Research article Topical Sections

Fluidity of biodegradable substrate regulates carcinoma cell behavior: A novel approach to cancer therapy

  • Received: 12 November 2015 Accepted: 10 January 2016 Published: 12 January 2016
  • Although various polymeric substrates with different stiffness have been applied for the regulation of cells’ fate, little attention has been given to the effects of substrates’ fluidity. Here, we implement for the first time biodegradable polymer with fluidic property for cancer therapy by investigating cell adhesion, proliferation, apoptosis/death, cycles of cancer cells as well as the anticancer drug efficacy. To achieve this, we prepared crosslinked and non-crosslinked copolymers of ɛ-caprolactone-co-D, L-lactide (P(CL-co-DLLA)). The tensile test showed the crosslinked P(CL-co-DLLA) substrate has the stiffness of 261 kPa while the loss modulus G’’ of the non-crosslinked substrate is always higher than the storage modulus G’ (G’’/G’=3.06), indicating a quasi-liquid state. Human lung epithelial adenocarcinoma cells on crosslinked substrate showed well- spread actin stress fibers and visible focal adhesion with an increased S phase (decreased G0/G1 phase). The cells on non-crosslinked substrate, on the other hand, showed rounded morphology without visible focal adhesion and an accumulated G0/G1 phase (decreased S phase). These results suggest that the behavior of cancer cells not only depends on stiffness but also the fluidity of P(CL-co-DLLA) substrate. In addition, the effects of substrate’s fluidity on anti-cancer drug efficacy were also investigated. The IC50 values of paclitaxel for cancer cells on crosslinked and non-crosslinked substrates are 5.46 and 2.86 nM, respectively. These results clearly indicate that the fluidity of polymeric materials should be considered as one of the crucial factors to study cellular functions and molecular mechanism of cancer progression.

    Citation: Sharmy S Mano, Koichiro Uto, Takao Aoyagi, Mitsuhiro Ebara. Fluidity of biodegradable substrate regulates carcinoma cell behavior: A novel approach to cancer therapy[J]. AIMS Materials Science, 2016, 3(1): 66-82. doi: 10.3934/matersci.2016.1.66

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  • Although various polymeric substrates with different stiffness have been applied for the regulation of cells’ fate, little attention has been given to the effects of substrates’ fluidity. Here, we implement for the first time biodegradable polymer with fluidic property for cancer therapy by investigating cell adhesion, proliferation, apoptosis/death, cycles of cancer cells as well as the anticancer drug efficacy. To achieve this, we prepared crosslinked and non-crosslinked copolymers of ɛ-caprolactone-co-D, L-lactide (P(CL-co-DLLA)). The tensile test showed the crosslinked P(CL-co-DLLA) substrate has the stiffness of 261 kPa while the loss modulus G’’ of the non-crosslinked substrate is always higher than the storage modulus G’ (G’’/G’=3.06), indicating a quasi-liquid state. Human lung epithelial adenocarcinoma cells on crosslinked substrate showed well- spread actin stress fibers and visible focal adhesion with an increased S phase (decreased G0/G1 phase). The cells on non-crosslinked substrate, on the other hand, showed rounded morphology without visible focal adhesion and an accumulated G0/G1 phase (decreased S phase). These results suggest that the behavior of cancer cells not only depends on stiffness but also the fluidity of P(CL-co-DLLA) substrate. In addition, the effects of substrate’s fluidity on anti-cancer drug efficacy were also investigated. The IC50 values of paclitaxel for cancer cells on crosslinked and non-crosslinked substrates are 5.46 and 2.86 nM, respectively. These results clearly indicate that the fluidity of polymeric materials should be considered as one of the crucial factors to study cellular functions and molecular mechanism of cancer progression.


    Dengue is a viral disease transmitted mainly by Aedes mosquitoes, including Aedes aegypti and Aedes albopictus. Infected human beings carrying dengue viruses may get high fever, headaches, and skin rash, which may progress to dengue shock syndrome or dengue hemorrhagic fever. The annually reported dengue cases increased sharply from 2.2 million in 2010 to 3.2 million in 2015 [1]. More than 500, 000 people with severe dengue are hospitalized annually, and the case fatality percentage is about 2.5%. Due to the lack of commercially available vaccines for dengue control, traditional methods have been focused on vector control by heavy applications of insecticides and environmental management [2]. However, these programs have not prevented the spread of these diseases due to the rapid development of insecticide resistance [3] and the continual creation of ubiquitous breeding sites.

    An innovational mosquito control method utilizes Wolbachia, an intracellular bacterium present in about 60% of insect species [4], including some mosquitoes. Wolbachia can induce cytoplasmic incompatibility (CI) in mosquitoes, which results in early embryonic death from matings between Wolbachia-infected males and females that are either uninfected or harbor a different Wolbachia strain [5,6]. In the world's largest "mosquito factory" located in Guangzhou, China, 20 million Wolbachia-infected males are produced each week. These mosquitoes have been released in several urban and suburb areas since 2015, and killed more than 95% mosquitoes on the Shazai island [7]. Motivated by the success and the challenge in field trials, the study on the complex Wolbachia spread dynamics has become a hot research topic. Various mathematical models have been developed, including models of ordinary differential equations [8,9,10], delay differential equations [9,11,12,13], impulsive differential equations [14], stochastic equations [15], and reaction-diffusion equations [16]. These models are mainly devoted to analyzing the threshold dynamics for population replacement with Wolbachia-infected female mosquitoes or CI-driven population suppression, which only included mosquito population into the models. However, it is usually a formidable task for complete population replacement or eradication in large-scale field trials, and we can only concede mosquito replacement/eradication to virus eradication by restricting the mosquito densities below the epidemic risk threshold.

