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A model for the emergence of drug resistance in the presence of asymptomatic infections. (English) Zbl 1281.92042
Summary: An analysis of a mathematical model, which describes the dynamics of an aerially transmitted disease, and the effects of the emergence of drug resistance after the introduction of treatment as an intervention strategy is presented. Under explicit consideration of asymptomatic and symptomatic infective individuals for the basic model without intervention the analysis shows that the dynamics of the epidemic is determined by a basic reproduction number \(R_0\). A disease-free and an endemic equilibrium exist and are locally asymptotically stable when \(R_0<1\) and \(R_0>1\), respectively. When a treatment is included the system has a basic reproduction number, which is the largest of the two reproduction numbers that characterise the drug-sensitive \((R_1)\) or resistant \((R_2)\) strains of the infectious agent. The system has a disease-free equilibrium, which is stable when both \(R_1\) and \(R_2\) are less than unity.
Two endemic equilibria also exist and are associated with the treatment and the development of drug resistance. An endemic equilibrium where only the drug-resistant strain persists exists and is stable when \(R_2>1\) and \(R_1<R_2\). A second endemic equilibrium exists when \(R_1>1\) and \(R_1>R_2\) and both drug-sensitive and drug-resistant strains are present. The analysis of the system provides insights about the conditions under which the infection will persist and whether sensitive and resistant strains will coexist or not.

MSC:
92C60 Medical epidemiology
92C50 Medical applications (general)
37N25 Dynamical systems in biology
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[1] Fraser, C.; Riley, S.; Anderson, R. M.; Ferguson, N. M., Factors that make an infectious disease outbreak controllable, PNAS, 101, 6146, (2004)
[2] Mathews, J. D.; McCaw, C. T.; McVernon, J.; McBryde, E. S.; McCaw, J. M., A biological model for influenza transmission: pandemic planning implications of asymptomatic infection and immunity, PLoS ONE, 2, e1220, (2007)
[3] Kelly, H. A.; Priest, P. C.; Mercer, G. N.; Dowse, G. K., We should not be complacent about our population-based public health response to the first influenza pandemic of the \(21^{\mathit{st}}\) century, BMC Public Health, 11, 78, (2011)
[4] Longini, I. M.; Halloran, M. E.; Nizam, A.; Yang, Y., Containing pandemic influenza with antiviral agents, Am. J. Epidemiol., 159, 623, (2004)
[5] Arino, J.; Brauer, F.; van den Driessche, P.; Watmough, J.; Wu, J., A model for influenza vaccination and antiviral treatment, J. Theor. Biol., 253, 118, (2008) · Zbl 1398.92147
[6] McCaw, J. M.; Wood, L. G.; McCaw, C. T.; McVernon, J., Impact of emerging antiviral drug resistance on influenza containment and spread: influence of subclinical infection and strategic use of a stockpile containing one or two drugs, PLoS ONE, 3, e2362, (2008)
[7] Stilianakis, N. I.; Perelson, A. S.; Hayden, F. G., Emergence of drug resistance during an influenza epidemic: insights from a mathematical model, J. Infect. Dis., 177, 863, (1998)
[8] Regoes, R. R.; Bonhoeffer, S., Emergence of drug-resistant influenza virus: population dynamical considerations, Science, 312, 380, (2006)
[9] Débarre, F.; Bonhoeffer, S.; Regoes, R. R., The effect of population structure on the emergence of drug resistance during influenza pandemics, J.R. Soc. Interface, 4, 893, (2007)
[10] Wu, J. T.; Riley, S.; Fraser, C.; Leung, G. M., Reducing the impact of the next influenza pandemic using household-based public health interventions, PLoS Med., 3, 1532, (2006)
[11] Moghadas, S. M.; Bowman, C. S.; Röst, G.; Wu, J., Population-wide emergence of antiviral resistance during pandemic influenza, PLoS ONE, 3, e1839, (2008)
