Document Zbl 1459.92128

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Difference Equ.</a> <a href="/?q=in%3A427324" title="Articles in this Issue">2019, Paper No. 368, 17 p. (2019)</a>. </div> <div class="abstract">Summary: Pulse vaccination is a repeated vaccination policy, which plays a tremendous role in the global fight against communicable diseases in terms of saving medical resources and decreasing the economic burden. In this article, we propose a dynamic model of dengue infection with periodic transmission functions and seasonality in vector population. Furthermore, we introduce a pulse vaccination strategy in the susceptible host population to examine how frequency and intensity of implementation of this strategy affect the dynamics of dengue infection. We successfully obtained the threshold dynamics by defining the basic reproduction number $R_0$, which is the spectral radius of the next generation operator and governs whether the disease dies out or not. It has been established that the infection-free periodic solution of the proposed impulsive system is globally asymptotically stable if $\mbox{R}_0<1$ and is unstable otherwise. Moreover, we found that the dengue infection is uniformly persistent for the proposed system if $\mbox{R}_0>1$. Finally, we execute the system numerically to illustrate the piecewise solutions of the proposed system with impulsive vaccination measure and to investigate the influence of different control parameters on the basic reproduction. The finding indicates that a frequent implementation of the vaccination strategy with great intensity and the use of mosquito nets can essentially lead to a decline of new infections.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7254382">Cited in <strong>10</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A92D30" title="MSC2020">92D30</a> </td> <td class="space"> Epidemiology </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A92D25" title="MSC2020">92D25</a> </td> <td class="space"> Population dynamics (general) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37N25" title="MSC2020">37N25</a> </td> <td class="space"> Dynamical systems in biology </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Adengue+fever">dengue fever</a>; <a href="/?q=ut%3Aperiodic+transmission">periodic transmission</a>; <a href="/?q=ut%3Aimpulsive+vaccination">impulsive vaccination</a>; <a href="/?q=ut%3Aasymptotical+stability">asymptotical stability</a>; <a href="/?q=ut%3Auniform+persistence">uniform persistence</a>; <a href="/?q=ut%3Anumerical+simulations">numerical simulations</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1459.92128" data-ciurl="/ci/07254382" data-biburl="/bibtex/07254382.bib" data-amsurl="/amsrefs/07254382.bib" data-xmlurl="/xml/07254382.