    To assess the efficacy of blocking dengue virus transmission by Wolbachia, a deterministic mathematical model of human and mosquito populations interfered by the circulation of a single dengue serotype was developed in the framework of SIER model [17]. This important study has inspired further development of compartmental models to analyze the transmission dynamics of dengue [18,19,20,21,22]. In this paper, we extend their efforts by incorporating the Wolbachia-infected male mosquito release into a compartmental model, where the infected males are maintained at a fixed ratio to the adult female mosquito population. We divide the mosquito population into three compartments: susceptible, exposed and infectious, and divide the human population into four compartments: susceptible, exposed, infectious and recovered. In particular, we include the respective extrinsic and intrinsic incubation periods (EIP and IIP) in the mosquito and human populations in our model. These periods have been shown to be crucial in clinical diagnosis, outbreak investigation, and dengue control [23], but have received relatively little attention [9,18]. Further, the release of Wolbachia-infected male mosquitoes is not effective immediately in sterilizing wild females due to the maturation delay between mating and emergence of adults [24]. To characterize the effect of EIP, IIP and the maturation delay on dengue transmission, we introduce three delays into our model.

    The model treats IIP in humans, EIP in mosquitoes, and the maturation delay as three delays. We aim to find the threshold values of the release ratio for mosquito or virus eradication under the proportional release policy. By analyzing the existence and stability of disease-free equilibria, we obtain the sufficient and necessary condition on the existence of the disease-endemic equilibrium. Two threshold values of the release ratio $ \theta $, denoted by $ \theta_1^* $ and $ \theta_2^* $ with $ \theta_1^* > \theta_2^* $ are explicitly expressed. When $ \theta > \theta_1^* $, the mosquito population will be eradicated eventually. If it fails for mosquito eradication but $ \theta_2^* < \theta < \theta_1^* $, virus eradication is ensured together with the persistence of susceptible mosquitoes. When $ \theta < \theta_2^* $, the disease-endemic equilibrium emerges that allows dengue virus to circulate between humans and mosquitoes through mosquito bites. Sensitivity analysis of the threshold values in terms of the model parameters, and numerical simulations on several possible control strategies with different release ratios confirm the public awareness that reducing mosquito bites and killing adult mosquitoes are the most effective strategy to control the epidemic. Our model will provide new insights on the effectiveness of Wolbachia in reducing dengue at a population level.

    We divide adult mosquitoes into subetaoups of susceptible ($ S_M $), exposed ($ E_M $) and infectious ($ I_M $). Let $ N_M = S_M+E_M+I_M $ be the total number of mosquitoes. The human population is divided into four subpopulations: susceptible ($ S_H $), exposed (infected but not infectious, $ E_H $), infectious ($ I_H $), and recovered ($ R_H $). The total number of humans is denoted by $ N_H = S_H+E_H+I_H+R_H $. Without the bothering of the dengue virus, i.e., $ E_M = I_M = E_H = I_H = R_H = 0 $, we assume that mosquito and human populations follow the logistic growth [25] satisfying

    $ \frac{dN_H}{dt} = r_HN_H(t)\left(1-\frac{N_H(t)}{\kappa_H}\right)-\mu_H N_H(t), $
    $ \frac{dN_M}{dt} = r_MN_M(t)\left(1-\frac{N_M(t)}{\kappa_M}\right)-\mu_M N_M(t), $

    where $ r_H $ and $ \mu_H $ are respectively the birth rate and the death rate of humans, and $ \kappa_H $ is a constant which leads the human carrying capacity to $ [(r_H-\mu_H)/r_H]\kappa_H $. Similarly, we set $ r_M $, $ \mu_M $, and $ \kappa_M $ as the birth rate, the death rate, and the carrying capacity parameter for mosquitoes. The prevalence of dengue virus disrupts the dynamics of humans and mosquitoes, which are split into $ S_H $, $ E_H $, $ I_H $, $ R_H $ and $ S_M $, $ E_M $, $ I_M $, respectively.

    Dengue viruses can be transmitted from infectious mosquitoes to susceptible humans through bites. Let $ b $ be the average daily biting rate per female mosquito, and $ \beta_{MH} $ be the fraction of transmission from mosquitoes to humans. Then, susceptible humans acquire the infection at the rate $ [b\beta_{MH}I_M(S_H/N_H)] $, and we have

    $ dSHdt=rHNH(t)(1NH(t)κH)bβMHIM(t)SH(t)NH(t)μHSH(t).
    $
    (2.1)

    Let $ \tau_H $ be the intrinsic incubation period (IIP) in humans between infection and the onset of infectiousness. IIP is an important determinant of dengue transmission dynamics, which varies from 3 to 14 days. Exposed humans become infectious at the rate

    $ [e^{-\mu_H\tau_H}b \beta_{MH} I_M(t-\tau_H) ( S_H(t-\tau_H) /N_H(t-\tau_H))]. $

    With further assumption that infected but not infectious humans have the same death rate as that of susceptible humans, $ E_H $ follows

    $ dEHdt=bβMHIM(t)SH(t)NH(t)eμHτHbβMHIM(tτH)SH(tτH)NH(tτH)μHEH(t).
    $
    (2.2)

    Let $ \gamma $ be the recovery rate of humans. Then for $ I_H $ and $ R_H $, we arrive at

    $ dIHdt=eμHτHbβMHIM(tτH)SH(tτH)NH(tτH)(γ+μH)IH(t),
    $
    (2.3)
    $ dRHdt=γIH(t)μHRH(t),
    $
    (2.4)

    where we disregard the negligible dengue mortality in humans [26] and set an identical mortality rate for humans.

