Modelling Wolbachia infection in a sexstructured mosquito population carrying West Nile virus
Abstract
Wolbachia is possibly the most studied reproductive parasite of arthropod species. It appears to be a promising candidate for biocontrol of some mosquito borne diseases. We begin by developing a sexstructured model for a Wolbachia infected mosquito population. Our model incorporates the key effects of Wolbachia infection including cytoplasmic incompatibility and male killing. We also allow the possibility of reduced reproductive output, incomplete maternal transmission, and different mortality rates for uninfected/infected male/female individuals. We study the existence and local stability of equilibria, including the biologically relevant and interesting boundary equilibria. For some biologically relevant parameter regimes there may be multiple coexistence steady states including, very importantly, a coexistence steady state in which Wolbachia infected individuals dominate. We also extend the model to incorporate West Nile virus (WNv) dynamics, using an SEI modelling approach. Recent evidence suggests that a particular strain of Wolbachia infection significantly reduces WNv replication in Aedes aegypti. We model this via increased time spent in the WNvexposed compartment for Wolbachia infected female mosquitoes. A basic reproduction number \(R_0\) is computed for the WNv infection. Our results suggest that, if the mosquito population consists mainly of Wolbachia infected individuals, WNv eradication is likely if WNv replication in Wolbachia infected individuals is sufficiently reduced.
Keywords
Wolbachia Sexstructure West Nile virus Epidemic StabilityMathematics Subject Classification
92D30 34D20 34C111 Introduction
Wolbachia is a maternally transmitted intracellular symbiont, and it is the most common reproductive parasite infecting a significant proportion of insect species, see e.g. O’Neill et al. (1997), Werren (1997). Wolbachia typically inhibits testes and ovaries of its host, and it is also present in its host’s eggs. It interferes with its host’s reproductive mechanism in a remarkable fashion. This allows Wolbachia to successfully establish itself in a number of arthropod species. Wellknown effects of Wolbachia infections include cytoplasmic incompatibility (CI for short) and feminization of genetic males also known as male killing (MK for short), see e.g. Caspari and Watson (1959), Hoffmann and Turelli (1997), Telschow et al. (2005a, b). Another important wellknown effect of Wolbachia infections is the inducement of parthenogenesis, see e.g. Engelstädter et al. (2004), Stouthamer (1997). All of these contribute to the fact that the mathematical modelling of Wolbachia infection dynamics is both interesting and challenging.
In recent decades a substantial number of mathematical modelling approaches have been applied to model different types of Wolbachia infections in a variety of arthropod species. Perhaps most frequently researchers have been focusing on the development of mathematical models for Wolbachia infections in mosquito species. Many of the earlier models took the form of discrete time matrix models, written for population frequencies, see e.g. Turelli (1994), Vautrin (2007), and the references therein. Using frequencytype models a number of researchers investigated the possibility of coexistence of multiple Wolbachia strains, each of which exhibits different types of the reproductive mechanisms mentioned earlier, see e.g. Engelstädter et al. (2004), Farkas and Hinow (2010), Keeling et al. (2003), Vautrin (2007). Among others, Wolbachia strains have been investigated as a potential biological control tool to eradicate mosquito borne diseases. Originally the focus has been on Wolbachia strains that induce lifeshortening of their hosts. This is because for many vector borne diseases only older mosquitoes are of interest from the point of view of disease transmission. Therefore the use of (discrete) agestructured population models has become increasingly prevalent, see e.g. Rasgon and Scott (2004) and the references therein. Fairly recently, in McMeniman (2009) the results of laboratory experiments were reported envisaging a successful introduction of a lifeshortening Wolbachia strain in the mosquito species Aedes aegypti. In McMeniman (2009) three key factors, namely, strong CI, low fitness cost and high maternal transmission rate, were identified as drivers of a successful introduction of the new Wolbachia strain into an Aedes population. To this end researchers have developed and analysed continuous agestructured population models for Wolbachia infection dynamics, which take the form of partial differential equations, see Farkas and Hinow (2010); which can often be recast as delay equations, see e.g. Hancock et al. (2011a, b).
