October  2017, 14(5&6): 1301-1316. doi: 10.3934/mbe.2017067

Modeling co-infection of Ixodes tick-borne 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

* Corresponding authorr

Received  August 06, 2016 Revised  December 30, 2016 Published  May 2017

Fund Project: YL is partially supported by NSFC (11301442) and RGC (PolyU 253004/14P). DG is partially supported by NSFC (11601336), Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning (TP2015050), Shanghai Gaofeng Project for University Academic Development Program.

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 co-infected with two or more pathogens, with a subsequent high likelihood of co-transmission to humans or animals. This paper is devoted to the modeling of co-infection of tick-borne pathogens, with special focus on the co-infection of Borrelia burgdorferi (agent of Lyme disease) and Babesia microti (agent of human babesiosis). Considering the effect of co-infection, we illustrate that co-infection 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, co-infection could facilitate range expansion of both pathogens.

Citation: Yijun Lou, Li Liu, Daozhou Gao. Modeling co-infection of Ixodes tick-borne pathogens. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1301-1316. doi: 10.3934/mbe.2017067
References:
[1]

M. E. AdelsonR. 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), 2799-2801.  doi: 10.1128/JCM.42.6.2799-2801.2004.  Google Scholar

[2]

F. R. AdlerJ. M. Pearce-Duvet and M. D. Dearing, How host population dynamics translate into time-lagged prevalence: An investigation of Sin Nombre virus in deer mice, Bull. Math. Biol., 70 (2008), 236-252.  doi: 10.1007/s11538-007-9251-8.  Google Scholar

[3]

C. AlveyZ. Feng and J. Glasser, A model for the coupled disease dynamics of HIV and HSV-2 with mixing among and between genders, Math. Biosci., 265 (2015), 82-100.  doi: 10.1016/j.mbs.2015.04.009.  Google Scholar

[4]

E. A. Belongia, Epidemiology and impact of coinfections acquired from Ixodes ticks, Vector-Borne and Zoonotic Dis., 2 (2002), 265-273.   Google Scholar

[5]

O. DiekmannJ. A. P. Heesterbeek and M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models, J. R. Soc. Interface, 5 (2009), 1-13.  doi: 10.1098/rsif.2009.0386.  Google Scholar

[6]

M. A. Diuk-WasserE. Vannier and P. J. Krause, Coinfection by Ixodes tick-borne pathogens: Ecological, epidemiological, and clinical consequences, Trends Para., 32 (2016), 30-42.  doi: 10.1016/j.pt.2015.09.008.  Google Scholar

[7]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[8]

G. FanY. LouH. R. Thieme and J. Wu, Stability and persistence in ODE models for populations with many stages, Math. Biosci. Eng., 12 (2015), 661-686.  doi: 10.3934/mbe.2015.12.661.  Google Scholar

[9]

G. FanH. R. Thieme and H. Zhu, Delay differential systems for tick population dynamics, J. Math. Biol., 71 (2015), 1017-1048.  doi: 10.1007/s00285-014-0845-0.  Google Scholar

[10]

D. GaoY. Lou and S. Ruan, A periodic Ross-Macdonald model in a patchy environment, Dis. Cont. Dyn. Syst.-B, 19 (2014), 3133-3145.  doi: 10.3934/dcdsb.2014.19.3133.  Google Scholar

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D. GaoT. C. Porco and S. Ruan, Coinfection dynamics of two diseases in a single host population, J. Math. Anal. Appl., 442 (2016), 171-188.  doi: 10.1016/j.jmaa.2016.04.039.  Google Scholar

[12]

E. J. GoldsteinC. ThompsonA. Spielman and P. J. Krause, Coinfecting deer-associated zoonoses: Lyme disease, babesiosis, and ehrlichiosis, Clin. Inf. Dis., 33 (2001), 676-685.   Google Scholar

[13]

L. HalosT. Jamal and R. Maillard, Evidence of Bartonella sp. in questing adult and nymphal Ixodes ricinus ticks from France and co-infection with Borrelia burgdorferi sensu lato and Babesia sp., Veterinary Res., 361 (2005), 79-87.  doi: 10.1051/vetres:2004052.  Google Scholar

