
Previous Article
A mathematical model for the seasonal transmission of schistosomiasis in the lake and marshland regions of China
 MBE Home
 This Issue

Next Article
An SEI infection model incorporating media impact
Modeling coinfection of Ixodes tickborne pathogens
1.  Department of Applied Mathematics, The Hong Kong Polytechnic University Hung Hom, Kowloon, Hong Kong, China 
2.  School of Information Engineering, Guangdong Medical University Dongguan, Guangdong 523808, China 
3.  Mathematics and Science College, Shanghai Normal University Shanghai 200234, China 
Ticks, including the Ixodes ricinus and Ixodes scapularis hard tick species, are regarded as the most common arthropod vectors of both human and animal diseases in Europe and the United States capable of transmitting a large number of bacteria, viruses and parasites. Since ticks in larval and nymphal stages share the same host community which can harbor multiple pathogens, they may be coinfected with two or more pathogens, with a subsequent high likelihood of cotransmission to humans or animals. This paper is devoted to the modeling of coinfection of tickborne pathogens, with special focus on the coinfection of Borrelia burgdorferi (agent of Lyme disease) and Babesia microti (agent of human babesiosis). Considering the effect of coinfection, we illustrate that coinfection with B. burgdorferi increases the likelihood of B. microti transmission, by increasing the basic reproduction number of B. microti below the threshold smaller than one to be possibly above the threshold for persistence. The study confirms a mechanism of the ecological fitness paradox, the establishment of B. microti which has weak fitness (basic reproduction number less than one). Furthermore, coinfection could facilitate range expansion of both pathogens.
References:
[1] 
M. E. Adelson, R. V. S. Rao and R. C. Tilton, Prevalence of Borrelia burgdorferi, Bartonella spp., Babesia microti, and Anaplasma phagocytophila in Ixodes scapularis ticks collected in Northern New Jersey, J. Cli. Micro., 42 (2004), 27992801. doi: 10.1128/JCM.42.6.27992801.2004. 
[2] 
F. R. Adler, J. M. PearceDuvet and M. D. Dearing, How host population dynamics translate into timelagged prevalence: An investigation of Sin Nombre virus in deer mice, Bull. Math. Biol., 70 (2008), 236252. doi: 10.1007/s1153800792518. 
[3] 
C. Alvey, Z. Feng and J. Glasser, A model for the coupled disease dynamics of HIV and HSV2 with mixing among and between genders, Math. Biosci., 265 (2015), 82100. doi: 10.1016/j.mbs.2015.04.009. 
[4] 
E. A. Belongia, Epidemiology and impact of coinfections acquired from Ixodes ticks, VectorBorne and Zoonotic Dis., 2 (2002), 265273. 
[5] 
O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts, The construction of nextgeneration matrices for compartmental epidemic models, J. R. Soc. Interface, 5 (2009), 113. doi: 10.1098/rsif.2009.0386. 
[6] 
M. A. DiukWasser, E. Vannier and P. J. Krause, Coinfection by Ixodes tickborne pathogens: Ecological, epidemiological, and clinical consequences, Trends Para., 32 (2016), 3042. doi: 10.1016/j.pt.2015.09.008. 
[7] 
P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 2948. doi: 10.1016/S00255564(02)001086. 
[8] 
G. Fan, Y. Lou, H. R. Thieme and J. Wu, Stability and persistence in ODE models for populations with many stages, Math. Biosci. Eng., 12 (2015), 661686. doi: 10.3934/mbe.2015.12.661. 
[9] 
G. Fan, H. R. Thieme and H. Zhu, Delay differential systems for tick population dynamics, J. Math. Biol., 71 (2015), 10171048. doi: 10.1007/s0028501408450. 
[10] 
D. Gao, Y. Lou and S. Ruan, A periodic RossMacdonald model in a patchy environment, Dis. Cont. Dyn. Syst.B, 19 (2014), 31333145. doi: 10.3934/dcdsb.2014.19.3133. 