In recent years there have been substantial modelling efforts to theoretically investigate the potential of biological control tools for limiting the impact of mosquito borne diseases. It is now widely recognised that biological control represents a viable alternative to established methods such as the use of insecticides and bed nets. Among others, the sterile insect technique has been investigated in the recent papers (Dufourd and Dumont 2012; Li 2011; Li and Yuan 2015). More recently, it was reported that particular strains of Wolbachia (completely or almost completely) block dengue virus replication inside the mosquito hosts, see for example (Blagrove 2012; Hoffmann 2011; Walker 2011). To this end Hughes and Britton (2013) developed a mathematical model for Wolbachia infection as a potential control tool for dengue fever. Their work suggests that Wolbachia may be effective as such a control measure in areas where the basic reproduction number \(R_0\) is not too large. These recent results underpin the possibility that Wolbachia may be a promising candidate for biocontrol of mosquito borne diseases, in general. Besides dengue, West Nile virus (WNv) is another wellknown mosquito borne disease of current interest. WNv infection cycles between mosquitoes (especially Culex species) and a number of species, particularly birds. Some infected birds develop high levels of virus in their bloodstream and mosquitoes can become infected by biting these infectious birds. After about a week, infected mosquitoes can transmit the virus to susceptible birds. Mosquitoes infected with West Nile virus also bite and infect people, horses, and other mammals. However, humans, horses, and other mammals are ‘dead end’ hosts. This virus was first isolated in the West Nile region of Uganda, and since then has spread rapidly, for example in North America during the past 12 years. Since there is no vaccine available, the emphasis has been mainly on controlling the vector mosquito species. Some recent experiments, see Hussain (2013), have confirmed that replication of the virus in orally fed mosquitoes was largely inhibited in the wMelPop strain of Wolbachia. Interestingly, in a recent paper, Dodson et al. (2014) demonstrated in laboratory experiments that the wAlbB Wolbachia strain in fact enhances WNv infection rates in the mosquito species Culex transalis. However, in Dodson et al. (2014) the Wolbachia was not a stable maternally inherited infection, but rather they infected transiently somatic mosquito tissues, and hence the wAlbB infection did not induce significant immune response in the mosquitoes. This is probably key to their findings. Here we will focus on modelling a maternally inherited Wolbachia infection in a population model, which hypothesizes a large number of successive generations. Nevertheless, the findings in Dodson et al. (2014) underpin the importance of Wolbachia research in general and highlight the importance of contrasting the findings of new theoretical, laboratory and field investigations.
In this work we introduce sexstructured models for Wolbachia infection dynamics in a mosquito population. This will allow us to incorporate and study the wellknown effects of CI and MK of particular Wolbachia infections, simultaneously. First we will treat a model which only involves the mosquito population itself. Then we will use this model as a basis for a much more complex scenario incorporating WNv dynamics in a Wolbachia infected mosquito population. The full WNv model will naturally include the bird population, too.
2 Model for a Wolbachia infected mosquito population without WNv
2.1 Model derivation
We start by introducing a model for a Wolbachia infection in a sexstructured mosquito population, incorporating sexstructure using a well established approach originally due to Kendall (1949). More recent papers of Hadeler (2012) and Hadeler et al. (1988) derive and discuss sexstructured pair formation models in depth. We only model (explicitly) the adult population of mosquitoes. Our model allows us to take into account the wellknown effects of cytoplasmic incompatibility (CI), incomplete maternal transmission, fertility cost of the Wolbachia infection to reproductive output, and male killing (MK), at the same time. We note that it was shown in Engelstädter et al. (2004) that a stable coexistence of MK and CI inducing Wolbachia strains is possible, in principle. Introduction of male killing Wolbachia strains in vector populations may have a significant effect on the disease dynamics, as typically only female mosquitoes are transmitting the disease. Also note that according to Walker (2011), those Wolbachia strains which cause greater disruption, as in the case of dengue transmission, confer greater fitness costs to the mosquitoes. This may well be the case for West Nile virus, hence we account for the reduced reproductive output in our model.
 (1)
\(m \times f\): create one pair of the same type (m, f).
 (2)
\(m \times f_w\): with probability \(\beta \), create no offspring. This reflects the fecundity reduction due to the Wolbachia infection. In the complementary case, with probability \((1\beta )\tau (1\gamma )\), create a new pair \((m_w, f_w)\), at the same time with probability \((1\beta )\tau \gamma \) create \((0,f_w)\), i.e. a female only brood. This accounts for male killing (MK). With probability \((1\beta )(1\tau )\) create a new pair (m, f).
 (3)
\(m_w\times f_w\) : same as above.
 (4)
\(m_w\times f\): with probability q, create no offspring. This is the effect of cytoplasmic incompatibility (CI). With probability \(1q\), create a new pair (m, f).