[14]

J. M. HeffernanY. Lou and J. Wu, Range expansion of Ixodes scapularis ticks and of Borrelia burgdorferi by migratory birds, Dis. Cont. Dyn. Syst.-B, 19 (2014), 3147-3167.  doi: 10.3934/dcdsb.2014.19.3147.  Google Scholar

[15]

M. H. Hersh, R. S. Ostfeld and D. J. McHenry et al., Co-infection of blacklegged ticks with Babesia microti and Borrelia burgdorferi is higher than expected and acquired from small mammal hosts, PloS one, 9 (2014), e99348. doi: 10.1371/journal.pone.0099348.  Google Scholar

[16]

M. W. HirschH. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity and strong repellors for semidynamical systems, J. Dyn. Differ. Equ., 13 (2001), 107-131.  doi: 10.1023/A:1009044515567.  Google Scholar

[17]

Y. Lou, J. Wu and X. Wu, Impact of biodiversity and seasonality on Lyme-pathogen transmission, Theo. Biol. Med. Modell., 11 (2014), 50. doi: 10.1186/1742-4682-11-50.  Google Scholar

[18]

Y. Lou and X.-Q. Zhao, Modelling malaria control by introduction of larvivorous fish, Bull. Math. Biol., 73 (2011), 2384-2407.  doi: 10.1007/s11538-011-9628-6.  Google Scholar

[19]

P. D. MitchellK. 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), 724-727.   Google Scholar

[20]

S. Moutailler, C. V. Moro and E. Vaumourin et al., Co-infection of ticks: The rule rather than the exception, PLoS Negl. Trop. Dis., 10 (2016), e0004539. doi: 10.1371/journal.pntd.0004539.  Google Scholar

[21]

J. M. MutuaF. B. Wang and N. K. Vaidya, Modeling malaria and typhoid fever co-infection dynamics, Math. Biosci., 264 (2015), 128-144.  doi: 10.1016/j.mbs.2015.03.014.  Google Scholar

[22]

N. H. Ogden, L. St-Onge 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. doi: 10.1186/1476-072X-7-24.  Google Scholar

[23]

A. R. Plourde and E. M. Bloch, A literature review of Zika virus, Emerg. Inf. Dise., 22 (2016), 1185-1192.  doi: 10.3201/eid2207.151990.  Google Scholar

[24]

H. C. Slater, M. Gambhir, P. E. Parham and E. Michael, Modelling co-infection with malaria and lymphatic filariasis, PLoS Comput. Biol., 9 (2013), e1003096, 14pp. doi: 10.1371/journal.pcbi.1003096.  Google Scholar

[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.  Google Scholar

[26]

F. E. SteinerR. R. Pinger and C. N. Vann, Infection and co-infection 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), 289-297.   Google Scholar

[27]

G. Stinco and S. Bergamo, Impact of co-infections in Lyme disease, Open Dermatology J., 10 (2016), 55-61.  doi: 10.2174/1874372201610010055.  Google Scholar

[28]

S. J. SwansonD. NeitzelK. D. Reed and E. A. Belongia, Coinfections acquired from Ixodes ticks, Cli. Micro. Rev., 19 (2006), 708-727.  doi: 10.1128/CMR.00011-06.  Google Scholar

[29]

B. Tang, Y. Xiao and J. Wu, Implication of vaccination against dengue for Zika outbreak, Sci. Rep., 6 (2016), 35623. doi: 10.1038/srep35623.  Google Scholar

[30]

H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.  Google Scholar

[31]

W. Wang and X.-Q. Zhao, Spatial invasion threshold of Lyme disease, SIAM J. Appl. Math., 75 (2015), 1142-1170.  doi: 10.1137/140981769.  Google Scholar

[32]

X. WuV. R. Duvvuri and Y. Lou, Developing a temperature-driven map of the basic reproductive number of the emerging tick vector of Lyme disease Ixodes scapularis in Canada, J. Theo. Biol., 319 (2013), 50-61.  doi: 10.1016/j.jtbi.2012.11.014.  Google Scholar

