The effect of sexual transmission on Zika virus dynamics. (English) Zbl 1404.92202
Summary: Zika virus is a human disease that may lead to neurological disorders in affected individuals, and may be transmitted vectorially (by mosquitoes) or sexually. A mathematical model of Zika virus transmission is formulated, taking into account mosquitoes, sexually active males and females, inactive individuals, and considering both vector transmission and sexual transmission from infectious males to susceptible females. Basic reproduction numbers are computed, and disease control strategies are evaluated. The effect of the incidence function used to model sexual transmission from infectious males to susceptible females is investigated. It is proved that for such functions that are sublinear, if the basic reproduction \(\mathcal{R}_0<1\), then the disease dies out and \(\mathcal{R}_0=1\) is a sharp threshold. Moreover, under certain conditions on model parameters and assuming mass action incidence for sexual transmission, it is proved that if \(\mathcal{R}_0>1\), there exists a unique endemic equilibrium that is globally asymptotically stable. However, under nonlinear incidence, it is shown that for certain functions backward bifurcation and Hopf bifurcation may occur, giving rise to subthreshold equilibria and periodic solutions, respectively. Numerical simulations for various parameter values are displayed to illustrate these behaviours.
MSC:
| 92D30 | Epidemiology |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34D23 | Global stability of solutions to ordinary differential equations |
References:
| [1] | Agusto, FB; Bewick, S.; Fagan, WF, Mathematical model for Zika virus dynamics with sexual transmission route, Ecol Complex, 29, 61-81, (2017) · doi:10.1016/j.ecocom.2016.12.007 |
| [2] | Agusto, FB; Bewick, S.; Fagan, WF, Mathematical model of Zika virus with vertical transmission, Infect Dis Model, 2, 244-267, (2017) |
| [3] | Atkinson, B.; Hearn, P.; Afrough, B.; Lumley, S.; Carter, D.; Aarons, E.; Simpson, AJ; Brooks, TJ; Hewson, R., Detection of Zika virus in semen, Emerg Infect Dis, 22, 940, (2016) · doi:10.3201/eid2205.160107 |
| [4] | Baca-Carrasco, D.; Velasco-Hernández, JX, Sex, mosquitoes and epidemics: an evaluation of Zika disease dynamics, Bull Math Biol, 78, 2228-2242, (2016) · Zbl 1357.92067 · doi:10.1007/s11538-016-0219-4 |
| [5] | Brooks, JT; Friedman, A.; Kachur, RE; LaFlam, M.; Peters, PJ; Jamieson, DJ, Update: interim guidance for prevention of sexual transmission of Zika virus-United States, July 2016, MMWR Morb Mortal Wkly Rep, 65, 745, (2016) · doi:10.15585/mmwr.mm6529e2 |
| [6] | Broutet, N.; Krauer, F.; Riesen, M.; Khalakdina, A.; Almiron, M.; Aldighieri, S.; Espinal, M.; Low, N.; Dye, C., Zika virus as a cause of neurologic disorders, New Engl J Med, 374, 1506-1509, (2016) · doi:10.1056/NEJMp1602708 |
| [7] | Campos, GS; Bandeira, AC; Sardi, SI, Zika virus outbreak, Bahia, Brazil, Emerg Infect Dis, 21, 1885-1886, (2015) · doi:10.3201/eid2110.150847 |
| [8] | Capasso, V.; Serio, G., A generalization of the Kermack-McKendrick deterministic epidemic model, Math Biosci, 42, 43-61, (1978) · Zbl 0398.92026 · doi:10.1016/0025-5564(78)90006-8 |
| [9] | Chitnis, N.; Hyman, JM; Cushing, JM, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull Math Biol, 70, 1272-1296, (2008) · Zbl 1142.92025 · doi:10.1007/s11538-008-9299-0 |