[11] 
D. Gao, T. C. Porco and S. Ruan, Coinfection dynamics of two diseases in a single host population, J. Math. Anal. Appl., 442 (2016), 171188. doi: 10.1016/j.jmaa.2016.04.039. 
[12] 
E. J. Goldstein, C. Thompson, A. Spielman and P. J. Krause, Coinfecting deerassociated zoonoses: Lyme disease, babesiosis, and ehrlichiosis, Clin. Inf. Dis., 33 (2001), 676685. 
[13] 
L. Halos, T. Jamal and R. Maillard, Evidence of Bartonella sp. in questing adult and nymphal Ixodes ricinus ticks from France and coinfection with Borrelia burgdorferi sensu lato and Babesia sp., Veterinary Res., 361 (2005), 7987. doi: 10.1051/vetres:2004052. 
[14] 
J. M. Heffernan, Y. Lou and J. Wu, Range expansion of Ixodes scapularis ticks and of Borrelia burgdorferi by migratory birds, Dis. Cont. Dyn. Syst.B, 19 (2014), 31473167. doi: 10.3934/dcdsb.2014.19.3147. 
[15] 
M. H. Hersh, R. S. Ostfeld and D. J. McHenry et al. , Coinfection of blacklegged ticks with Babesia microti and Borrelia burgdorferi is higher than expected and acquired from small mammal hosts PloS one, 9 (2014), e99348. 
[16] 
M. W. Hirsch, H. L. Smith and X.Q. Zhao, Chain transitivity, attractivity and strong repellors for semidynamical systems, J. Dyn. Differ. Equ., 13 (2001), 107131. doi: 10.1023/A:1009044515567. 
[17] 
Y. Lou, J. Wu and X. Wu, Impact of biodiversity and seasonality on Lymepathogen transmission Theo. Biol. Med. Modell. , 11 (2014), 50. 
[18] 
Y. Lou and X.Q. Zhao, Modelling malaria control by introduction of larvivorous fish, Bull. Math. Biol., 73 (2011), 23842407. doi: 10.1007/s1153801196286. 
[19] 
P. D. Mitchell, K. D. Reed and J. M. Hofkes, Immunoserologic evidence of coinfection with Borrelia burgdorferi, Babesia microti, and human granulocytic Ehrlichia species in residents of Wisconsin and Minnesota, J. Cli. Biol., 34 (1996), 724727. 
[20] 
S. Moutailler, C. V. Moro and E. Vaumourin et al. , Coinfection of ticks: The rule rather than the exception PLoS Negl. Trop. Dis. , 10 (2016), e0004539. 
[21] 
J. M. Mutua, F. B. Wang and N. K. Vaidya, Modeling malaria and typhoid fever coinfection dynamics, Math. Biosci., 264 (2015), 128144. doi: 10.1016/j.mbs.2015.03.014. 
[22] 
N. H. Ogden, L. StOnge and I. K. Barker et al. , Risk maps for range expansion of the Lyme disease vector, Ixodes scapularis in Canada now and with climate change, Int. J. Heal. Geog. , 7 (2008), 24. 
[23] 
A. R. Plourde and E. M. Bloch, A literature review of Zika virus, Emerg. Inf. Dise., 22 (2016), 11851192. doi: 10.3201/eid2207.151990. 
[24] 
H. C. Slater, M. Gambhir, P. E. Parham and E. Michael, Modelling coinfection with malaria and lymphatic filariasis PLoS Comput. Biol. , 9 (2013), e1003096, 14pp. 
[25] 
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems Providence, RI: Math. Surveys Monogr. , 41, AMS, 1995. 
[26] 
F. E. Steiner, R. R. Pinger and C. N. Vann, Infection and coinfection rates of Anaplasma phagocytophilum variants, Babesia spp., Borrelia burgdorferi, and the rickettsial endosymbiont in Ixodes scapularis (Acari: Ixodidae) from sites in Indiana, Maine, Pennsylvania, and Wisconsin, J. Med. Entom., 45 (2008), 289297. 
