bipartite networks, PloS ONE, 5 (2010), e13796. doi: 10.1371/journal.pone.0013796. [6] T. Britton and D. Lindenstrand, Epidemic modelling: Aspects where stochasticity matters, Mathematical Biosciences, 222 (2009), 109-116. doi: 10.1016/j.mbs.2009.10.001. [7] D. V. Canyon, J. L. K. Hii and R. Muller, The frequency of host biting and its effect on oviposition and survival in Aedes aegypti (Diptera: Culicidae), Bulletin of Entomological Research, 89 (1999), 35-39. doi: 10.1017/S000748539900005X. [8] N. Chitnis, J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bulletin of Mathematical Biology, 70 (2008), 1272-1296. doi: 10.1007/s11538-008-9299-0. [9] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases, Wiley, Chichester, 2000. [10] O. Diekmann, J. A. P. Heesterbeek and 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, Journal of Mathematical Biology, 28 (1990), 365-382. doi: 10.1007/BF00178324. [11] B. J. Erasmus and J. A. W. Coetzer, The symptomatology and pathology of Rift Valley fever in domestic animals, Contributions to Epidemiology and Biostatistics, 3 (1981), 77-82. [12] N. G. Gratz, Emerging and resurging vector-borne diseases, Annual Review of Entomology, 44 (1999), 51-75. doi: 10.1146/annurev.ento.44.1.51. [13] R. O. Hayes, C. H. Tempelis, A. D. Hess and W. C. Reeves, Mosquito host preference studies in Hale County, Texas, American Journal of Tropical Medicine and Hygiene, 22 (1973), 270-277. [14] G. R. Hosack, P. A. Rossignol and P. V. den Driessche, The control of vector-borne disease epidemics, Journal of Theoretical Biology, 255 (2008), 16-25. doi: 10.1016/j.jtbi.2008.07.033. [15] C. J. Jones and J. E. Lloyd, Mosquitos feeding on sheep in southeastern Wyoming, Journal of the American Mosquito Control Association, 1 (1985), 530-532. [16] R. Kao, Networks and models with heterogeneous population structure in epidemiology, in Network Science (eds. E. Estrada, M. Fox, D. Higham and G. Oppo), Springer, (2010), 51-84. doi: 10.1007/978-1-84996-396-1_4. [17] M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2008. [18] G. E. Lahodny Jr and L. J. Allen, Probability of a disease outbreak in stochastic multipatch epidemic models, Bulletin of Mathematical Biology, 75 (2013), 1157-1180. doi: 10.1007/s11538-013-9848-z. [19] A. L. Lloyd, J. Zhang and A. M. Root, Stochasticity and heterogeneity in host-vector models, Journal of the Royal Society Interface, 4 (2007), 851-863. doi: 10.1098/rsif.2007.1064. [20] L. A. Magnarelli, Host feeding patterns of Connecticut mosquitos (Diptera: Culicidae), American Journal of Tropical Medicine and Hygiene, 26 (1977), 547-552. [21] C. G. Moore, R. G. McLean, C. J. Mitchell, R. S. Nasci, T. F. Tsai, C. H. Caslisher, A. A. Marfin, P. S. Moorse and D. J. Gubler, Guidelines for Arbovirus Surveillance Programs in the United States, Centers for Disease Control and Prevention, 1993. [22] F. Natale, A. Giovannini, L. Savini, D. Palma, L. Possenti, G. Fiore and P. Calistri, Network analysis of Italian cattle trade patterns and evaluation of risks for potential disease spread, Preventive Veterinary Medicine, 92 (2009), 341-350. doi: 10.1016/j.prevetmed.2009.08.026. [23] S. Pénisson, Conditional Limit Theorems for Multitype Branching Processes and Illustration in Epidemiological Risk Analysis, PhD thesis, Institut für Mathematik der Unversität Potsdam, Germany, 2010. [24] C. J. Peters and K. J. Linthicum, Rift Valley fever, in Handbook of Zoonoses (ed. G. Beran), $2^{nd}$ edition, Section B: Viral, CRC Press, Inc., Boca Raton, Fl, (1994), 125-138. [25] H. D. Pratt and C. G. Moore, Vector-borne Disease Control: Mosquitoes of Public Health Importance and Their Control, U.S. Department of Health and Human Services, Atlanta, GA, 1993. [26] O. M. Radostits, Herd Healthy: Food Animal Production Medicine, Saunders, 2001. [27] M. G. Roberts and J. A. P. Heesterbeek, A new method for estimating the effort required to control an infectious disease, Proceedings of the Royal Society B: Biological Sciences, 270 (2003), 