[33]

X. -Q. Zhao, Dynamical Systems in Population Biology, New York: Springer, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

[34]

X.-Q. Zhao and Z. Jing, Global asymptotic behavior in some cooperative systems of functional-differential equations, Canad. Appl. Math. Quart., 4 (1996), 421-444.   Google Scholar

show all references

References:
[1]

M. E. AdelsonR. 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), 2799-2801.  doi: 10.1128/JCM.42.6.2799-2801.2004.  Google Scholar

[2]

F. R. AdlerJ. M. Pearce-Duvet and M. D. Dearing, How host population dynamics translate into time-lagged prevalence: An investigation of Sin Nombre virus in deer mice, Bull. Math. Biol., 70 (2008), 236-252.  doi: 10.1007/s11538-007-9251-8.  Google Scholar

[3]

C. AlveyZ. Feng and J. Glasser, A model for the coupled disease dynamics of HIV and HSV-2 with mixing among and between genders, Math. Biosci., 265 (2015), 82-100.  doi: 10.1016/j.mbs.2015.04.009.  Google Scholar

[4]

E. A. Belongia, Epidemiology and impact of coinfections acquired from Ixodes ticks, Vector-Borne and Zoonotic Dis., 2 (2002), 265-273.   Google Scholar

[5]

O. DiekmannJ. A. P. Heesterbeek and M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models, J. R. Soc. Interface, 5 (2009), 1-13.  doi: 10.1098/rsif.2009.0386.  Google Scholar

[6]

M. A. Diuk-WasserE. Vannier and P. J. Krause, Coinfection by Ixodes tick-borne pathogens: Ecological, epidemiological, and clinical consequences, Trends Para., 32 (2016), 30-42.  doi: 10.1016/j.pt.2015.09.008.  Google Scholar

[7]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[8]

G. FanY. LouH. R. Thieme and J. Wu, Stability and persistence in ODE models for populations with many stages, Math. Biosci. Eng., 12 (2015), 661-686.  doi: 10.3934/mbe.2015.12.661.  Google Scholar

[9]

G. FanH. R. Thieme and H. Zhu, Delay differential systems for tick population dynamics, J. Math. Biol., 71 (2015), 1017-1048.  doi: 10.1007/s00285-014-0845-0.  Google Scholar

[10]

D. GaoY. Lou and S. Ruan, A periodic Ross-Macdonald model in a patchy environment, Dis. Cont. Dyn. Syst.-B, 19 (2014), 3133-3145.  doi: 10.3934/dcdsb.2014.19.3133.  Google Scholar

[11]

D. GaoT. C. Porco and S. Ruan, Coinfection dynamics of two diseases in a single host population, J. Math. Anal. Appl., 442 (2016), 171-188.  doi: 10.1016/j.jmaa.2016.04.039.  Google Scholar

[12]

E. J. GoldsteinC. ThompsonA. Spielman and P. J. Krause, Coinfecting deer-associated zoonoses: Lyme disease, babesiosis, and ehrlichiosis, Clin. Inf. Dis., 33 (2001), 676-685.   Google Scholar

[13]

L. HalosT. Jamal and R. Maillard, Evidence of Bartonella sp. in questing adult and nymphal Ixodes ricinus ticks from France and co-infection with Borrelia burgdorferi sensu lato and Babesia sp., Veterinary Res., 361 (2005), 79-87.  doi: 10.1051/vetres:2004052.  Google Scholar

[14]

J. M. HeffernanY. Lou and J. Wu, Range expansion of Ixodes scapularis ticks and of Borrelia burgdorferi by migratory birds, Dis. Cont. Dyn. Syst.-B, 19 (2014), 3147-3167.  doi: 10.3934/dcdsb.2014.19.3147.  Google Scholar

[15]

M. H. Hersh, R. S. Ostfeld and D. J. McHenry et al., Co-infection of blacklegged ticks with Babesia microti and Borrelia burgdorferi is higher than expected and acquired from small mammal hosts, PloS one, 9 (2014), e99348. doi: 10.1371/journal.pone.0099348.  Google Scholar

[16]