    With the release of Wolbachia-infected male mosquitoes, denoted by $ R(t) $ at time $ t $, the birth rate of mosquitoes is reduced from $ r_M $ to

    $ r_M(1-\text{Probability of complete CI occurrence}). $

    Under random mating and equal sex determination, the probability of CI occurrence at time $ t $ is the proportion of Wolbachia-infected male mosquitoes among all male mosquitoes, i.e., $ R(t)/(R(t)+N_M(t)) $. However, there is a delay between the release of Wolbachia-infected males and the reduction of the wild mosquitoes which is caused by the maturation delay between mating and emergence of adult mosquitoes [24], denoted by $ \tau_e $. Hence, the number of susceptible mosquitoes that survive the maturation period is

    $ r_Me^{-\mu_M \tau_e}\left[1-\frac{R(t-\tau_e)}{N_M(t-\tau_e)+R(t-\tau_e)}\right] \cdot N_M(t-\tau_e)\left(1-\frac{N_M(t-\tau_e)}{\kappa_M}\right). $

    The susceptible mosquitoes become exposed at a rate of $ [b\beta_{HM}S_M(I_H/N_H)] $, where $ \beta_{HM} $ is the fraction of virus transmission from infectious humans to susceptible mosquitoes through blood meals. Taking these considerations into account, we have

    $ dSMdt=rMeμMτeNM(tτe)(1NM(tτe)κM)NM(tτe)NM(tτe)+R(tτe)bβHMSM(t)IH(t)NH(t)μMSM(t).
    $
    (2.5)

    Exposed mosquitoes spend the extrinsic incubation period (EIP), denoted by $ \tau_M $, which is the viral incubation period between the time when a female mosquito takes a viraemia blood meal from an infectious human and the time when that mosquito becomes infectious [27], typically 8~12 days. EIP has been frequently recognized as a crucial component of dengue virus transmission dynamics [23]. In view of this point, the rate that exposed mosquitoes become infectious per unit time is $ [e^{-\mu_M\tau_M}b\beta_{HM}S_M(t-\tau_M) (I_H(t-\tau_M)/N_H(t-\tau_M))] $. Then the whole set of equations in our model ends with

    $ dEMdt=bβHMSM(t)IH(t)NH(t)eμMτMbβHMSM(tτM)IH(tτM)NH(tτM)μMEM(t),
    $
    (2.6)
    $ dIMdt=eμMτMbβHMSM(tτM)IH(tτM)NH(tτM)μMIM(t)
    $
    (2.7)

    for exposed and infectious mosquitoes, respectively.

    Our purpose is to find the threshold values in terms of the release ratio for mosquito or virus eradication under the proportional release policy, where the Wolbachia-infected male mosquitoes is maintained in a fixed proportion to the adult female mosquito population, i.e.,

    $ R(t)=θNM(t).
    $
    (2.8)

    The explicit expressions of the threshold values are obtained as follows:

    $ \theta_1^* = \frac{r_Me^{-\mu_M\tau_e}}{\mu_M}-1, \quad \theta_2^* = \frac{r_Me^{-\mu_M\tau_e}}{\mu_M}-1-\frac{r_Me^{-\mu_M\tau_e}\kappa_H(r_H-\mu_H)(\gamma+\mu_H)}{ r_H\kappa_M b^2\beta_{MH}\beta_{HM}e^{-\mu_H\tau_H-\mu_M\tau_M} } $

    with $ \theta_1^* > \theta_2^* $. We get the following theorem.

    Theorem 1. If $ \theta > \theta_1^* $, then eradication of mosquitoes occurs. If $ \theta_2^* < \theta < \theta_1^* $, then eradication of virus occurs. If $ \theta < \theta_2^* $, then there exists a unique disease-endemic equilibrium of system (2.1)-(2.7).

    The proof of Theorem 1 is embodied in the analysis of the existence and stability of equilibrium points of system (2.1)-(2.7) in the following two sections, and we omit it here.

    To determine the steady-state solutions of system (2.1)-(2.7), we set the right sides of (2.1)-(2.7) to zero and ignore the time lags to arrive at

    $ rHNH(1NHκH)bβMHIMSHNHμHSH=0,
    $
    (3.1a)
    $ bβMHIMSHNHeμHτHbβMHIMSHNHμHEH=0,
    $
    (3.1b)
    $ eμHτHbβMHIMSHNH(γ+μH)IH=0,
    $
    (3.1c)
    $ γIHμHRH=0,
    $
    (3.1d)
    $ rMeμMτeNM(1NMκM)11+θbβHMSMIHNHμMSM=0,
    $
    (3.1e)
    $ bβHMSMIHNHeμMτMbβHMSMIHNHμMEM=0,
    $
    (3.1f)
    $ eμMτMbβHMSMIHNHμMIM=0.
    $
    (3.1g)

    Adding (3.1a) to (3.1d) together, we have

    $ r_HN_H\Big(1-\frac{N_H}{\kappa_H}\Big) = \mu_HN_H, $

    which yields

    $ NH=0orNH=κH(1μHrH):=NH,
    $
    (3.2)

    provided that $ \mu_H < r_H $ holds. Similarly, adding (3.1e) to (3.1g) together, we have

    $ r_Me^{-\mu_M\tau_e}N_M\Big(1-\frac{N_M}{\kappa_M}\Big)\cdot\frac{1}{1+\theta} = \mu_MN_M, $

    which leads to

    $ NM=0orNM=κM[1μM(1+θ)eμMτerM]:=NM.
    $
    (3.3)

    Define the net reproductive numbers for humans and mosquitoes respectively by

    $ R_0^H: = \frac{r_H}{\mu_H}, \quad R_0^M: = \frac{r_M}{\mu_M(1+\theta)e^{\mu_M\tau_e}}. $

    Theorem 2. Assume that

    $ RH0>1 and  RM0>1.
    $
    (3.4)

    Besides the zero equilibrium, system (2.1)-(2.7) admits two disease-free equilibrium points

    $ E_{01}^*: = (N_H^*, 0, 0, 0, 0, 0, 0), \quad E_{02}^*: = (N_H^*, 0, 0, 0, N_M^*, 0, 0). $

    Remark 1. The equilibrium point $ E_{01}^* $ corresponds to mosquito eradication as well as the infectious-free state for humans. The equilibrium point $ E_{02}^* $ biologically corresponds to the coexistence of mosquitoes and humans, without the infection of dengue virus, which more closely fits the actual situation. In the following discussion, we always assume that $ R_0^H > 1 $.

    To find the disease-endemic equilibrium point of system (2.1)-(2.7), denoted by

    $ E^*: = (S_H^*, E_H^*, I_H^*, R_H^*, S_M^*, E_M^*, I_M^*), $

    we need to solve the algebraic equations (3.1a)-(3.1g). If we define the basic reproduction number by

    $ R_0: = \frac{b^2\beta_{MH}\beta_{HM}e^{-\mu_H\tau_H-\mu_M\tau_M}N_M^*}{\mu_M(\gamma+\mu_H)N_H^*}, $

    then we obtain the existence of the disease-endemic equilibrium as follows.