| [10] | Cushing, JM; Diekmann, O., The many guises of R0 (a didactic note), J Theor Biol, 404, 295-302, (2016) · Zbl 1343.92400 · doi:10.1016/j.jtbi.2016.06.017 |
| [11] | Davidson, A.; Slavinski, S.; Komoto, K.; Rakeman, J.; Weiss, D., Suspected female-to-male sexual transmission of Zika virus-New York City, 2016, MMWR Morb Mortal Wkly Rep, 65, 716, (2016) · doi:10.15585/mmwr.mm6528e2 |
| [12] | Faria, NR; Silva Azevedo, RdS; Kraemer, MUG; Souza, R.; Cunha, MS; Hill, SC; Thézé, J.; Bonsall, MB; Bowden, TA; Rissanen, I.; etal., Zika virus in the Americas: early epidemiological and genetic findings, Science, 352, 345-349, (2016) · doi:10.1126/science.aaf5036 |
| [13] | Gao, D.; Lou, Y.; He, D.; Porco, TC; Kuang, Y.; Chowell, G.; Ruan, S., Prevention and control of Zika as a mosquito-borne and sexually transmitted disease: a mathematical modeling analysis, Sci Rep, (2016) · doi:10.1038/srep28070 |
| [14] | Hadeler, KP; Castillo-Chávez, C., A core group model for disease transmission, Math Biosci, 128, 41-55, (1995) · Zbl 0832.92021 · doi:10.1016/0025-5564(94)00066-9 |
| [15] | Heesterbeek, JAP; Roberts, MG, The type-reproduction number \(T\) in models for infectious disease control, Math Biosci, 206, 3-10, (2007) · Zbl 1124.92043 · doi:10.1016/j.mbs.2004.10.013 |
| [16] | Hethcote, HW; Driessche, P., Some epidemiological models with nonlinear incidence, J Math Biol, 29, 271-287, (1991) · Zbl 0722.92015 · doi:10.1007/BF00160539 |
| [17] | Korobeinikov, A.; Maini, PK, Non-linear incidence and stability of infectious disease models, Math Med Biol, 22, 113-128, (2005) · Zbl 1076.92048 · doi:10.1093/imammb/dqi001 |
| [18] | Kucharski, AJ; Funk, S.; Eggo, RM; Mallet, HP; Edmunds, WJ; Nilles, EJ, Transmission dynamics of Zika virus in island populations: a modelling analysis of the 2013-14 French Polynesia outbreak, PLoS Negl Trop Dis, 10, e0004,726, (2016) · doi:10.1371/journal.pntd.0004726 |
| [19] | LaSalle JP (1976) The stability of dynamical systems. In: CBMS-NSF regional conference series in applied mathematics, vol 25. SIAM · Zbl 0364.93002 |
| [20] | Liu, WL; Levin, SA; Iwasa, Y., Influence of nonlinear incidence rates upon the behavior of \(SIRS\) epidemiological models, J Math Biol, 23, 187-204, (1986) · Zbl 0582.92023 · doi:10.1007/BF00276956 |
| [21] | Liu, WL; Hethcote, HW; Levin, SA, Dynamical behavior of epidemiological models with nonlinear incidence rates, J Math Biol, 25, 359-380, (1987) · Zbl 0621.92014 · doi:10.1007/BF00277162 |
| [22] | Manore, C.; Hickmann, KS; Xu, S.; Hyman, HJ, Comparing dengue and chikungunya emergence and endemic transmission in A. aegypti and A. albopictus, J Theor Biol, 356, 174-191, (2014) · Zbl 1412.92292 · doi:10.1016/j.jtbi.2014.04.033 |
| [23] | Martcheva M (2015) An introduction to mathematical epidemiology, texts in applied mathematics, vol 61. Springer, Berlin · Zbl 1333.92006 · doi:10.1007/978-1-4899-7612-3 |
| [24] | Maxian, O.; Neufeld, A.; Talis, EJ; Childs, LM; Blackwood, JC, Zika virus dynamics: When does sexual transmission matter?, Epidemics, 21, 48-55, (2017) · doi:10.1016/j.epidem.2017.06.003 |
| [25] | McCallum, H.; Barlow, N.; Hone, J., How should pathogen transmission be modelled?, Trends Ecol Evol, 16, 295-300, (2001) · doi:10.1016/S0169-5347(01)02144-9 |