M. W. HirschH. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity and strong repellors for semidynamical systems, J. Dyn. Differ. Equ., 13 (2001), 107-131.  doi: 10.1023/A:1009044515567.  Google Scholar

[17]

Y. Lou, J. Wu and X. Wu, Impact of biodiversity and seasonality on Lyme-pathogen transmission, Theo. Biol. Med. Modell., 11 (2014), 50. doi: 10.1186/1742-4682-11-50.  Google Scholar

[18]

Y. Lou and X.-Q. Zhao, Modelling malaria control by introduction of larvivorous fish, Bull. Math. Biol., 73 (2011), 2384-2407.  doi: 10.1007/s11538-011-9628-6.  Google Scholar

[19]

P. D. MitchellK. 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), 724-727.   Google Scholar

[20]

S. Moutailler, C. V. Moro and E. Vaumourin et al., Co-infection of ticks: The rule rather than the exception, PLoS Negl. Trop. Dis., 10 (2016), e0004539. doi: 10.1371/journal.pntd.0004539.  Google Scholar

[21]

J. M. MutuaF. B. Wang and N. K. Vaidya, Modeling malaria and typhoid fever co-infection dynamics, Math. Biosci., 264 (2015), 128-144.  doi: 10.1016/j.mbs.2015.03.014.  Google Scholar

[22]

N. H. Ogden, L. St-Onge 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. doi: 10.1186/1476-072X-7-24.  Google Scholar

[23]

A. R. Plourde and E. M. Bloch, A literature review of Zika virus, Emerg. Inf. Dise., 22 (2016), 1185-1192.  doi: 10.3201/eid2207.151990.  Google Scholar

[24]

H. C. Slater, M. Gambhir, P. E. Parham and E. Michael, Modelling co-infection with malaria and lymphatic filariasis, PLoS Comput. Biol., 9 (2013), e1003096, 14pp. doi: 10.1371/journal.pcbi.1003096.  Google Scholar

[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.  Google Scholar

[26]

F. E. SteinerR. R. Pinger and C. N. Vann, Infection and co-infection 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), 289-297.   Google Scholar

[27]

G. Stinco and S. Bergamo, Impact of co-infections in Lyme disease, Open Dermatology J., 10 (2016), 55-61.  doi: 10.2174/1874372201610010055.  Google Scholar

[28]

S. J. SwansonD. NeitzelK. D. Reed and E. A. Belongia, Coinfections acquired from Ixodes ticks, Cli. Micro. Rev., 19 (2006), 708-727.  doi: 10.1128/CMR.00011-06.  Google Scholar

[29]

B. Tang, Y. Xiao and J. Wu, Implication of vaccination against dengue for Zika outbreak, Sci. Rep., 6 (2016), 35623. doi: 10.1038/srep35623.  Google Scholar

[30]

H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.  Google Scholar

[31]

W. Wang and X.-Q. Zhao, Spatial invasion threshold of Lyme disease, SIAM J. Appl. Math., 75 (2015), 1142-1170.  doi: 10.1137/140981769.  Google Scholar

[32]

X. WuV. R. Duvvuri and Y. Lou, Developing a temperature-driven map of the basic reproductive number of the emerging tick vector of Lyme disease Ixodes scapularis in Canada, J. Theo. Biol., 319 (2013), 50-61.  doi: 10.1016/j.jtbi.2012.11.014.  Google Scholar

[33]

X. -Q. Zhao, Dynamical Systems in Population Biology, New York: Springer, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

[34]

X.-Q. Zhao and Z. Jing, Global asymptotic behavior in some cooperative systems of functional-differential equations, Canad. Appl. Math. Quart., 4 (1996), 421-444.   Google Scholar