[27] 
G. Stinco and S. Bergamo, Impact of coinfections in Lyme disease, Open Dermatology J., 10 (2016), 5561. doi: 10.2174/1874372201610010055. 
[28] 
S. J. Swanson, D. Neitzel, K. D. Reed and E. A. Belongia, Coinfections acquired from Ixodes ticks, Cli. Micro. Rev., 19 (2006), 708727. doi: 10.1128/CMR.0001106. 
[29] 
B. Tang, Y. Xiao and J. Wu, Implication of vaccination against dengue for Zika outbreak Sci. Rep. , 6 (2016), 35623. 
[30] 
H. R. Thieme, Convergence results and a PoincareBendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755763. 
[31] 
W. Wang and X.Q. Zhao, Spatial invasion threshold of Lyme disease, SIAM J. Appl. Math., 75 (2015), 11421170. doi: 10.1137/140981769. 
[32] 
X. Wu, V. R. Duvvuri and Y. Lou, Developing a temperaturedriven map of the basic reproductive number of the emerging tick vector of Lyme disease Ixodes scapularis in Canada, J. Theo. Biol., 319 (2013), 5061. doi: 10.1016/j.jtbi.2012.11.014. 
[33] 
X. Q. Zhao, Dynamical Systems in Population Biology New York: Springer, 2003. 
[34] 
X.Q. Zhao and Z. Jing, Global asymptotic behavior in some cooperative systems of functionaldifferential equations, Canad. Appl. Math. Quart., 4 (1996), 421444. 
show all references
References:
[1] 
M. E. Adelson, R. V. S. Rao and R. C. Tilton, Prevalence of Borrelia burgdorferi, Bartonella spp., Babesia microti, and Anaplasma phagocytophila in Ixodes scapularis ticks collected in Northern New Jersey, J. Cli. Micro., 42 (2004), 27992801. doi: 10.1128/JCM.42.6.27992801.2004. 
[2] 
F. R. Adler, J. M. PearceDuvet and M. D. Dearing, How host population dynamics translate into timelagged prevalence: An investigation of Sin Nombre virus in deer mice, Bull. Math. Biol., 70 (2008), 236252. doi: 10.1007/s1153800792518. 
[3] 
C. Alvey, Z. Feng and J. Glasser, A model for the coupled disease dynamics of HIV and HSV2 with mixing among and between genders, Math. Biosci., 265 (2015), 82100. doi: 10.1016/j.mbs.2015.04.009. 
[4] 
E. A. Belongia, Epidemiology and impact of coinfections acquired from Ixodes ticks, VectorBorne and Zoonotic Dis., 2 (2002), 265273. 
[5] 
O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts, The construction of nextgeneration matrices for compartmental epidemic models, J. R. Soc. Interface, 5 (2009), 113. doi: 10.1098/rsif.2009.0386. 
[6] 
M. A. DiukWasser, E. Vannier and P. J. Krause, Coinfection by Ixodes tickborne pathogens: Ecological, epidemiological, and clinical consequences, Trends Para., 32 (2016), 3042. doi: 10.1016/j.pt.2015.09.008. 
[7] 
P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 2948. doi: 10.1016/S00255564(02)001086. 
[8] 
G. Fan, Y. Lou, H. R. Thieme and J. Wu, Stability and persistence in ODE models for populations with many stages, Math. Biosci. Eng., 12 (2015), 661686. doi: 10.3934/mbe.2015.12.661. 
[9] 
G. Fan, H. R. Thieme and H. Zhu, Delay differential systems for tick population dynamics, J. Math. Biol., 71 (2015), 10171048. doi: 10.1007/s0028501408450. 
[10] 
D. Gao, Y. Lou and S. Ruan, A periodic RossMacdonald model in a patchy environment, Dis. Cont. Dyn. Syst.B, 19 (2014), 31333145. doi: 10.3934/dcdsb.2014.19.3133. 
[11] 
D. Gao, T. C. Porco and S. Ruan, Coinfection dynamics of two diseases in a single host population, J. Math. Anal. Appl., 442 (2016), 171188. doi: 10.1016/j.jmaa.2016.04.039. 