    Theorem 3. The unique disease-endemic equilibrium of system (2.1)-(2.7) exists if and only if $ R_0 > 1 $ and $ R_0^M > 1 $.

    Proof. From (3.1d), we have

    $ RH=γμHIH.
    $
    (3.5)

    From (3.1b) and (3.1c), we respectively have

    $ \mu_HE^*_H = b \beta_{MH} I^*_M \frac{S^*_H }{N^*_H}\cdot (1-e^{-\mu_H\tau_H}), \quad\quad (\gamma+\mu_H) I^*_H = b \beta_{MH} I^*_M \frac{S^*_H }{N^*_H }\cdot e^{-\mu_H\tau_H}. $

    Hence we get the relation between $ E_H^* $ and $ I_H^* $ as

    $ \frac{\mu_HE^*_H}{(\gamma+\mu_H)I^*_H} = \frac{1-e^{-\mu_H\tau_H}}{e^{-\mu_H\tau_H}}, $

    i.e.,

    $ EH=(eμHτH1)(γ+μH)μHIH.
    $
    (3.6)

    Recall that at any steady state, we have

    $ r_HN_H^*\Big(1-\frac{N_H^*}{\kappa_H}\Big) = \mu_H N_H^*. $

    Combining (3.1a) and (3.1c), one has

    $ \mu_H(N_H^*-S_H^*) = b\beta_{MH} I_M^* \frac{S_H^*}{N_H^*} = e^{\mu_H\tau_H}(\gamma+\mu_H) I_H^*, $

    and hence

    $ SH=NHeμHτH(γ+μH)μHIH.
    $
    (3.7)

    From (3.1f) by (3.1g), we have

    $ \mu_M E^*_M = b\beta_{HM}S^*_M \frac{I^*_H }{N^*_H }\cdot (1-e^{-\mu_M\tau_M}), $
    $ \mu_M I^*_M = b\beta_{HM}S^*_M \frac{I^*_H }{N^*_H }\cdot e^{-\mu_M\tau_M}. $

    Hence, we get the relation between $ E_M^* $ and $ I_M^* $ which reads as

    $ EM=(eμMτM1)IM.
    $
    (3.8)

    Again, notice that at any steady state, we have

    $ r_Me^{-\mu_M\tau_e} N_M^* \Big(1-\frac{N_M^*}{\kappa_M}\Big)\frac{1}{1+\theta} = \mu_M N_M^*. $

    Then from (3.1e) and (3.1g), we arrive at

    $ \mu_M(N_M^*-S_M^*) = b\beta_{HM}S_M^*\frac{I_H^*}{N_H^*} = \mu_Me^{\mu_M\tau_M}I_M^*, $

    which leads to

    $ SM=NMeμMτMIM.
    $
    (3.9)

    From (3.5) to (3.9), to get the explicit expression of the disease-endemic equilibrium point, we only need to solve for $ I_H^* $ and $ I_M^* $. To this end, combining (3.1c) and (3.7), we have

    $ IH=eμHτHbβMH(γ+μH)NH[NHeμHτH(γ+μH)μHIH]IM=(eμHτHbβMHγ+μHbβMHμHNHIH)IM.
    $
    (3.10)

    Similarly, from (3.1g) and (3.9), we have

    $ IM=eμMτMbβHMμMNH(NMeμMτMIM)IH=(eμMτMbβHMNMμMNHbβHMμMNHIM)IH.
    $
    (3.11)

    Tedious but direct computations from (3.10)-(3.11) offers a linear relation between $ I_M^* $ and $ I_H^* $ as

    $ (bβHMμMNH+bβMHμHNHeμMτMbβHMNMμMNH)IH=(bβMHμHNH+bβHMμMNHeμHτHbβMHγ+μH)IM.
    $
    (3.12)

    Plugging (3.12) into (3.10) and (3.11) yields the unique $ I_H^* $ and $ I_M^* $ as

    $ IH=[eμHτHμMτMb2βMHβHMNM(γ+μH)μMNH]μHNHbβHM(eμMτMbβMHNM+μHNH)(γ+μH),
    $
    (3.13)
    $ IM=[eμHτHμMτMb2βMHβHMNM(γ+μH)μMNH]μHbβMH[eμHτHbβHMμH+(γ+μH)μM].
    $
    (3.14)

    It is easy to see from (3.5), (3.6), and (3.8) that $ R_H^* $, $ E_H^* $, and $ E_M^* $ are positive provided that $ I^*_M $ and $ I_H^* $ are positive. From (3.7) and (3.13), we have

    $ SH=NHeμHτH[eμHτHμMτMb2βMHβHMNM(γ+μH)μMNH]NHbβHM(eμMτMbβMHNM+μHNH)=NHbβHM(eμMτMbβMHNM+μHNH)[bβHMμH+(γ+μH)μMeμHτH]NH>0
    $

    always holds provided $ N^*_M > 0 $ and $ N_H^* > 0 $. Similarly, to guarantee $ S_M^* > 0 $, (3.9) requires again $ N^*_M > 0 $. From (3.9) and (3.14), with $ N^*_H > 0 $, we have

    $ SM=NMeμMτM[eμHτHμMτMb2βMHβHMNM(γ+μH)μMNH]μHbβMH[eμHτHbβHMμH+(γ+μH)μM]=1bβMH[eμHτHbβHMμH+(γ+μH)μM]μM(γ+μH)(NM+eμMτMμHNH)>0.
    $

    Hence, the disease-endemic equilibrium point exists if and only if

    $ e^{-\mu_H\tau_H-\mu_M\tau_M}b^2\beta_{MH}\beta_{HM}N_M^*-(\gamma+\mu_H)\mu_MN_H^* \gt 0, $

    i.e., $ R_0 > 1 $. With the conclusion of Theorem 2, we complete the proof.