1359-1364. doi: 10.1098/rspb.2003.2339. [28] R. Ross, Some quantitative studies in epidemiology, Nature, 87 (1911), 466-467. doi: 10.1038/087466a0. [29] Z. Shuai, J. A. P. Heesterbeek and P. van den Driessche, Extending the type reproduction number to infectious disease control targeting contacts between types, Journal of mathematical biology, 67 (2013), 1067-1082. doi: 10.1007/s00285-012-0579-9. [30] M. J. Turell and C. L. Bailey, Transmission studies in mosquitoes (Diptera: Culicidae) with disseminated Rift Valley fever virus infections, Journal of Medical Entomology, 24 (1987), 11-18. [31] M. J. Turell, C. L. Bailey and J. R. Beaman, Vector competence of a Houston, Texas strain of Aedes Albopictus for Rift Valley fever virus, Journal of the American Mosquito Control Association, 4 (1988), 94-96. [32] M. J. Turell, M. E. Faran, M. Cornet and C. L. Bailey, Vector competence of senegalese Aedes fowleri (Diptera: Culicidae) for Rift Valley fever virus, Journal of Medical Entomology, 25 (1988), 262-266. [33] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [34] M. C. Vernon and M. J. Keeling, Representing the UK's cattle herd as static and dynamic networks, Proceedings of the Royal Society B: Biological Sciences, 276 (2009), 469-476. doi: 10.1098/rspb.2008.1009. [35] L. Xue and C. Scoglio, The network level reproduction number for infectious diseases with both vertical and horizontal transmission, Mathematical Biosciences, 243 (2013), 67-80. doi: 10.1016/j.mbs.2013.02.004.

show all references

##### References:
 [1] L. J. Allen and G. E. Lahodny Jr, Extinction thresholds in deterministic and stochastic epidemic models, Journal of Biological Dynamics, 6 (2012), 590-611. doi: 10.1080/17513758.2012.665502. [2] L. J. Allen and P. van den Driessche, Relations between deterministic and stochastic thresholds for disease extinction in continuous- and discrete-time infectious disease models, Mathematical Biolosciences, 243 (2013), 99-108. doi: 10.1016/j.mbs.2013.02.006. [3] J. Arino and P. van den Driessche, The basic reproduction number in a multi-city compartmental epidemic model, Lecture Notes in Control and Information Sciences, 294 (2003), 135-142. doi: 10.1007/978-3-540-44928-7_19. [4] M. Bates, The natural history of mosquitoes, American Journal of Public Health, 39 (1949), p1592. [5] D. Bisanzio, L. Bertolotti, L. Tomassone, G. Amore, C. Ragagli, A. Mannelli, M. Giacobini and P. Provero, Modeling the spread of vector-borne diseases on bipartite networks, PloS ONE, 5 (2010), e13796. doi: 10.1371/journal.pone.0013796. [6] T. Britton and D. Lindenstrand, Epidemic modelling: Aspects where stochasticity matters, Mathematical Biosciences, 222 (2009), 109-116. doi: 10.1016/j.mbs.2009.10.001. [7] D. V. Canyon, J. L. K. Hii and R. Muller, The frequency of host biting and its effect on oviposition and survival in Aedes aegypti (Diptera: Culicidae), Bulletin of Entomological Research, 89 (1999), 35-39. doi: 10.1017/S000748539900005X. [8] N. Chitnis, J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bulletin of Mathematical Biology, 70 (2008), 1272-1296. doi: 10.1007/s11538-008-9299-0. [9] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases, Wiley, Chichester, 2000. [10] O. Diekmann, J. A. P. Heesterbeek and 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, Journal of Mathematical Biology, 28 (1990), 365-382. doi: 10.1007/BF00178324. [11] B. J. Erasmus and J. A. W. Coetzer, The symptomatology and pathology of Rift Valley fever in domestic animals, Contributions to Epidemiology and Biostatistics, 3 (1981), 77-82. [12] N. G. Gratz, Emerging and resurging vector-borne diseases, Annual Review of Entomology, 44 (1999), 51-75. doi: 10.1146/annurev.ento.44.1.51. [13] R. O. Hayes, C. H. Tempelis, A. D. Hess and W. C. Reeves, Mosquito host preference studies in Hale County, Texas, American Journal of Tropical Medicine and Hygiene, 22 (1973), 270-277. [14] G. R. Hosack, P. A. Rossignol and P. V. den Driessche, The control of vector-borne disease epidemics, Journal