# American Institute of Mathematical Sciences

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
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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
A schematic diagram of co-infection in mice $M$ with subscripts denoting the infectious status for each pathogen
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
The state variables for the co-infection model. Bo and Ba represent Borrelia and Babesia, respectively
 Variable Meaning $E$ 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
 Variable Meaning $E$ 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
Definitions and corresponding values of the model parameters with the daily timescale. Abbreviations: Bo: Borrelia; Ba: Babesia; TP: transmission probability; AS: assumed parameter values
 Symbol Description Value Ref $\mu_M$ mortality rate of mice 0.01 [2] $b_M$ birth rate of mice 0.02 [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 eggs 0.0025 [17] $\mu_{L\!Q}$ mortality rate of questing larvae 0.006 [17] $\mu_{L\!F}$ mortality rate of feeding larvae 0.038 [17] $\mu_{N\!Q}$ mortality rate of questing nymphs 0.006 [17] $\mu_{N\!F}$ mortality rate of feeding nymphs 0.028 [17] $\mu_A$ mortality rate of adults 0.01 [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 $L\!Q$ 0.45 AS $\beta_{32}$ 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 $M_{0}$ 0.6 AS $\gamma_{31}$ 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
 Symbol Description Value Ref $\mu_M$ mortality rate of mice 0.01 [2] $b_M$ birth rate of mice 0.02 [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 eggs 0.0025 [17] $\mu_{L\!Q}$ mortality rate of questing larvae 0.006 [17] $\mu_{L\!F}$ mortality rate of feeding larvae 0.038 [17] $\mu_{N\!Q}$ mortality rate of questing nymphs 0.006 [17] $\mu_{N\!F}$ mortality rate of feeding nymphs 0.028 [17] $\mu_A$ mortality rate of adults 0.01 [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 $L\!Q$ 0.45 AS $\beta_{32}$ 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 $M_{0}$ 0.6 AS $\gamma_{31}$ 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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