[12] 
E. J. Goldstein, C. Thompson, A. Spielman and P. J. Krause, Coinfecting deerassociated zoonoses: Lyme disease, babesiosis, and ehrlichiosis, Clin. Inf. Dis., 33 (2001), 676685. 
[13] 
L. Halos, T. Jamal and R. Maillard, Evidence of Bartonella sp. in questing adult and nymphal Ixodes ricinus ticks from France and coinfection with Borrelia burgdorferi sensu lato and Babesia sp., Veterinary Res., 361 (2005), 7987. doi: 10.1051/vetres:2004052. 
[14] 
J. M. Heffernan, Y. Lou and J. Wu, Range expansion of Ixodes scapularis ticks and of Borrelia burgdorferi by migratory birds, Dis. Cont. Dyn. Syst.B, 19 (2014), 31473167. doi: 10.3934/dcdsb.2014.19.3147. 
[15] 
M. H. Hersh, R. S. Ostfeld and D. J. McHenry et al. , Coinfection of blacklegged ticks with Babesia microti and Borrelia burgdorferi is higher than expected and acquired from small mammal hosts PloS one, 9 (2014), e99348. 
[16] 
M. W. Hirsch, H. L. Smith and X.Q. Zhao, Chain transitivity, attractivity and strong repellors for semidynamical systems, J. Dyn. Differ. Equ., 13 (2001), 107131. doi: 10.1023/A:1009044515567. 
[17] 
Y. Lou, J. Wu and X. Wu, Impact of biodiversity and seasonality on Lymepathogen transmission Theo. Biol. Med. Modell. , 11 (2014), 50. 
[18] 
Y. Lou and X.Q. Zhao, Modelling malaria control by introduction of larvivorous fish, Bull. Math. Biol., 73 (2011), 23842407. doi: 10.1007/s1153801196286. 
[19] 
P. D. Mitchell, K. D. Reed and J. M. Hofkes, Immunoserologic evidence of coinfection with Borrelia burgdorferi, Babesia microti, and human granulocytic Ehrlichia species in residents of Wisconsin and Minnesota, J. Cli. Biol., 34 (1996), 724727. 
[20] 
S. Moutailler, C. V. Moro and E. Vaumourin et al. , Coinfection of ticks: The rule rather than the exception PLoS Negl. Trop. Dis. , 10 (2016), e0004539. 
[21] 
J. M. Mutua, F. B. Wang and N. K. Vaidya, Modeling malaria and typhoid fever coinfection dynamics, Math. Biosci., 264 (2015), 128144. doi: 10.1016/j.mbs.2015.03.014. 
[22] 
N. H. Ogden, L. StOnge and I. K. Barker et al. , Risk maps for range expansion of the Lyme disease vector, Ixodes scapularis in Canada now and with climate change, Int. J. Heal. Geog. , 7 (2008), 24. 
[23] 
A. R. Plourde and E. M. Bloch, A literature review of Zika virus, Emerg. Inf. Dise., 22 (2016), 11851192. doi: 10.3201/eid2207.151990. 
[24] 
H. C. Slater, M. Gambhir, P. E. Parham and E. Michael, Modelling coinfection with malaria and lymphatic filariasis PLoS Comput. Biol. , 9 (2013), e1003096, 14pp. 
[25] 
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems Providence, RI: Math. Surveys Monogr. , 41, AMS, 1995. 
[26] 
F. E. Steiner, R. R. Pinger and C. N. Vann, Infection and coinfection rates of Anaplasma phagocytophilum variants, Babesia spp., Borrelia burgdorferi, and the rickettsial endosymbiont in Ixodes scapularis (Acari: Ixodidae) from sites in Indiana, Maine, Pennsylvania, and Wisconsin, J. Med. Entom., 45 (2008), 289297. 
[27] 
G. Stinco and S. Bergamo, Impact of coinfections in Lyme disease, Open Dermatology J., 10 (2016), 5561. doi: 10.2174/1874372201610010055. 