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07254382.pdf" title="Zbl 1459.92128 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1186/s13662-019-2314-y" aria-label="DOI for “Effect of pulse vaccination on dynamics of dengue with periodic transmission functions”" title="10.1186/s13662-019-2314-y">DOI</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Center for Disease Control and Prevention. Dengue, Accessed January 14, 2013. http://www.cdc.gov/dengue/</td> </tr><tr> <td>[2]</td> <td class="space">Constenla, D., Armien, B., Arredondo, J., et al.: Costing dengue fever cases and outbreaks: recommendations from a costing dengue working group in the Americas. Value Health Reg. Issues 2015(8), 80-91 (2015) · <a href="https://doi.org/10.1016/j.vhri.2015.06.001" class="nowrap">doi:10.1016/j.vhri.2015.06.001</a></td> </tr><tr> <td>[3]</td> <td class="space">Dengue Signs and Clinical Symptoms. Cbwinfo.com. http://www.cbwinfo.com/Biological/Pathogens/DENV.html [cited April 23, 2010]</td> </tr><tr> <td>[4]</td> <td class="space">Teixeira, M.G., Barreto, M.L.: Diagnosis and management of dengue. Br. Med. J. 339, 4338 (2009) · <a href="https://doi.org/10.1136/bmj.b4338" class="nowrap">doi:10.1136/bmj.b4338</a></td> </tr><tr> <td>[5]</td> <td class="space">Gubler, D.J.: Dengue and dengue hemorrhagic fever. Clin. Microbiol. Rev. 11(3), 480-496 (1998) · <a href="https://doi.org/10.1128/CMR.11.3.480" class="nowrap">doi:10.1128/CMR.11.3.480</a></td> </tr><tr> <td>[6]</td> <td class="space">Guzman, M.G., Kouri, G., Bravo, J., Valdes, L., Vasquez, S., Halstead, S.B.: Effect of age on outcome of secondary dengue 2 infections. Int. J. Infect. Dis. 6(2), 118-124 (2002) · <a href="https://doi.org/10.1016/S1201-9712(02)90072-X" class="nowrap">doi:10.1016/S1201-9712(02)90072-X</a></td> </tr><tr> <td>[7]</td> <td class="space">Sabin, A.B.: Research on dengue during World War II. Am. J. Trop. Med. Hyg. 1(1), 30-50 (1952) · <a href="https://doi.org/10.4269/ajtmh.1952.1.30" class="nowrap">doi:10.4269/ajtmh.1952.1.30</a></td> </tr><tr> <td>[8]</td> <td class="space">Siler, J.F., Hall, M.W., Hitchens, A.P.: Dengue: its history, epidemiology, mechanism of transmission, etiology, clinical manifestations, immunity, and prevention. Philipp. J. Sci. 29(20), 1-304 (1952)</td> </tr><tr> <td>[9]</td> <td class="space">Rigau-Perez, J.G.: Severe dengue: the need for new case definitions. Lancet Infect. Dis. 6(5), 297-302 (2006) · <a href="https://doi.org/10.1016/S1473-3099(06)70465-0" class="nowrap">doi:10.1016/S1473-3099(06)70465-0</a></td> </tr><tr> <td>[10]</td> <td class="space">World Health Organisation. Impact of dengue. http://www.who.int/csr/disease/dengue/impact/en/ (2012)</td> </tr><tr> <td>[11]</td> <td class="space">Halstead, S.B.: Dengue. Lancet 370, 1644-1652 (2007) · <a href="https://doi.org/10.1016/S0140-6736(07)61687-0" class="nowrap">doi:10.1016/S0140-6736(07)61687-0</a></td> </tr><tr> <td>[12]</td> <td class="space">Kautner, I., Robinson, M.J., Kuhnle, U.: Dengue virus infection: epidemiology, pathogenesis, clinical presentation, diagnosis, and prevention. J. Pediatr. 131, 516-524 (1997) · <a href="https://doi.org/10.1016/S0022-3476(97)70054-4" class="nowrap">doi:10.1016/S0022-3476(97)70054-4</a></td> </tr><tr> <td>[13]</td> <td class="space">Rodrigues, H.S., Monteiro, M.T.T., Torres, D.F.M.: Vaccination models and optimal control strategies to dengue. J. Math. Biosci. 