of Theoretical Biology, 255 (2008), 16-25. doi: 10.1016/j.jtbi.2008.07.033. [15] C. J. Jones and J. E. Lloyd, Mosquitos feeding on sheep in southeastern Wyoming, Journal of the American Mosquito Control Association, 1 (1985), 530-532. [16] R. Kao, Networks and models with heterogeneous population structure in epidemiology, in Network Science (eds. E. Estrada, M. Fox, D. Higham and G. Oppo), Springer, (2010), 51-84. doi: 10.1007/978-1-84996-396-1_4. [17] M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2008. [18] G. E. Lahodny Jr and L. J. Allen, Probability of a disease outbreak in stochastic multipatch epidemic models, Bulletin of Mathematical Biology, 75 (2013), 1157-1180. doi: 10.1007/s11538-013-9848-z. [19] A. L. Lloyd, J. Zhang and A. M. Root, Stochasticity and heterogeneity in host-vector models, Journal of the Royal Society Interface, 4 (2007), 851-863. doi: 10.1098/rsif.2007.1064. [20] L. A. Magnarelli, Host feeding patterns of Connecticut mosquitos (Diptera: Culicidae), American Journal of Tropical Medicine and Hygiene, 26 (1977), 547-552. [21] C. G. Moore, R. G. McLean, C. J. Mitchell, R. S. Nasci, T. F. Tsai, C. H. Caslisher, A. A. Marfin, P. S. Moorse and D. J. Gubler, Guidelines for Arbovirus Surveillance Programs in the United States, Centers for Disease Control and Prevention, 1993. [22] F. Natale, A. Giovannini, L. Savini, D. Palma, L. Possenti, G. Fiore and P. Calistri, Network analysis of Italian cattle trade patterns and evaluation of risks for potential disease spread, Preventive Veterinary Medicine, 92 (2009), 341-350. doi: 10.1016/j.prevetmed.2009.08.026. [23] S. Pénisson, Conditional Limit Theorems for Multitype Branching Processes and Illustration in Epidemiological Risk Analysis, PhD thesis, Institut für Mathematik der Unversität Potsdam, Germany, 2010. [24] C. J. Peters and K. J. Linthicum, Rift Valley fever, in Handbook of Zoonoses (ed. G. Beran), $2^{nd}$ edition, Section B: Viral, CRC Press, Inc., Boca Raton, Fl, (1994), 125-138. [25] H. D. Pratt and C. G. Moore, Vector-borne Disease Control: Mosquitoes of Public Health Importance and Their Control, U.S. Department of Health and Human Services, Atlanta, GA, 1993. [26] O. M. Radostits, Herd Healthy: Food Animal Production Medicine, Saunders, 2001. [27] M. G. Roberts and J. A. P. Heesterbeek, A new method for estimating the effort required to control an infectious disease, Proceedings of the Royal Society B: Biological Sciences, 270 (2003), 1359-1364. doi: 10.1098/rspb.2003.2339. [28] R. Ross, Some quantitative studies in epidemiology, Nature, 87 (1911), 466-467. doi: 10.1038/087466a0. [29] Z. Shuai, J. A. P. Heesterbeek and P. van den Driessche, Extending the type reproduction number to infectious disease control targeting contacts between types, Journal of mathematical biology, 67 (2013), 1067-1082. doi: 10.1007/s00285-012-0579-9. [30] M. J. Turell and C. L. Bailey, Transmission studies in mosquitoes (Diptera: Culicidae) with disseminated Rift Valley fever virus infections, Journal of Medical Entomology, 24 (1987), 11-18. [31] M. J. Turell, C. L. Bailey and J. R. Beaman, Vector competence of a Houston, Texas strain of Aedes Albopictus for Rift Valley fever virus, Journal of the American Mosquito Control Association, 4 (1988), 94-96. [32] M. J. Turell, M. E. Faran, M. Cornet and C. L. Bailey, Vector competence of senegalese Aedes fowleri (Diptera: Culicidae) for Rift Valley fever virus, Journal of Medical Entomology, 25 (1988), 262-266. [33] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [34] M. C. Vernon and M. J. Keeling, Representing the UK's cattle herd as static and dynamic networks, Proceedings of the Royal Society B: Biological Sciences, 276 (2009), 469-476. doi: 10.1098/rspb.2008.1009. [35] L. Xue and C. Scoglio, The network level reproduction number for infectious diseases with both vertical and horizontal transmission, Mathematical Biosciences, 243 (2013), 67-80. doi: 10.1016/j.mbs.2013.02.004.
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