[12] Alexander, M. E.; Bowman, C. S.; Feng, Z.; Gardam, M.; Moghadas, S. M.; Röst, G.; Wu, J.; Yan, P., Emergence of drug resistance: implications for antiviral control of pandemic influenza, Proc. R. Soc. B, 274, 1675, (2007)
[13] Keeling, M.; Rohani, P., Modelling infectious diseases in humans and animals, (2008), Princeton University Press · Zbl 1279.92038
[14] Kemper, J. T., The effects of asymptomatic attacks on the spread of infectious disease: a deterministic model, Bull. Math. Biol., 40, 707, (1978) · Zbl 0406.92021
[15] Anderson, R. M.; May, R. M., Infectious diseases of humans: dynamics and control, (1991), Oxford University Press
[16] van den Driessche, P.; Watmough, J., Further notes on the basic reproductive number, (Mathematical Epidemiology, (2008), Springer New York), 159 · Zbl 1206.92038
[17] van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. BioSci., 180, 29, (2001) · Zbl 1015.92036
[18] Diekmann, O.; Heesterbeek, J. A.P., Mathematical epidemiology of infectious diseases: model building, analysis and interpretation, (2000), Wiley Chichester, UK · Zbl 0997.92505
[19] Diekmann, O.; Heesterbeek, J. A.P.; Roberts, M. G., The construction of next-generation matrices for compartmental epidemic models, J. R. Soc. Interface, 7, 873, (2010)
[20] Gu, Y.; Komiya, N.; Kamiya, H.; Yasuo, Y.; Taniguchi, K.; Okabe, N., Pandemic (H1N1) 2009 transmission during presymptomatic phase, Japan, Emerg. Infect. Dis., 17, 1737, (2011)
[21] P. Marquez, D. Tereshita, L. Engish, Pre-symptomatic health care worker transmission of pandemic (H1N1) 2009 influenza in acute care settings, Acute Communicable Disease Control Program, Los Angeles, California, Special Studies Report, 2009, Los Angeles County Department of Public Health. <http://www.ph.lacounty.gov/acd/reports/annual/2009SpecialStudies.pdf>.
[22] Lau, L. H.L.; Cowling, B. J.; Fang, V. J.; Chan, K. H.; Lau, E. H.Y.; Lipsitch, M.; Cheng, C. K.Y.; Nouck, P. M.; Uyeki, T. M.; Peiris, J. S.M.; Leung, G. M., Viral shedding and clinical illness in naturally acquired influenza virus infections, J. Infect. Dis., 201, 1509, (2010)
[23] Sheat, K., An investigation into an explosive outbreak of influenza - new plymouth, Commun. Dis. N. Z., 92, 18, (1992)
[24] Webb, G. F.; Hsieh, Y.-H.; Wu, J.; Blaser, M. J., Pre-symptomatic influenza transmission surveillance and school closings: implications for novel influenza A (H1N1), Math. Model. Nat. Phenom., 5, 191, (2010) · Zbl 1187.92065
[25] World Health Organisation Writing Group, Nonpharmaceutical interventions for pandemic influenza, international measures, Emerg. Infect. Dis., 12, 81, (2006)
[26] Ferguson, N. M.; Mallett, S.; Jackson, H.; Roberts, N.; Ward, P., A population-dynamic model for evaluating the potential spread of resistant influenza virus infections during community-based use of antivirals, J. Antimicrob. Chemother., 51, 977, (2003)
[27] Patrozou, E.; Mermel, L. A., Does infuenza transmission occur from asymptomatic infection or prior to symptom onset?, Public Health Rep., 124, 193, (2009)
[28] Donnelly, C. A.; Finelli, L.; Cauchemez, S., Serial intervals and the temporal distribution of secondary infections within households of 2009 pandemic influenza A (H1N1): implications for influenza control recommendations, Clin. Infect. Dis., 52, S1, S123, (2011)
[29] Elvebeack, L. R.; Fox, J. P.; Ackerman, E.; Langworthy, A.; Boyd, M.; Gatewood, L., An influenza simulation model for immunization studies, Am. J. Epidemiol., 103, 152, (1976)
[30] Carrat, F.; Vergu, E.; Ferguson, N. M.; Lemaitre, M.; Cauchemez, S.; Leach, S.; Valleron, A. J., Time lines of infection and disease in human influenza: a review of volunteer challenge studies, Am. J. Epidemiol., 167, 775, (2008)
[31] Ghani, A. C.; Baguelin, M.; Griffin, J., The early transmission dynamics of H1N1pdm influenza in the united kingdom, PLoS Curr. Influenza, RRN1130, (2009)