247, 1-12 (2014) · <a href="/1282.92022" class="nowrap">Zbl 1282.92022</a> · <a href="https://doi.org/10.1016/j.mbs.2013.10.006" class="nowrap">doi:10.1016/j.mbs.2013.10.006</a></td> </tr><tr> <td>[14]</td> <td class="space">Fenner, F., Henderson, D.A., Arita, I., Jezek, Z., Ladnyi, I.D.: Smallpox and its eradication. J. Commun. Health 13(4), 245-592 (1988)</td> </tr><tr> <td>[15]</td> <td class="space">Hethcote, H.W.: Oscillations in an endemic model for pertussis. Can. Appl. Math. Q. 6(1), 61-88 (1998) · <a href="/0919.92032" class="nowrap">Zbl 0919.92032</a></td> </tr><tr> <td>[16]</td> <td class="space">Tian, X.H.: Stability analysis of a delayed sirs epidemic model with vaccination and nonlinear incidence. Int. J. Biomath. 5(6), 1250050 (2012) · <a href="/1297.92084" class="nowrap">Zbl 1297.92084</a> · <a href="https://doi.org/10.1142/S1793524512500507" class="nowrap">doi:10.1142/S1793524512500507</a></td> </tr><tr> <td>[17]</td> <td class="space">Garba, S.M., Gumel, A.B., Abu Bakar, M.R.: Backward bifurcations in dengue transmission dynamics. Math. Biosci. 215, 11-25 (2008) · <a href="/1156.92036" class="nowrap">Zbl 1156.92036</a> · <a href="https://doi.org/10.1016/j.mbs.2008.05.002" class="nowrap">doi:10.1016/j.mbs.2008.05.002</a></td> </tr><tr> <td>[18]</td> <td class="space">Pongsumpun, P., Tang, I.M.: Transmission of dengue hemorrhagic fever in an age structured population. Math. Comput. Model. 37, 949-961 (2003) · <a href="/1045.92040" class="nowrap">Zbl 1045.92040</a> · <a href="https://doi.org/10.1016/S0895-7177(03)00111-0" class="nowrap">doi:10.1016/S0895-7177(03)00111-0</a></td> </tr><tr> <td>[19]</td> <td class="space">Supriatna, A.K., Soewono, E., Van Gils, S.A.: A two-age-classes dengue transmission model. Math. Biosci. 216, 114-121 (2008) · <a href="/1151.92019" class="nowrap">Zbl 1151.92019</a> · <a href="https://doi.org/10.1016/j.mbs.2008.08.011" class="nowrap">doi:10.1016/j.mbs.2008.08.011</a></td> </tr><tr> <td>[20]</td> <td class="space">De la Sen, M., Agarwal, R.P., Ibeas, A., Alonso-Quesada, S.: On the existence of equilibrium points, boundedness, oscillating behavior and positivity of a sveirs epidemic model under constant and impulsive vaccination. Adv. Differ. Equ. 2011, 748608 (2011) · <a href="/1219.34066" class="nowrap">Zbl 1219.34066</a></td> </tr><tr> <td>[21]</td> <td class="space">Billings, L., Fiorillo, A., Schwartz, I.B.: Vaccinations in disease models with antibody-dependent enhancement. Math. Biosci. 211, 265-281 (2008) · <a href="/1131.92040" class="nowrap">Zbl 1131.92040</a> · <a href="https://doi.org/10.1016/j.mbs.2007.08.004" class="nowrap">doi:10.1016/j.mbs.2007.08.004</a></td> </tr><tr> <td>[22]</td> <td class="space">Koopman, J.S., Prevots, D.R., Vaca Marin, M.A., Gomez Dantes, H., Zarate Aquino, M.L., Longini, I.M.J., Sepulveda, A.J.: Determinants and predictors of dengue infection in Mexico. Can. Appl. Math. Q. 133(11), 1168-1178 (1991)</td> </tr><tr> <td>[23]</td> <td class="space">Esteva, L., Vargas, C.: Analysis of a dengue disease transmission model. Math. Biosci. 150, 131-151 (1998) · <a href="/0930.92020" class="nowrap">Zbl 0930.92020</a> · <a href="https://doi.org/10.1016/S0025-5564(98)10003-2" class="nowrap">doi:10.1016/S0025-5564(98)10003-2</a></td> </tr><tr> <td>[24]</td> <td class="space">Esteva, L., Vargas, C.: A model for dengue disease with variable human population. J. Math. Biol. 38(3), 220-240 (1999) · <a href="/0981.92016" class="nowrap">Zbl 0981.92016</a> · <a href="https://doi.org/10.1007/s002850050147" class="nowrap">doi:10.1007/s002850050147</a></td> </tr><tr> <td>[25]</td> <td class="space">Esteva, L., Vargas, C.: Influence of vertical and mechanical transmission on the dynamics of dengue disease. Math. Biosci. 