[28] 
S. J. Swanson, D. Neitzel, K. D. Reed and E. A. Belongia, Coinfections acquired from Ixodes ticks, Cli. Micro. Rev., 19 (2006), 708727. doi: 10.1128/CMR.0001106. 
[29] 
B. Tang, Y. Xiao and J. Wu, Implication of vaccination against dengue for Zika outbreak Sci. Rep. , 6 (2016), 35623. 
[30] 
H. R. Thieme, Convergence results and a PoincareBendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755763. 
[31] 
W. Wang and X.Q. Zhao, Spatial invasion threshold of Lyme disease, SIAM J. Appl. Math., 75 (2015), 11421170. doi: 10.1137/140981769. 
[32] 
X. Wu, V. R. Duvvuri and Y. Lou, Developing a temperaturedriven map of the basic reproductive number of the emerging tick vector of Lyme disease Ixodes scapularis in Canada, J. Theo. Biol., 319 (2013), 5061. doi: 10.1016/j.jtbi.2012.11.014. 
[33] 
X. Q. Zhao, Dynamical Systems in Population Biology New York: Springer, 2003. 
[34] 
X.Q. Zhao and Z. Jing, Global asymptotic behavior in some cooperative systems of functionaldifferential equations, Canad. Appl. Math. Quart., 4 (1996), 421444. 
Variable  Meaning 
 number of eggs 
 number of questing larvae 
 number of feeding larvae susceptible to both Ba and Bo 
 number of feeding larvae infected with Bo only 
 number of feeding larvae infected with Ba only 
 number of feeding larvae coinfected with Ba and Bo 
 number of questing nymphs susceptible to both Ba and Bo 
 number of questing nymphs infected with Bo only 
 number of questing nymphs infected with Ba only 
number of questing nymphs coinfected with Ba and Bo  
 number of feeding nymphs susceptible to both Ba and Bo 
 number of feeding nymphs infected with Bo only 
 number of feeding nymphs infected with Ba only 
 number of feeding nymphs coinfected with Ba and Bo 
 number of adults susceptible to both Ba and Bo 
 number of adults infected with Bo only 
 number of adults infected with Ba only 
 number of adults coinfected with Ba and Bo 
 number of mice susceptible to both Ba and Bo 
 number of mice infected with Bo only 
 number of mice infected with Ba only 
 number of mice coinfected with Ba and Bo 
Variable  Meaning 
 number of eggs 
 number of questing larvae 
 number of feeding larvae susceptible to both Ba and Bo 
 number of feeding larvae infected with Bo only 
 number of feeding larvae infected with Ba only 
 number of feeding larvae coinfected with Ba and Bo 
 number of questing nymphs susceptible to both Ba and Bo 
 number of questing nymphs infected with Bo only 
 number of questing nymphs infected with Ba only 
number of questing nymphs coinfected with Ba and Bo  
 number of feeding nymphs susceptible to both Ba and Bo 
 number of feeding nymphs infected with Bo only 
 number of feeding nymphs infected with Ba only 
 number of feeding nymphs coinfected with Ba and Bo 
 number of adults susceptible to both Ba and Bo 
 number of adults infected with Bo only 
 number of adults infected with Ba only 
 number of adults coinfected with Ba and Bo 
 number of mice susceptible to both Ba and Bo 
 number of mice infected with Bo only 
 number of mice infected with Ba only 
 number of mice coinfected with Ba and Bo 
Symbol  Description  Value  Ref 
 mortality rate of mice  0.01  [2] 
 birth rate of mice  0.02  [2] 
 densitydependent death rate of mice   AS 