Figure 1.  A schematic diagram of co-infection in the tick population. Here $E$ (eggs), $L\!Q$ (questing larvae), $L\!F$ (feeding larvae), $N\!Q$ (questing nymphs), $N\!F$ (feeding nymphs) and $A$ (adults) represent the stages of tick population with subscripts denoting the infectious status for each pathogen. Subscript $0$: no pathogen in ticks; $1$: Borrelia only; $2$: Babesia only; $3$: both pathogens
Figure 2.  A schematic diagram of co-infection in mice $M$ with subscripts denoting the infectious status for each pathogen
Figure 3.  Solution simulations with the model parameters in Table 2. Solutions through different initial values converge to the constant level for ticks (a), constant infected ticks for Borrelia infection only (b) and Babesia transmission cycle can not establish without the co-infection (c). However, on the scenario of coinfection, both pathogens can get established ((d), (e) and (f)). More interestingly, some ticks becomes infected with only Babesia or Borrelia while some others get infected with both pathogens
Table 1.  The state variables for the co-infection model. Bo and Ba represent Borrelia and Babesia, respectively
VariableMeaning
$E$number of eggs
$L\!Q$number of questing larvae
$L\!F_{0}$number of feeding larvae susceptible to both Ba and Bo
$L\!F_{1}$number of feeding larvae infected with Bo only
$L\!F_{2}$number of feeding larvae infected with Ba only
$L\!F_{3}$number of feeding larvae co-infected with Ba and Bo
$N\!Q_{0}$number of questing nymphs susceptible to both Ba and Bo
$N\!Q_{1}$number of questing nymphs infected with Bo only
$N\!Q_{2}$number of questing nymphs infected with Ba only
$N\!Q_{3}$number of questing nymphs co-infected with Ba and Bo
$N\!F_{0}$number of feeding nymphs susceptible to both Ba and Bo
$N\!F_{1}$number of feeding nymphs infected with Bo only
$N\!F_{2}$number of feeding nymphs infected with Ba only
$N\!F_{3}$number of feeding nymphs co-infected with Ba and Bo
$A_{0}$number of adults susceptible to both Ba and Bo
$A_{1}$number of adults infected with Bo only
$A_{2}$number of adults infected with Ba only
$A_3$number of adults co-infected with Ba and Bo
$M_{0}$number of mice susceptible to both Ba and Bo
$M_{1}$number of mice infected with Bo only
$M_{2}$number of mice infected with Ba only
$M_{3}$number of mice co-infected with Ba and Bo
VariableMeaning
$E$number of eggs
$L\!Q$number of questing larvae
$L\!F_{0}$number of feeding larvae susceptible to both Ba and Bo
$L\!F_{1}$number of feeding larvae infected with Bo only
$L\!F_{2}$number of feeding larvae infected with Ba only
$L\!F_{3}$number of feeding larvae co-infected with Ba and Bo
$N\!Q_{0}$number of questing nymphs susceptible to both Ba and Bo
$N\!Q_{1}$number of questing nymphs infected with Bo only
$N\!Q_{2}$number of questing nymphs infected with Ba only
$N\!Q_{3}$number of questing nymphs co-infected with Ba and Bo