167(1), 51-64 (2000) · <a href="/0970.92011" class="nowrap">Zbl 0970.92011</a> · <a href="https://doi.org/10.1016/S0025-5564(00)00024-9" class="nowrap">doi:10.1016/S0025-5564(00)00024-9</a></td> </tr><tr> <td>[26]</td> <td class="space">Derouich, M., Boutayeb, A., Twizell, E.H.: A model of dengue fever. Biomed. Eng. Online 2, 4 (2003) · <a href="https://doi.org/10.1186/1475-925X-2-4" class="nowrap">doi:10.1186/1475-925X-2-4</a></td> </tr><tr> <td>[27]</td> <td class="space">Syafruddin, S., Noorani, S.M.: Seir model for transmission of dengue fever in Selangor, Malaysia. Int. J. Mod. Phys. Conf. Ser. 9, 380-389 (2012) · <a href="https://doi.org/10.1142/S2010194512005454" class="nowrap">doi:10.1142/S2010194512005454</a></td> </tr><tr> <td>[28]</td> <td class="space">Feng, Z., Jorge, X.: Velasco-Hernandez, competitive exclusion in a vectorhost model for the dengue fever. J. Math. Biol. 35, 523-544 (1997) · <a href="/0878.92025" class="nowrap">Zbl 0878.92025</a> · <a href="https://doi.org/10.1007/s002850050064" class="nowrap">doi:10.1007/s002850050064</a></td> </tr><tr> <td>[29]</td> <td class="space">Sidarto, K.A.: Mathematical model of dengue disease transmission with severe DHF compartment. Bull. Malays. Math. Sci. Soc. 30(2), 143-157 (2007) · <a href="/1132.92023" class="nowrap">Zbl 1132.92023</a></td> </tr><tr> <td>[30]</td> <td class="space">Tang, B., Xiao, Y., Tang, S., Wu, J.: Modelling weekly vector control against dengue in the Guangdong Province of China. J. Theor. Biol. 410, 65-76 (2016) · <a href="/1348.92162" class="nowrap">Zbl 1348.92162</a> · <a href="https://doi.org/10.1016/j.jtbi.2016.09.012" class="nowrap">doi:10.1016/j.jtbi.2016.09.012</a></td> </tr><tr> <td>[31]</td> <td class="space">Amaku, M., Coutinho, F.A., Raimundo, S.M., Lopez, L.F., Nascimento Burattini, M., Massad, E.: A comparative analysis of the relative eficacy of vector-control strategies against dengue fever. Bull. Math. Biol. 76(3), 697-717 (2014) · <a href="/1329.92124" class="nowrap">Zbl 1329.92124</a> · <a href="https://doi.org/10.1007/s11538-014-9939-5" class="nowrap">doi:10.1007/s11538-014-9939-5</a></td> </tr><tr> <td>[32]</td> <td class="space">Tewa, J.J., Dimi, J.L., Bowang, S.: Lypaunov functions for a dengue disease transmission model. Chaos Solitons Fractals (2007). https://doi.org/10.1016/j.chaos.2007.01.069 · <a href="/1197.34099" class="nowrap">Zbl 1197.34099</a> · <a href="https://doi.org/10.1016/j.chaos.2007.01.069" class="nowrap">doi:10.1016/j.chaos.2007.01.069</a></td> </tr><tr> <td>[33]</td> <td class="space">Yang, Y., Xiao, Y.: The effects of population dispersal and pulse vaccination on disease control. Math. Comput. Model. 52, 1591-1604 (2010) · <a href="/1205.34008" class="nowrap">Zbl 1205.34008</a> · <a href="https://doi.org/10.1016/j.mcm.2010.06.024" class="nowrap">doi:10.1016/j.mcm.2010.06.024</a></td> </tr><tr> <td>[34]</td> <td class="space">Zhang, X.B., Xiang, H., Huo, H.F., Meng, X.Y., Xin, Y.: An SIRS epidemic models with pulse vaccinations and non-monotonic incidence rate. Nonlinear Anal. Hybrid Syst. 8, 13-21 (2013) · <a href="/1257.93014" class="nowrap">Zbl 1257.93014</a> · <a