 egg reproduction rate   [17] 
 mortality rate of eggs  0.0025  [17] 
 mortality rate of questing larvae  0.006  [17] 
 mortality rate of feeding larvae  0.038  [17] 
 mortality rate of questing nymphs  0.006  [17] 
 mortality rate of feeding nymphs  0.028  [17] 
 mortality rate of adults  0.01  [17] 
 development rate of eggs   [17] 
 development rate of larvae   [17] 
 development rate of nymphs   [17] 
 feeding rate of larvae   [17] 
 feeding rate of nymphs   [17] 
 densitydependent mortality rate of   AS 
 densitydependent mortality rate of   AS 
 TP of Bo from  0.6  [17] 
 TP of Bo from   AS 
 TP of Ba from  0.45  AS 
 TP of Ba from   AS 
 TP of both pathogens from   AS 
 TP of Bo from   AS 
 TP of Bo from   AS 
 TP of Ba from   AS 
 TP of Ba from   AS 
 TP of both pathogens from   AS 
 TP of Ba from   AS 
 TP of both pathogens from   AS 
 TP of Bo from   AS 
 TP of both pathogens from   AS 
 TP of Bo from  0.6  AS 
 TP of Bo from   AS 
 TP of Ba from   AS 
 TP of Ba from   AS 
 TP of both pathogen from   AS 
 TP of Ba from   AS 
 TP of Ba from   AS 
 TP of Bo from   AS 
 TP of Bo from   AS 
Symbol  Description  Value  Ref 
 mortality rate of mice  0.01  [2] 
 birth rate of mice  0.02  [2] 
 densitydependent death rate of mice   AS 
 egg reproduction rate   [17] 
 mortality rate of eggs  0.0025  [17] 
 mortality rate of questing larvae  0.006  [17] 
 mortality rate of feeding larvae  0.038  [17] 
 mortality rate of questing nymphs  0.006  [17] 
 mortality rate of feeding nymphs  0.028  [17] 
 mortality rate of adults  0.01  [17] 
 development rate of eggs   [17] 
 development rate of larvae   [17] 
 development rate of nymphs   [17] 
 feeding rate of larvae   [17] 
 feeding rate of nymphs   [17] 
 densitydependent mortality rate of   AS 
 densitydependent mortality rate of   AS 
 TP of Bo from  0.6  [17] 
 TP of Bo from   AS 
 TP of Ba from  0.45  AS 
 TP of Ba from   AS 
 TP of both pathogens from   AS 
 TP of Bo from   AS 
 TP of Bo from   AS 
 TP of Ba from   AS 
 TP of Ba from   AS 
 TP of both pathogens from   AS 
 TP of Ba from   AS 
 TP of both pathogens from   AS 
 TP of Bo from   AS 
 TP of both pathogens from   AS 
 TP of Bo from  0.6  AS 
 TP of Bo from   AS 
 TP of Ba from   AS 
 TP of Ba from   AS 
 TP of both pathogen from   AS 
 TP of Ba from   AS 
 TP of Ba from   AS 
 TP of Bo from   AS 
 TP of Bo from   AS 
[1] 
Zindoga Mukandavire, Abba B. Gumel, Winston Garira, Jean Michel Tchuenche. Mathematical analysis of a model for HIVmalaria coinfection. Mathematical Biosciences & Engineering, 2009, 6 (2) : 333362. doi: 10.3934/mbe.2009.6.333 
[2] 
Holly Gaff. Preliminary analysis of an agentbased model for a tickborne disease. Mathematical Biosciences & Engineering, 2011, 8 (2) : 463473. doi: 10.3934/mbe.2011.8.463 
[3] 
Shangbing Ai. Global stability of equilibria in a tickborne disease model. Mathematical Biosciences & Engineering, 2007, 4 (4) : 567572. doi: 10.3934/mbe.2007.4.567 
[4] 
Georgi Kapitanov. A double agestructured model of the coinfection of tuberculosis and HIV. Mathematical Biosciences & Engineering, 2015, 12 (1) : 2340. doi: 10.3934/mbe.2015.12.23 
[5] 
Holly Gaff, Robyn Nadolny. Identifying requirements for the invasion of a tick species and tickborne pathogen through TICKSIM. Mathematical Biosciences & Engineering, 2013, 10 (3) : 625635. doi: 10.3934/mbe.2013.10.625 