M: number of uninfected male mosquitoes.

F: number of uninfected female mosquitoes.

\(M_w\): number of Wolbachia infected male mosquitoes.

\(F_w\): number of Wolbachia infected female mosquitoes.

\(M_{total} = M+M_w\), total number of male mosquitoes.

\(F_{total} = F + F_w\), total number of female mosquitoes.

\(N=M_{total}+F_{total}\), total number of mosquitoes.

\(\beta \): reduction in reproductive output of Wolbachia infected females.

\(\tau \): maternal transmission probability for Wolbachia infection.

q: probability of cytoplasmic incompatibility (CI).

\(\gamma \): probability of male killing (MK) induced by Wolbachia infection.

\(\lambda (F_{total})\): average egg laying rate, which depends on the total number of female mosquitoes.

\(\mu _m\): percapita mortality rate of uninfected male mosquitoes.

\(\mu _f\): percapita mortality rate of uninfected female mosquitoes.

\(\mu _{mw}\): percapita mortality rate of Wolbachia infected male mosquitoes.

\(\mu _{fw}\): percapita mortality rate of Wolbachia infected female mosquitoes.
2.2 Positivity and boundedness
First we begin by establishing positivity and boundedness of solutions of model (2.2)–(2.5).
Proposition 2.1
Proof
2.3 Boundary equilibria and their stability
Note that there is no Wolbachia infected boundary equilibrium unless \(\tau =1\), a case that we shall treat separately later.
Theorem 2.1
Proof
Theorem 2.2
Proof
Next we prove that, under certain conditions, both infected and uninfected mosquitoes die out. Note, however, that (0, 0, 0, 0) is not technically an equilibrium of (2.2)–(2.5).
Theorem 2.3
Proof
2.4 Existence of strictly positive steady states
In this section we examine the possible existence of coexistence steady states \((M,F,M_w,F_w)\) of model (2.2)–(2.5), i.e. steady states in which each component is strictly positive. It turns out that in some parameter regimes multiple coexistence steady states may exist while, in others, there is just one or none at all. An understanding of these properties helps us to understand how one might exploit Wolbachia infection in mosquitoes to effectively control WNv. In this section we simplify by assuming that \(\gamma =0\), i.e. that there is no male killing.
Theorem 2.4
Suppose that \(\gamma =0\), \(\tau =1\), \(\mu _f(1\beta )>\mu _{fw}\), \(\mu _m\mu _{fw}\ne \mu _f\mu _{mw}\), and that \(\lambda \) is a strictly positive decreasing function with \(\lambda (\infty )=\lambda _{min}\) (and \(\lambda _{min}\) sufficiently small). Then system (2.2)–(2.5) has no coexistence equilibrium with \(M^*,F^*,M_w^*,F_w^*>0\).
If the foregoing hypotheses hold, except that \(\tau \) is slightly less than 1, then system (2.2)–(2.5) has precisely one coexistence equilibrium in which Wolbachia uninfected mosquitoes exist in very small numbers relative to Wolbachia infected ones.
Proof
Note that from the proof of Theorem 2.4 we can see that at the equilibrium the male/female ratio for Wolbachia uninfected mosquitoes is given approximately by \(\frac{\mu _f}{\mu _m}\).
Circumstances under which (2.43) is likely to hold include that q is sufficiently close to 1 and, at the same time, \(\beta \) is sufficiently close to 1 or \(\frac{\mu _f}{\mu _{fw}}\) is less than 1. The biological interpretation is clear: for the existence of a coexistence steady state, the fertility cost (or mortality increase) due to Wolbachia infection should be sufficiently large.