href="https://doi.org/10.1016/j.nahs.2012.08.001" class="nowrap">doi:10.1016/j.nahs.2012.08.001</a></td> </tr><tr> <td>[35]</td> <td class="space">Xiao, Y., Zhao, T., Tang, S.: Dynamics of an infectious diseases with media/psychology induced non-smooth incidence. Math. Biosci. Eng. 10, 455-461 (2013) · <a href="/1259.92061" class="nowrap">Zbl 1259.92061</a></td> </tr><tr> <td>[36]</td> <td class="space">Song, P., Lou, Y., Xiao, Y.: A spatial SEIRS reaction – diffusion model in heterogeneous environment. J. Differ. Equ. 267(9), 5084-5114 (2019) · <a href="/1440.35101" class="nowrap">Zbl 1440.35101</a> · <a href="https://doi.org/10.1016/j.jde.2019.05.022" class="nowrap">doi:10.1016/j.jde.2019.05.022</a></td> </tr><tr> <td>[37]</td> <td class="space">Zhang, X.: Modelling the effects of augmentation strategies on the control of dengue fever with an impulsive differential equation. Bull. Math. Biol. 78(10), 1968-2010 (2016) · <a href="/1361.92075" class="nowrap">Zbl 1361.92075</a> · <a href="https://doi.org/10.1007/s11538-016-0208-7" class="nowrap">doi:10.1007/s11538-016-0208-7</a></td> </tr><tr> <td>[38]</td> <td class="space">De la Sen, M., Agarwal, R.P., Ibeas, A., Alonso-Quesada, S.: On a generalized time-varying seir epidemic model with mixed point and distributed time-varying delays and combined regular and impulsive vaccination controls. Adv. Differ. Equ. 2010, 281612 (2010) · <a href="/1219.34104" class="nowrap">Zbl 1219.34104</a> · <a href="https://doi.org/10.1186/1687-1847-2010-281612" class="nowrap">doi:10.1186/1687-1847-2010-281612</a></td> </tr><tr> <td>[39]</td> <td class="space">Song, X., Jiang, Y., Wei, H.: Analysis of a saturation incidence SVEIRS epidemic model with pulse and two time delays. Appl. Math. Comput. 214(2), 381-390 (2009) · <a href="/1168.92326" class="nowrap">Zbl 1168.92326</a></td> </tr><tr> <td>[40]</td> <td class="space">Tang, S., Cheke, R.: Stage-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences. J. Math. Biol. 50, 275-292 (2005) · <a href="/1080.92067" class="nowrap">Zbl 1080.92067</a></td> </tr><tr> <td>[41]</td> <td class="space">Anokhin, A., Berezansky, L., Braverman, E.: Exponential stability of linear delay impulsive differential equations. J. Math. Anal. Appl. 193(3), 923-941 (1995) · <a href="/0837.34076" class="nowrap">Zbl 0837.34076</a> · <a href="https://doi.org/10.1006/jmaa.1995.1275" class="nowrap">doi:10.1006/jmaa.1995.1275</a></td> </tr><tr> <td>[42]</td> <td class="space">Martinez, M.E.: The calendar of epidemics: seasonal cycles of infectious diseases. PLoS Pathogens 14(11), 1007327 (2018) · <a href="https://doi.org/10.1371/journal.ppat.1007327" class="nowrap">doi:10.1371/journal.ppat.1007327</a></td> </tr><tr> <td>[43]</td> <td class="space">Cull, P.: Global stability for population models. Bull. Math. Biol. 43, 47-58 (1981) · <a href="/0451.92011" class="nowrap">Zbl 0451.92011</a> · <a href="https://doi.org/10.1007/BF02460938" class="nowrap">doi:10.1007/BF02460938</a></td> </tr><tr> <td>[44]</td> <td class="space">Zhang, F., Zhao, X.Q.: A periodic epidemic model in a patchy environment. J. Math. Anal. Appl. 325, 496-516 (2007) · <a href="/1101.92046" class="nowrap">Zbl 1101.92046</a> · <a href="https://doi.org/10.1016/j.jmaa.2006.01.085" class="nowrap">doi:10.1016/j.jmaa.2006.01.085</a></td> </tr><tr> <td>[45]</td> <td