[32] Mills, C. E.; Robins, J. M.; Lipsitch, M., Transmissibility of 1918 pandemic influenza, Nature, 432, 904, (2004)
[33] Boëlle, P. Y.; Ansart, S.; Cori, A.; Vallerona, A. J., Transmission parameters of the A/H1N1 (2009) influenza virus pandemic: a review, Influenza Other Respir. Viruses, 5, 306, (2011)
[34] Ferguson, N. M.; Mallett, S.; Jackson, H.; Roberts, N.; Ward, P., A population-dynamic model for evaluating the potential spread of drug-resistant influenza virus infections during community-based use of antivirals, J. Antimicrob. Chemother., 51, 977, (2003)
[35] Nichol, K. L.; Tummers, K.; Hoyer-Leitzel, A.; Marsh, J.; Moynihan, M.; McKelvey, S., Modeling seasonal influenza outbreak in a closed college campus: impact of pre-season vaccination in-season vaccination and holidays/breaks, PLoS ONE, 5, e9548, (2010)
[36] Yang, Y.; Sugimoto, J. D.; Halloran, M. E.; Basta, N. E.; Chao, D. L.; Matrajt, L.; Potter, G.; Kenah, E.; Longini, I. M., The transmissibility and control of pandemic influenza A (H1N1) virus, Science, 326, 729, (2009)
[37] Chowell, G.; Miller, M. A.; Viboud, C., Seasonal influenza in the united states France and Australia: transmission and prospects for control, Epidemiol. Infect., 136, 852, (2008)
[38] Dushoff, J.; Plotkin, J. B.; Levin, S. A.; Earn, D. J.D., Dynamical resonance can account for seasonality of influenza epidemics, PNAS, 101, 16915, (2004)
[39] Van Voris, L. P.; Betts, R. F.; Hayden, F. G.; Christmas, W. A.; Douglas Jr, R. F., Successful treatment of naturally occurring influenza A/USSR/77 H1N1, JAMA, 245, 1128, (1981)
[40] Couch, R. B.; Kasel, J. A.; Glezen, W. P.; Cate, T. R.; Six, H. R.; Taber, L. H.; Frank, A. L.; Greenberg, S. B.; Zahradnik, J. M.; Keitel, W. A., Influenza: its control in person and populations, J. Infect. Dis., 153, 431, (1986)
[41] Hurt, A. C.; Deng, Y. M.; Ernest, J.; Caldwell, N.; Leang, L.; Iannello, P.; Komadina, N.; Shaw, R.; Smith, D.; Dwyer, D. E.; Tramontana, A. R.; Lin, R. T.; Freeman, K.; Kelso, A.; Barr, I. G., Oseltamivir-resistant influenza viruses circulating during the first year of the influenza A(H1N1) 2009 pandemic in the Asia-Pacific region, March 2009 to March 2010, Euro Surveill, 16, 3, 19770, (2011)
[42] Stephenson, I.; Democratis, J.; Lackenby, A.; McNally, T.; Smith, J.; Pareek, M.; Ellis, J.; Bermingham, A.; Nicholson, K.; Zambon, M., Neuraminidase inhibitor resistance after oseltamivir treatment of acute influenza A and B in children, Clin. Infect. Dis., 4, 389, (2009)
[43] Lackenby, A.; Gilad, J. M.; Pebody, R.; Miah, S.; Calatayud, L.; Bolotin, S.; Vipond, I.; Muir, P.; Guiver, M.; McMenamin, J.; Reynolds, A.; Moore, C.; Gunson, R.; Thompson, C. I.; Galiano, M.; Bermingham, A.; Ellis, J.; Zambon, M., Continued emergence and changing epidemiology of oseltamivir-resistant influenza A(H1N1) 2009 virus, united kingdom, winter 2010/11, Euro Surveill, 5, 19784, (2011)
[44] Castillo-Chavez, C.; Feng, Z., To treat or not to treat: the case of tuberculosis, J. Math. Biol., 35, 629, (1997) · Zbl 0895.92024
[45] Feng, Z.; Iannelli, M.; Milner, F. A., A two-strain tuberculosis model with age of infection, SIAM J. Appl. Math., 62, 1634, (2002) · Zbl 1017.35066
[46] Castillo-Chavez, C.; Song, B., Dynamical models of tuberculosis and their applications, Math. BioSci. Eng., 1, 361, (2004) · Zbl 1060.92041
[47] Balicer, R. D.; Huerta, M.; Davidovitch, N.; Grotto, I., Cost-benefit of stockpiling drugs for influenza pandemic, Emerg. Infect. Dis., 11, 1280, (2005)
[48] Lugner, A. K.; Postma, M. J., Investment decisions in influenza pandemic contingency planning: cost-effectiveness of stockpiling antiviral drugs, Eur. J. Public Health, 19, 516, (2009)
[49] Arino, J.; Bowman, C. S.; Moghadas, S. M., Antiviral resistance during pandemic influenza: implication for stockpiling and drug use, BMC Infect. Dis., 9, 8, (2009)
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