$N\!F_{0}$number of feeding nymphs susceptible to both Ba and Bo
$N\!F_{1}$number of feeding nymphs infected with Bo only
$N\!F_{2}$number of feeding nymphs infected with Ba only
$N\!F_{3}$number of feeding nymphs co-infected with Ba and Bo
$A_{0}$number of adults susceptible to both Ba and Bo
$A_{1}$number of adults infected with Bo only
$A_{2}$number of adults infected with Ba only
$A_3$number of adults co-infected with Ba and Bo
$M_{0}$number of mice susceptible to both Ba and Bo
$M_{1}$number of mice infected with Bo only
$M_{2}$number of mice infected with Ba only
$M_{3}$number of mice co-infected with Ba and Bo
Table 2.  Definitions and corresponding values of the model parameters with the daily timescale. Abbreviations: Bo: Borrelia; Ba: Babesia; TP: transmission probability; AS: assumed parameter values
SymbolDescriptionValue Ref
$\mu_M$mortality rate of mice0.01[2]
$b_M$birth rate of mice0.02[2]
$D_M$density-dependent death rate of mice $5\times 10^{-5}$AS
$b_E$egg reproduction rate $\frac{16657}{365}$[17]
$\mu_E$mortality rate of eggs0.0025[17]
$\mu_{L\!Q}$mortality rate of questing larvae0.006[17]
$\mu_{L\!F}$mortality rate of feeding larvae0.038[17]
$\mu_{N\!Q}$mortality rate of questing nymphs0.006[17]
$\mu_{N\!F}$mortality rate of feeding nymphs0.028[17]
$\mu_A$mortality rate of adults0.01[17]
$d_E$development rate of eggs $\frac{2.4701}{365}$[17]
$d_L$development rate of larvae $\frac{2.2571}{365}$[17]
$d_N$development rate of nymphs $\frac{1.7935}{365}$[17]
$f_L$feeding rate of larvae $\frac{1.0475}{365}$[17]
$f_N$feeding rate of nymphs $\frac{1.0475}{365}$[17]
$D_L$density-dependent mortality rate of $LF$ $\frac{0.01}{\text{200}}$AS
$D_N$density-dependent mortality rate of $LN$ $\frac{0.01}{\text{200}}$AS
$\beta_{11}$TP of Bo from $M_{1}$ to $L\!Q$0.6[17]
$\beta_{31}$TP of Bo from $M_{3}$ to $L\!Q$ $1.5*\beta_{11}-\beta_{33}$AS
$\beta_{22}$TP of Ba from $M_{2}$ to $L\!Q$0.45AS
$\beta_{32}$TP of Ba from $M_{3}$ to $L\!Q$ $1.5*\beta_{22}-\beta_{33}$AS
$\beta_{33}$TP of both pathogens from $M_{3}$ to $L\!Q$ $\beta_{22}$AS
$\bar{\beta}_{11}$TP of Bo from $M_{1}$ to $N\!Q_{0}$ $\beta_{11}$AS
$\bar{\beta}_{31}$TP of Bo from $M_{3}$ to $N\!Q_{0}$ $\beta_{31}$AS
$\bar{\beta}_{22}$TP of Ba from $M_{2}$ to $N\!Q_{0}$ $\beta_{22}$AS
$\bar{\beta}_{32}$TP of Ba from $M_{3}$ to $N\!Q_{0}$ $\beta_{32}$AS
$\bar{\beta}_{33}$TP of both pathogens from $M_{3}$ to $N\!Q_{0}$ $\beta_{33}$AS
$\beta^{N\!Q_{1}}_{23}$TP of Ba from $M_{2}$ to $N\!Q_{1}$ $\beta_{22}$AS
$\beta^{N\!Q_{1}}_{33}$TP of both pathogens from $M_{3}$ to $N\!Q_{1}$ $\beta_{33}$AS
$\beta^{N\!Q_{2}}_{13}$TP of Bo from $M_{1}$ to $N\!Q_{2}$ $\beta_{11}$AS
$\beta^{N\!Q_{2}}_{33}$TP of both pathogens from $M_{3}$ to $N\!Q_{2}$ $\beta_{33}$AS
$\gamma_{11}$TP of Bo from $N\!F_{1}$ to $M_{0}$0.6AS
$\gamma_{31}$TP of Bo from $N\!F_3$ to $M_0$ $\beta_{31}$AS
$\gamma_{22}$TP of Ba from $N\!F_2$ to $M_0$ $\beta_{22}$AS