class="space">Zhao, X.Q.: Dynamical Systems in Population Biology. Springer, New York (2003) · <a href="/1023.37047" class="nowrap">Zbl 1023.37047</a> · <a href="https://doi.org/10.1007/978-0-387-21761-1" class="nowrap">doi:10.1007/978-0-387-21761-1</a></td> </tr><tr> <td>[46]</td> <td class="space">Li, J., Blakeley, D., Smith R.J.: The failure of r $0r_0$. Comput. Math. Methods Med. 13, 224-234 (2011) · <a href="/1227.92045" class="nowrap">Zbl 1227.92045</a></td> </tr><tr> <td>[47]</td> <td class="space">Smith, H.L., Waltman, P.: The Theory of the Chemostat. Cambridge University Press, Cambridge (1995) · <a href="/0860.92031" class="nowrap">Zbl 0860.92031</a> · <a href="https://doi.org/10.1017/CBO9780511530043" class="nowrap">doi:10.1017/CBO9780511530043</a></td> </tr><tr> <td>[48]</td> <td class="space">KKM.2007. Health Facts (2007)</td> </tr><tr> <td>[49]</td> <td class="space">KKRI: dengue fever is still high in South Sulawesi. Tribun Timur Makassar, Newspaper fact (2009)</td> </tr><tr> <td>[50]</td> <td class="space">Wang, W., Zhao, X.Q.: Threshold dynamics for compartmental epidemic models in periodic environments. J. Dyn. Differ. Equ. 20(3), 699-717 (2008) · <a href="/1157.34041" class="nowrap">Zbl 1157.34041</a> · <a href="https://doi.org/10.1007/s10884-008-9111-8" class="nowrap">doi:10.1007/s10884-008-9111-8</a></td> </tr><tr> <td>[51]</td> <td class="space">Hersh, B.S., Tambini, G., Norgueira, A.C., Carrasco, P., de Quadros, C.A.: Review of regional measles surveilleance in the Americas, 1996-1999. Lancet 355, 1943-1948 (2000) · <a href="https://doi.org/10.1016/S0140-6736(00)02325-4" class="nowrap">doi:10.1016/S0140-6736(00)02325-4</a></td> </tr><tr> <td>[52]</td> <td class="space">D’Onofrio, A.: Mixed pulse vaccination strategy in epidemic model with realistic distributed infectious and latent times. Appl. Math. Comput. 151, 181-187 (2004) · <a href="/1043.92033" class="nowrap">Zbl 1043.92033</a></td> </tr><tr> <td>[53]</td> <td class="space">Nokes, D.J., Swinton, J.: The control of a childhood viral infection by pulse vaccination. IMA J. Math. Appl. Med. Biol. 12, 29-53 (1995) · <a href="/0832.92024" class="nowrap">Zbl 0832.92024</a> · <a href="https://doi.org/10.1093/imammb/12.1.29" class="nowrap">doi:10.1093/imammb/12.1.29</a></td> </tr><tr> <td>[54]</td> <td class="space">Nokes, D.J., Swinton, J.: A strategy for global eradication of measles and polio? Trends Microbiol. 5, 14-19 (1997) · <a href="https://doi.org/10.1016/S0966-842X(97)81769-6" class="nowrap">doi:10.1016/S0966-842X(97)81769-6</a></td> </tr><tr> <td>[55]</td> <td class="space">Shulgin, B., Stone, L., Agur, Z.: Theoretical examinations of pulse vaccination policy in the SIR epidemic model. Math. Comput. Model. 30, 207-215 (2000) · <a href="/1043.92527" class="nowrap">Zbl 1043.92527</a></td> </tr><tr> <td>[56]</td> <td class="space">Shulgin, B., Stone, L., Agur, Z.: Pulse vaccination strategy in the SIR epidemic model. Bull. Math. Biol. 60, 1123-1148 (1998) · <a href="/0941.92026" class="nowrap">Zbl 0941.92026</a> · <a href="https://doi.org/10.1016/S0092-8240(98)90005-2" class="nowrap">doi:10.1016/S0092-8240(98)90005-2</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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