| [26] | McGrath, EL; Rossi, SL; Gao, J.; Widen, SG; Grant, AC; Dunn, TJ; Azar, SR; Roundy, CM; Xiong, Y.; Prusak, DJ; etal., Differential responses of human fetal brain neural stem cells to Zika virus infection, Stem Cell Rep, 8, 715-727, (2017) · doi:10.1016/j.stemcr.2017.01.008 |
| [27] | McMeniman, CJ; Lane, RV; Cass, BN; Fong, AW; Sidhu, M.; Wang, YF; O’Neill, SL, Stable introduction of a life-shortening Wolbachia infection into the mosquito Aedes aegypti, Science, 323, 141-144, (2009) · doi:10.1126/science.1165326 |
| [28] | Nicastri E, Castilletti C, Liuzzi G, Iannetta M, Capobianchi MR, Ippolito G (2016) Persistent detection of Zika virus RNA in semen for six months after symptom onset in a traveller returning from Haiti to Italy, February 2016. Eurosurveillance 21(32):30314 |
| [29] | Parra, B.; Lizarazo, J.; Jiménez-Arango, JA; Zea-Vera, AF; González-Manrique, G.; Vargas, J.; Angarita, JA; Zuñiga, G.; Lopez-Gonzalez, R.; Beltran, CL; etal., Guillain-Barré syndrome associated with Zika virus infection in Colombia, New Engl J Med, 375, 1513-1523, (2016) · doi:10.1056/NEJMoa1605564 |
| [30] | Petersen, EE; Meaney-Delman, D.; Neblett-Fanfair, R.; Havers, F.; Oduyebo, T.; Hills, SL; Rabe, IB; Lambert, A.; Abercrombie, J.; Martin, SW; etal., Update: interim guidance for preconception counseling and prevention of sexual transmission of Zika virus for persons with possible Zika virus exposure-United States, september 2016, MMWR Morb Mortal Wkly Rep, 65, 1077, (2016) · doi:10.15585/mmwr.mm6539e1 |
| [31] | Rasmussen, SA; Jamieson, DJ; Honein, MA; Petersen, LR, Zika virus and birth defects reviewing the evidence for causality, New Engl J Med, 374, 1981-1987, (2016) · doi:10.1056/NEJMsr1604338 |
| [32] | Roberts, MG; Heesterbeek, JAP, A new method for estimating the effort required to control an infectious disease, Proc R Soc Lond B Biol Sci, 270, 1359-1364, (2003) · doi:10.1098/rspb.2003.2339 |
| [33] | Saad-Roy, CM; Driessche, P.; Ma, J., Estimation of Zika virus prevalence by appearance of microcephaly, BMC Infect Dis, 16, 754, (2016) · doi:10.1186/s12879-016-2076-z |
| [34] | Shuai, Z.; Driessche, P., Global stability of infectious disease models using Lyapunov functions, SIAM J Appl Math, 73, 1513-1532, (2013) · Zbl 1308.34072 · doi:10.1137/120876642 |
| [35] | Shuai, Z.; Heesterbeek, JAP; Driessche, P., Extending the type reproduction number to infectious disease control targeting contacts between types, J Math Biol, 67, 1067-1082, (2013) · Zbl 1278.92031 · doi:10.1007/s00285-012-0579-9 |
| [36] | Towers, S.; Brauer, F.; Castillo-Chavez, C.; Falconar, AKI; Mubayi, A.; Romero-Vivas, CME, Estimate of the reproduction number of the 2015 Zika virus outbreak in Barranquilla, Colombia, and estimation of the relative role of sexual transmission, Epidemics, 17, 50-55, (2016) · doi:10.1016/j.epidem.2016.10.003 |
| [37] | Driessche, P.; Watmough, J., A simple SIS epidemic model with a backward bifurcation, J Math Biol, 40, 525-540, (2000) · Zbl 0961.92029 · doi:10.1007/s002850000032 |
| [38] | Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math Biosci, 180, 29-48, (2002) · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6 |
| [39] | Zhang, W.; Wahl, LM; Yu, P., Backward bifurcations, turning points and rich dynamics in simple disease models, J Math Biol, 73, 947-976, (2016) · Zbl 1354.92098 · doi:10.1007/s00285-016-0976-6 |
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. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