    Because we are dealing with a system of delay differential equations, the characteristic equation has an infinite number of roots satisfying

    $ det(J+eλτHJτH+eλτMJτM+eλτeJτeλI)=0,
    $
    (4.1)

    where $ I $ is the identity matrix and the matrices $ J $, $ J_{\tau_H} $, $ J_{\tau_M} $ and $ J_{\tau_e} $ have entries that are the partial derivatives of the right sides of (2.1)-(2.7) with respect to, respectively,

    $ (S_H(t), E_H(t), I_H(t), R_H(t), S_M(t), E_M(t), I_M(t)), $
    $ (S_H(t-\tau_H), E_H(t-\tau_H), I_H(t-\tau_H), R_H(t-\tau_H), S_M(t-\tau_H), E_M(t-\tau_H), I_M(t-\tau_H)), $
    $ (S_H(t-\tau_M), E_H(t-\tau_M), I_H(t-\tau_M), R_H(t-\tau_M), S_M(t-\tau_M), E_M(t-\tau_M), I_M(t-\tau_M)), $
    $ (S_H(t-\tau_e), E_H(t-\tau_e), I_H(t-\tau_e), R_H(t-\tau_e), S_M(t-\tau_e), E_M(t-\tau_e), I_M(t-\tau_e)). $

    Theorem 4. The disease-free equilibria $ E_{01}^* $ is locally asymptotically stable if and only if $ R_0^M < 1 $.

    Proof. The matrices in (4.1) at $ E_{01}^* $ are

    $ J=(μHrH2μHrH2μHrH2μHrH00bβMH0μH0000bβMH00(γ+μH)000000γμH0000000μM0000000μM0000000μM),
    $
    (4.2)
    $ J_{\tau_H} = \left(0000000000000eμHτHbβMH000000eμHτHbβMH0000000000000000000000000000
    \right), \ J_{\tau_M} = 0, $
    $ J_{\tau_e} = \left(00000000000000000000000000000000rMeμMτe1+θrMeμMτe1+θrMeμMτe1+θ00000000000000
    \right). $

    The characteristic matrix at $ E_{01}^* $ is $ \left(A01(λ)B010C01(λ)

    \right) $ with

    $ A01(λ)=(μHrHλ2μHrH0μHλ)
    $
    (4.3)
    $ B01=(2μHrH2μHrH00bβMH0000bβMHe(λ+μH)τHbβMH)
    $
    (4.4)
    $ C01(λ)=((γ+μH)λ000e(λ+μH)τHbβMHγμHλ00000μMλ+e(λ+μM)τerM1+θe(λ+μM)τerM1+θe(λ+μM)τerM1+θ000μMλ00000μMλ).
    $
    (4.5)

    We firstly notice that when $ R_0^H > 1 $, $ \mu_H-r_H < 0 $. To prove the conclusion, we only need to prove that all roots of

    $ μMλ+e(λ+μM)τerM1+θ=0
    $
    (4.6)

    have negative real parts if and only if $ R_0^M < 1 $. Recall Theorem 4.7 in Smith [28] which states that $ \lambda = a+be^{- \lambda \tau} $ has no roots with non-negative real parts if $ a+b < 0 $ and $ b\ge a $, irrelevant of the value of $ \tau > 0 $. By this theorem, we see that when

    $ μM+rMeμMτe1+θ<0,
    $
    (4.7)

    i.e., $ R_0^M < 1 $, all roots of (4.6) have negative real parts.

    Next we prove that the condition (4.7) is also sufficient to exclude the possibility for (4.6) to have roots with non-negative real parts. To proceed, assume that $ \lambda = z_1+iz_2 $ with $ z_1\ge 0 $ is a solution of (4.6). Then

    $ rMeμMτe1+θeτez1cos(τez2)=μM+z1,
    $
    (4.8)
    $ rMeμMτe1+θeτez1sin(τez2)=z2,
    $
    (4.9)

    It is easy to see that if $ (z_1, z_2) $ satisfies (4.8)-(4.9), so does $ (z_1, -z_2) $. Without loss of generality, we assume that $ z_2\ge 0 $. Taking square of (4.8) and (4.9) and adding them together, we have

    $ r2M(1+θ)2e2τez12μMτe=(μM+z1)2+z22.
    $
    (4.10)

    If $ z_2 = 0 $, then $ z_1 > 0 $ due to (4.7), and from (4.10), we have

    $ \mu_M^2 \lt (\mu_M+z_1)^2 = \frac{r_M^2}{(1+\theta)^2}e^{-2\tau_e z_1-2\mu_M\tau_e} \lt \frac{r_M^2}{(1+\theta)^2}e^{ -2\mu_M\tau_e}, $

    a contradiction to (4.7). If $ z_2 > 0 $, from (4.10), we have

    $ \mu_M^2 \lt (\mu_M+z_1)^2+z_2^2 = \frac{r_M^2}{(1+\theta)^2}e^{-2\tau_e z_1-2\mu_M\tau_e}\leq \frac{r_M^2}{(1+\theta)^2}e^{ -2\mu_M\tau_e}, $

    also a contradiction to (4.7). This completes the proof.

    Remark 2. Theorem 4 implies that if the proportion of Wolbachia-infected males to wild mosquito population, $ \theta $, satisfies $ \theta > \theta_1^* $, then wild mosquitoes will be eradicated eventually, which has proved the first conclusion of Theorem 1.