[6] 
Kazeem Oare Okosun, Robert Smith?. Optimal control analysis of malaria–schistosomiasis coinfection dynamics. Mathematical Biosciences & Engineering, 2017, 14 (2) : 377405. doi: 10.3934/mbe.2017024 
[7] 
A. K. Misra, Anupama Sharma, Jia Li. A mathematical model for control of vector borne diseases through media campaigns. Discrete & Continuous Dynamical Systems  B, 2013, 18 (7) : 19091927. doi: 10.3934/dcdsb.2013.18.1909 
[8] 
Marcello Delitala, Tommaso Lorenzi. Emergence of spatial patterns in a mathematical model for the coculture dynamics of epitheliallike and mesenchymallike cells. Mathematical Biosciences & Engineering, 2017, 14 (1) : 7993. doi: 10.3934/mbe.2017006 
[9] 
Xuejuan Lu, Lulu Hui, Shengqiang Liu, Jia Li. A mathematical model of HTLVI infection with two time delays. Mathematical Biosciences & Engineering, 2015, 12 (3) : 431449. doi: 10.3934/mbe.2015.12.431 
[10] 
Suxia Zhang, Xiaxia Xu. A mathematical model for hepatitis B with infectionage structure. Discrete & Continuous Dynamical Systems  B, 2016, 21 (4) : 13291346. doi: 10.3934/dcdsb.2016.21.1329 
[11] 
Expeditho Mtisi, Herieth Rwezaura, Jean Michel Tchuenche. A mathematical analysis of malaria and tuberculosis codynamics. Discrete & Continuous Dynamical Systems  B, 2009, 12 (4) : 827864. doi: 10.3934/dcdsb.2009.12.827 
[12] 
Wandi Ding. Optimal control on hybrid ODE Systems with application to a tick disease model. Mathematical Biosciences & Engineering, 2007, 4 (4) : 633659. doi: 10.3934/mbe.2007.4.633 
[13] 
Laura Fumanelli, Pierre Magal, Dongmei Xiao, Xiao Yu. Qualitative analysis of a model for coculture of bacteria and amoebae. Mathematical Biosciences & Engineering, 2012, 9 (2) : 259279. doi: 10.3934/mbe.2012.9.259 
[14] 
Jinliang Wang, Xiu Dong. Analysis of an HIV infection model incorporating latency age and infection age. Mathematical Biosciences & Engineering, 2018, 15 (3) : 569594. doi: 10.3934/mbe.2018026 
[15] 
José Ignacio Tello. Mathematical analysis of a model of morphogenesis. Discrete & Continuous Dynamical Systems  A, 2009, 25 (1) : 343361. doi: 10.3934/dcds.2009.25.343 
[16] 
Avner Friedman, Chuan Xue. A mathematical model for chronic wounds. Mathematical Biosciences & Engineering, 2011, 8 (2) : 253261. doi: 10.3934/mbe.2011.8.253 
[17] 
Yuming Chen, Junyuan Yang, Fengqin Zhang. The global stability of an SIRS model with infection age. Mathematical Biosciences & Engineering, 2014, 11 (3) : 449469. doi: 10.3934/mbe.2014.11.449 
[18] 
Fred Brauer, Zhisheng Shuai, P. van den Driessche. Dynamics of an ageofinfection cholera model. Mathematical Biosciences & Engineering, 2013, 10 (5/6) : 13351349. doi: 10.3934/mbe.2013.10.1335 
[19] 
Jinliang Wang, Ran Zhang, Toshikazu Kuniya. A note on dynamics of an ageofinfection cholera model. Mathematical Biosciences & Engineering, 2016, 13 (1) : 227247. doi: 10.3934/mbe.2016.13.227 
[20] 
Avner Friedman, Najat Ziyadi, Khalid Boushaba. A model of drug resistance with infection by health care workers. Mathematical Biosciences & Engineering, 2010, 7 (4) : 779792. doi: 10.3934/mbe.2010.7.779 
2016 Impact Factor: 1.035
Tools
Metrics
Other articles
by authors
[Back to Top]