$\gamma_{32}$TP of Ba from $N\!F_3$ to $M_0$ $\beta_{32}$AS
$\gamma_{33}$TP of both pathogen from $N\!F_3$ to $M_0$ $\beta_{33}$AS
$\bar{\gamma}_{23}$TP of Ba from $N\!F_2$ to $M_1$ $\beta_{22}$AS
$\bar{\gamma}_{33}$TP of Ba from $N\!F_3$ to $M_1$ $\beta_{22}$AS
$\tilde{\gamma}_{13}$TP of Bo from $N\!F_1$ to $M_2$ $\beta_{11}$AS
$\tilde{\gamma}_{33}$TP of Bo from $N\!F_3$ to $M_2$ $\beta_{11}$AS
SymbolDescriptionValue Ref
$\mu_M$mortality rate of mice0.01[2]
$b_M$birth rate of mice0.02[2]
$D_M$density-dependent death rate of mice $5\times 10^{-5}$AS
$b_E$egg reproduction rate $\frac{16657}{365}$[17]
$\mu_E$mortality rate of eggs0.0025[17]
$\mu_{L\!Q}$mortality rate of questing larvae0.006[17]
$\mu_{L\!F}$mortality rate of feeding larvae0.038[17]
$\mu_{N\!Q}$mortality rate of questing nymphs0.006[17]
$\mu_{N\!F}$mortality rate of feeding nymphs0.028[17]
$\mu_A$mortality rate of adults0.01[17]
$d_E$development rate of eggs $\frac{2.4701}{365}$[17]
$d_L$development rate of larvae $\frac{2.2571}{365}$[17]
$d_N$development rate of nymphs $\frac{1.7935}{365}$[17]
$f_L$feeding rate of larvae $\frac{1.0475}{365}$[17]
$f_N$feeding rate of nymphs $\frac{1.0475}{365}$[17]
$D_L$density-dependent mortality rate of $LF$ $\frac{0.01}{\text{200}}$AS
$D_N$density-dependent mortality rate of $LN$ $\frac{0.01}{\text{200}}$AS
$\beta_{11}$TP of Bo from $M_{1}$ to $L\!Q$0.6[17]
$\beta_{31}$TP of Bo from $M_{3}$ to $L\!Q$ $1.5*\beta_{11}-\beta_{33}$AS
$\beta_{22}$TP of Ba from $M_{2}$ to $L\!Q$0.45AS
$\beta_{32}$TP of Ba from $M_{3}$ to $L\!Q$ $1.5*\beta_{22}-\beta_{33}$AS
$\beta_{33}$TP of both pathogens from $M_{3}$ to $L\!Q$ $\beta_{22}$AS
$\bar{\beta}_{11}$TP of Bo from $M_{1}$ to $N\!Q_{0}$ $\beta_{11}$AS
$\bar{\beta}_{31}$TP of Bo from $M_{3}$ to $N\!Q_{0}$ $\beta_{31}$AS
$\bar{\beta}_{22}$TP of Ba from $M_{2}$ to $N\!Q_{0}$ $\beta_{22}$AS
$\bar{\beta}_{32}$TP of Ba from $M_{3}$ to $N\!Q_{0}$ $\beta_{32}$AS
$\bar{\beta}_{33}$TP of both pathogens from $M_{3}$ to $N\!Q_{0}$ $\beta_{33}$AS
$\beta^{N\!Q_{1}}_{23}$TP of Ba from $M_{2}$ to $N\!Q_{1}$ $\beta_{22}$AS
$\beta^{N\!Q_{1}}_{33}$TP of both pathogens from $M_{3}$ to $N\!Q_{1}$ $\beta_{33}$AS
$\beta^{N\!Q_{2}}_{13}$TP of Bo from $M_{1}$ to $N\!Q_{2}$ $\beta_{11}$AS
$\beta^{N\!Q_{2}}_{33}$TP of both pathogens from $M_{3}$ to $N\!Q_{2}$ $\beta_{33}$AS
$\gamma_{11}$TP of Bo from $N\!F_{1}$ to $M_{0}$0.6AS
$\gamma_{31}$TP of Bo from $N\!F_3$ to $M_0$ $\beta_{31}$AS
$\gamma_{22}$TP of Ba from $N\!F_2$ to $M_0$ $\beta_{22}$AS
$\gamma_{32}$TP of Ba from $N\!F_3$ to $M_0$ $\beta_{32}$AS
$\gamma_{33}$TP of both pathogen from $N\!F_3$ to $M_0$ $\beta_{33}$AS
$\bar{\gamma}_{23}$TP of Ba from $N\!F_2$ to $M_1$ $\beta_{22}$AS
$\bar{\gamma}_{33}$TP of Ba from $N\!F_3$ to $M_1$ $\beta_{22}$AS
$\tilde{\gamma}_{13}$TP of Bo from $N\!F_1$ to $M_2$ $\beta_{11}$AS
$\tilde{\gamma}_{33}$TP of Bo from $N\!F_3$ to $M_2$ $\beta_{11}$AS
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