    Next we consider the stability of $ E_{02}^* $. The Jacobian matrices in (4.1) at $ E_{02}^* $ are

    $ J=(μHrH2μHrH2μHrH2μHrH00bβMH0μH0000bβMH00(γ+μH)000000γμH00000bβHMNMNH0μM0000bβHMNMNH00μM0000000μM),
    $
    (4.11)
    $ J_{\tau_H} = \left(0000000000000eμHτHbβMH000000eμHτHbβMH0000000000000000000000000000
    \right), \quad J_{\tau_M} = \left(0000000000000000000000000000000000000eμMτMbβHMNMNH000000eμMτMbβHMNMNH0000
    \right), $
    $ J_{\tau_e} = \left(000000000000000000000000000000002μMrMeμMτe1+θ2μMrMeμMτe1+θ2μMrMeμMτe1+θ00000000000000
    \right). $

    The characteristic matrix at $ E_{02}^* $ is $ \left(A02(λ)B02C02D02(λ)

    \right) $ with

    $ A02(λ)=(μHrHλ2μHrH2μHrH0μHλ000(γ+μH)λ)
    $
    (4.12)
    $ B02=(2μHrH00bβMH000bβMHe(λ+μH)τHbβMH000e(λ+μH)τHbβMH),
    $
    (4.13)
    $ C02=(00γ00bβHMNMNH00bβHMNMNHe(λ+μM)τMbβMHNMNH00e(λ+μM)τMbβMHNMNH),
    $
    (4.14)
    $ D02(λ)=(μHλ0000S1(λ)eλτe(2μMrMeμMτe1+θ)eλτe(2μMrMeμMτe1+θ)00μMλ0000μMλ),
    $
    (4.15)

    where

    $ S1(λ)=μMλ+eλτe(2μMrMeμMτe1+θ).
    $
    (4.16)

    By direct computation, roots of (4.1) at $ E_{02}^* $ are $ \mu_H-r_H, \ -\mu_H $ together with roots of

    $ (μHλ)(μMλ)S1(λ)S2(λ)=0,
    $
    (4.17)

    where

    $ S2(λ)=(γ+μH+λ)(μM+λ)b2βMHβHMeμHτHμMτMe(τH+τM)λNMNH.
    $
    (4.18)

    On the roots of $ S_1(\lambda) = 0 $ and $ S_2(\lambda) = 0 $, we have the following two lemmas.

    Lemma 4.1. All roots of $ S_1(\lambda) = 0 $ have negative real parts if and only if $ R_0^M > 1 $.

    Proof. Notice that $ S_1(\lambda)\to -\infty $ as $ \lambda\to +\infty $. Then it is necessary to have $ S_1(0) < 0 $ to guarantee that all roots of $ S_1(\lambda) = 0 $ have negative real parts. Since

    $ S_1(0) = \mu_M-\frac{r_Me^{-\mu_M\tau_e}}{1+\theta} = \mu_M(1-R_0^M), $

    $ S_1(0) < 0 $ if $ R_0^M > 1 $. We claim that the condition $ R_0^M > 1 $ is also sufficient. In fact, assume that $ \lambda = \alpha+ i\beta $ satisfies $ S_1(\lambda) = 0 $. Then

    $ -\mu_M- \alpha- i\beta+\left(2\mu_M-\frac{r_Me^{-\mu_M\tau_e}}{1+\theta}\right) e^{- \alpha \tau_e} [\cos(\beta\tau_e)-i\sin(\beta\tau_e)]. $

    Separating the real and imaginary parts yields

    $ eατecos(βτe)=μM+α2μMrMeμMτe1+θ,
    $
    (4.19)
    $ eατesin(βτe)=β2μMrMeμMτe1+θ.
    $
    (4.20)

    If $ (\alpha, \beta) $ satisfies (4.19)-(4.20), so does $ (z_1, -z_2) $. Without loss of generality, we assume that $ \beta\ge 0 $. If $ \alpha\ge 0 $, then $ e^{- \alpha \tau_e}\leq 1 $. Equation (4.19) implies that

    $ \mu_M+ \alpha\leq 2\mu_M-\frac{r_Me^{-\mu_M\tau_e}}{1+\theta}, $

    and hence

    $ \alpha\leq \mu_M-\frac{r_Me^{-\mu_M\tau_e}}{1+\theta} = \mu_M(1-R_0^M) \lt 0, $

    a contradiction, which completes the proof.

    Lemma 4.2. All roots of $ S_2(\lambda) = 0 $ have negative real parts if and only if $ R_0 < 1 $.

    Proof. Notice that $ S_2(\lambda) $ is increasing with $ \lambda $, and $ S_2(\lambda)\to +\infty $ as $ \lambda\to +\infty $. Hence to make sure that all roots of $ S_2(\lambda) = 0 $ have negative real parts, the necessary condition is that $ S_2(0) > 0 $, that is,

    $ b2βMHβHMeμHτHμMτMNMμM(γ+μH)NH<1,
    $
    (4.21)

    i.e., $ R_0 < 1 $. We claim that condition (4.21) is also sufficient to guarantee that all roots of $ S_2(\lambda) = 0 $ have negative real parts. For notation simplicity, we define

    $ B: = b^2\beta_{MH}\beta_{HM}e^{-\mu_H\tau_H-\mu_M\tau_M}\frac{N_M^*}{N_H^*}, \quad \tau: = \tau_H+\tau_M. $

    Assume that $ \lambda = z_1+iz_2 $ is a solution of $ S_2(\lambda) = 0 $. Then

    $ Beτz1cos(τz2)=z21z22+(γ+μH+μM)z1+(γ+μH)μM,
    $
    (4.22a)
    $ Beτz1sin(τz2)=2z1z2(γ+μH+μM)z2.
    $
    (4.22b)

    It is easy to see that if $ (z_1, z_2) $ is a solution of (4.22a)-(4.22b), so does $ (z_1, -z_2) $. Thus we can assume $ z_2 > 0 $. Next we prove that $ z_1 < 0 $ holds. If not, assume $ z_1 = 0 $. Then (4.22a)-(4.22b) are reduced to

    $ Bcos(τz2)=z22+(γ+μH)μM,
    $
    (4.23a)
    $ Bsin(τz2)=(γ+μH+μM)z2,
    $
    (4.23b)

    which produces

    $ B^2 = \Big[z_2^2+(\gamma+\mu_H)^2\Big] (z_2^2+\mu_M^2). $

    Therefore,

    $ \frac{B^2}{\mu_M^2(\gamma+\mu_H)^2} = \left(1+\frac{z_2^2}{\mu_M^2}\right)\left[1+\frac{z_2^2}{(\gamma+\mu_H)^2}\right] \gt 1, $

    a contradiction to (4.21). On the other hand, if $ z_1 > 0 $, then from (4.22a)-(4.22b), we have

    $ B2e2τz1=[z21z22+(γ+μH+μM)z1+(γ+μH)μM]2+[2z1z2+(γ+μH+μM)z2]2=(z21+z22)2+z1(γ+μH+μM)[2z21+2z22+(γ+μH+μM)z1+2μM(γ+μH)z1]+2(γ+μH)μMz21+[(γ+μH)2+μ2M]z22+(γ+μH)2μ2M>(γ+μH)2μ2M.
    $

    Therefore,

    $ \frac{B^2}{\mu_M^2(\gamma+\mu_H)^2} \gt e^{2\tau z_1} \gt 1, $

    also a contradiction to (4.21). This completes the proof.

    From Lemmas 4.1 and 4.2, we have

    Theorem 5. The disease-free equilibrium point $ E_{02}^* $, if exists, is locally asymptotically stable.

    Results on the existence and stability of equilibria for system (2.1)-(2.2) are summarized in Table 1. Noticing the fact that

    $ R_0^M \lt 1\Leftrightarrow \theta \gt \theta_1^*, \quad R_0 \lt 1\Leftrightarrow \theta \gt \theta_2^*, $
    Table 1.  Conditions for the existence and stability of equilibria of system (2.1)-(2.7) with $ R_0^H > 1 $.
    Condition on $ R_0^M $ Condition on $ R_0 $ Equilibria and stability
    $ R_0^M < 1 $ $ R_0 < 1 $ $ E_{01}^* $ stable, $ E_{02}^* $(NA), $ E^* $(NA)
    $ R_0^M < 1 $ $ R_0 > 1 $ $ E_{01}^* $ stable, $ E_{02}^* $(NA), $ E^* $ (NA)
    $ R_0^M > 1 $ $ R_0 < 1 $ $ E_{01}^* $ unstable, $ E_{02}^* $ stable, $ E^* $(NA)
    $ R_0^M > 1 $ $ R_0 > 1 $ $ E_{01}^* $ unstable, $ E_{02}^* $ unstable, $ E^* $ exists

     | Show Table
    DownLoad: CSV

    we conclude that Theorem 1 holds.

    The basic reproduction number $ R_0 $ is a function of the parameters listed in Table 2.

    Table 2.  Parameter description.
    Parameter Description Baseline Range References
    $ r_H $ Birth rate of humans 12.43‰ year$ ^{-1} $ [29]
    $ r_M $ Birth rate of mosquitoes 2 day$ ^{-1} $ 0$ \sim $11.2 [19]
    $ \mu_H $ Death rate of humans 7.11‰ year$ ^{-1} $ [29]
    $ \mu_M $ Death rate of mosquitoes 1/14 day$ ^{-1} $ 1/30$ \sim $1/10 [19]
    $ b $ Average daily biting rate 0.63 day$ ^{-1} $ 0.5$ \sim $3 [30]
    $ \gamma $ Recovery rate of humans 1/5 day$ ^{-1} $ 1/14$ \sim $1/3 [26]
    $ \beta_{HM} $ Fraction of transmission from humans to mosquitoes 0.33 0.2$ \sim $1 [31]
    $ \beta_{MH} $ Fraction of transmission from mosquitoes to humans 0.33 0.2$ \sim $1 [31]
    $ r_{MH}=\kappa_M/\kappa_H $ The carrying capacity ratio of mosquitoes to humans 0.5 0$ \sim $1 Modelled
    $ \tau_H $ Intrinsic incubation period of humans 7 days $ 3\sim 14 $ [32]
    $ \tau_M $ Extrinsic incubation period of mosquitoes 9 days $ 8\sim 12 $ [17,32]
    $ \tau_e $ Delay between mating and emergence of adult mosquitoes 10 days $ 7\sim 19 $ [24]
    $ \theta $ The release ratio 1 1$ \sim $ 10 Modelled

     | Show Table
    DownLoad: CSV

    To estimate the response of $ R_0 $ to different parameters, following the procedure in [20], we define the relative sensitivity index of $ R_0 $ with respect to each parameter $ p $ in Table 2 by

    $ S_p^{R_0} = \frac{p^*}{R_0^*} \times \frac{\partial R_0}{\partial p}\Big |_{p = p^*}, $

    where $ p^* $ is taken as the baseline value in Table 2 which also yields $ R_0^* = 1.5869 $. The relative sensitivity indices are shown in Figure 1 which are ranked in their absolute values. It turns out that avoiding mosquito bites through physical and chemical means or a combination of both is the most direct and effective method to reduce the transmission of dengue virus, which has comparable performance of elevating the death rate of mosquitoes, $ \mu_M $. Compared with the parameter $ \mu_M $, the birth rate of mosquitoes, $ r_M $, has much less contribution to $ R_0 $. The elevation of $ R_0 $ due to the increase of $ \mu_H $ can be almost equally offset by the decrease of $ r_H $. Parameters $ \beta_{MH} $, $ \beta_{HM} $, the ratio of the carrying capacities of mosquitoes to humans, $ r_{MH} $, and the recovery rate of humans $ \gamma $, contribute equally to $ R_0 $. Among the three delay parameters, the extrinsic incubation period of mosquitoes, $ \tau_M $, is the most sensitive parameter, while the intrinsic incubation period of humans, $ \tau_H $, is the least sensitive one. Unexpectedly, the release ratio, $ \theta $, only plays a minor role in controlling the basic reproduction number. The most likely reason is that the baseline value of $ \theta $ is set as 1, much smaller than the release ratio in field trials. Coincidentally, further computation shows that when the release ratio is increased from 5 to 6, the basic reproduction number is decreased from 1.0447 to 0.9092$ (<1) $. This observation is in line with our claim that for mosquito eradication or virus eradication, the optimal release ratio of Wolbachia-infected male mosquitoes to wild males is about 5 to 1 [9,33].

    Figure 1.  The relative sensitivity indices of $ R_0 $ changing with parameters in Table 2.

    Theorem 2.1 shows that $ \theta_1^* $ is the threshold value for mosquito eradication. If mosquito eradication fails, then $ \theta_2^* $ is the threshold value for virus eradication. Dynamics of $ \theta_1^* $ and $ \theta_2^* $ are shown in Figure 2, which are in terms of the designated parameter while letting other parameters in system (2.1)-(2.7) be fixed as the baseline values in Table 2. The threshold values $ \theta_1^* $ and $ \theta_2^* $ increase as $ r_M $ increase; see Figure 2(a), which decrease as $ \mu_M $ or $ \tau_e $ decrease; see Figure 2(b) and (c). The threshold value $ \theta_1^* $ is fixed at 12.7072 when $ r_M $, $ \mu_M $ and $ \tau_e $ are fixed. In terms of $ b $, $ \beta_{HM} $, $ \beta_{MH} $, and $ r_{MH} $, the threshold $ \theta_2^* $ presents a quasi-logistic growth mode; see Figure 2(d), (e), and (f). The threshold $ \theta_2^* $ is a linear decreasing function of $ \gamma $ or $ \tau_M $; see Figure 2(g) and (i). When $ \tau_H $ lies between 3 and 14, it only brings a negligible change of $ \theta_2^* $; see Figure 2(h) which is consistent with the observation in Figure 1 that the relative sensitivity index of $ \tau_H $ is very close to 0.

    Figure 2.  Dynamics of the threshold values $ \theta_1^* $ and $ \theta_2^* $.

    Given the baseline values in Table 2, we have

    $ \theta_1^* = 12.7072, \quad \theta_2^* = 5.3298 $

    which offer the threshold values of mosquito and virus eradication, respectively. We initiate system (2.1)-(2.7) with one infectious human, and a mosquito population size about $ 25, 000 $ with one infectious mosquito. Figure 3 shows that if we take the release ratio $ \theta = 13 > \theta_1^* $, then both the virus and mosquito will eventually be eradicated. When the production of Wolbachia-infected male mosquitoes fails to meet $ \theta_1^* $, we can only concede mosquito eradication to virus eradication.

    Figure 3.  The release ratio greater than $ \theta_1^* $ is capable of wiping both virus and mosquito.

    Figure 4 shows that the release ratio $ \theta = 6 $ successfully clears the virus, together with the persistence of mosquito populations. However, further lower of the release ratio to $ 2 $ which is less than the threshold value $ \theta_2^* $ can neither clear virus nor eradicate mosquitoes. To see this, we initiate system (2.1)-(2.7) near the disease-endemic equilibrium point $ E^* $, with

    $ E^*\approx (29492.13, 1.8144, 1.2958, 13304.4331, 39095.3777, 1.6320, 1.8095). $
    Figure 4.  The release ratio lying in $ (\theta_1^*, \theta_2^*) $ clears virus, while leaving mosquito population size at a nonzero steady states eventually.

    When $ \theta = 2 $, Figure 5 shows that the number of infectious humans oscillates in the vicinity of their steady-state $ I_H^*\approx 1.2958 $, and the number of infectious mosquitoes oscillates in the vicinity of their steady-state $ I_M^*\approx 1.8095 $.

    Figure 5.  The release ratio less than $ \theta_2^* $ can neither clear virus nor eradicate mosquitoes.

    Dengue fever is one of the most common mosquito-borne viral diseases. Due to the lack of commercially available vaccines and efficient clinical cures, traditional methods have been focused on vector control by heavy applications of insecticides and environmental management. However, these programs have not prevented the spread of these diseases due to the rapid development of insecticide resistance and the continual creation of ubiquitous larval breeding sites. One novel dengue control method involves the intracellular bacterium Wolbachia, whose infection in Aedes aegypti or Aedes albopictus, the major mosquito vector of dengue virus, can greatly reduce the virus replication in mosquitoes. Wolbachia can also induce cytoplasmic incompatibility (CI) in mosquitoes, which results in early embryonic death from matings between Wolbachia-infected males and females that are either uninfected or harbor a different Wolbachia strain.

    CI mechanism drives Wolbachia-infected male mosquitoes as a new weapon to suppress or eradicate wild females. However, complete population eradication in large-scale field trials is usually formidable, and we can only concede mosquito eradication to virus eradication by restricting the mosquito densities below the epidemic critical threshold. Motivated by the success and the challenge in field trials, we developed a deterministic mathematical model of human and mosquito populations interfered by the circulation of a single dengue serotype in the framework of SIER model to assess the efficacy of blocking dengue virus transmission by Wolbachia. We extended the SIER model by incorporating the Wolbachia-infected male mosquito release, where the infected males are maintained at a fixed ratio to the adult female mosquito population. Furthermore, the extrinsic incubation period in mosquito (EIP), the intrinsic incubation period in human (IIP), and the maturation delay between mating and emergence of adult mosquitoes were embedded as three delays in our model.

    The threshold values of the release ratio $ \theta $ for mosquito or virus eradication were found by seeking the sufficient and necessary condition on the existence of the disease-endemic equilibrium. Two explicit expressions on threshold values of $ \theta $, denoted by $ \theta_1^* $ and $ \theta_2^* $ with $ \theta_1^* > \theta_2^* $, were obtained. When $ \theta > \theta_1^* $, the mosquito population will be eradicated eventually. If it fails for mosquito eradication but $ \theta_2^* < \theta < \theta_1^* $, virus eradication is ensured together with the persistence of susceptible mosquitoes. When $ \theta < \theta_2^* $, the emergence of the disease-endemic equilibrium makes dengue virus circulation between humans and mosquitoes possible. Sensitivity analysis of the threshold values showed that avoiding mosquito bites through physical and chemical means is the most direct and effective method to reduce the transmission of dengue virus, which has comparable performance of elevating the death rate of mosquitoes. Results from numerical simulations also confirmed our previous claim that for mosquito eradication or virus eradication, the optimal release ratio of Wolbachia-infected male mosquitoes to wild males is about 5 to 1.

    This work was supported by National Natural Science Foundation of China (11826302, 11631005, 11871174), Program for Changjiang Scholars and Innovative Research Team in University (IRT_16R16) and Science and Technology Program of Guangzhou (201707010337).

    The authors have declared that no competing interests exist.

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