It is clear from (2.41) that we can never have more than two coexistence steady states. On the other hand it is interesting to show that for some realistic parameter values it is possible to have two coexistence steady states.
Proposition 2.2
In the case when \(\gamma =0\), \(q=1\) and \(\frac{\mu _f}{\mu _m}=\frac{\mu _{fw}}{\mu _{mw}}\), there exists a set of values for the remaining parameters such that system (2.2)–(2.5) admits two coexistence steady states.
In summary, we have shown that our Wolbachia model (2.2)–(2.5) may exhibit all of the three qualitatively different possible scenarios, i.e. when there are 0, 1 or 2 coexistence steady states.
3 Model incorporating West Nile virus (WNv)
There has been considerable recent interest in West Nile virus (WNv), with the great majority of mathematical papers on the topic having appeared in the last 15 years. Numerous types of models have appeared, some including spatial effects and others giving consideration to issues such as agestructure in hosts, optimal control or backward bifurcation. See, for example, Blayneh et al. (2010), Bowman et al. (2005), Gourley et al. (2006/07), Lewis et al. (2006) and Wonham and Lewis (2008).

\(\alpha _f\): biting rate of female Wolbachia uninfected mosquitoes;

\(\alpha _{fw}\): biting rate of female Wolbachia infected mosquitoes;

\(p_{bf}\): transmission probability of WNv from infectious birds to WNvsusceptible female mosquitoes;

\(p_{fb}\): transmission probability of WNv from WNvinfectious female mosquitoes to susceptible birds;

\(\nu _f\): percapita transition rate of WNvexposed female Wolbachia uninfected mosquitoes to the infectious stage of WNv;

\(\varepsilon \in [0,1]\): small parameter modelling increased time that Wolbachia infected mosquitoes spend in the latent stage of WNv, due to the tendency of Wolbachia infection to hamper the replication of WNv in mosquitoes;

\(\nu _b\): percapita transition rate of WNvexposed birds to the infectious stage of WNv;

\(\nu _i\): percapita rate at which infectious birds recover;

\(\mu _b\): percapita natural death rate for birds;

\(\mu _{bi}\): percapita WNvinduced death rate for infectious birds.
Birds acquire WNv from bites by WNvinfectious mosquitoes, which may or may not have Wolbachia infection as well. Thus there are two infection rates for birds, these can be found in the right hand side of the seventh equation of (3.46), and also in the eighth equation since birds initially enter the exposed stage of WNv. This has a mean duration of \(1/\nu _b\) for birds, after which they become WNvinfectious. Birds may recover from WNv, at a percapita rate \(\nu _i\). Note that, for birds, death due to WNv is modelled using a separate parameter \(\mu _{bi}\) to distinguish from natural death, accounted for by \(\mu _b\). The function \(\Pi (B_{total})\) is the birth rate function for birds.
The approach we use here to model the latency stage of WNv (in either birds or mosquitoes) is not the only possible approach. Our approach permits individuals to spend different amounts of time in the latency stage, and we may only speak of the mean time spent in that stage. There are other approaches in which all individuals of a particular status (for example, all Wolbachia uninfected mosquitoes) spend the same amount of time in the latent stage of WNv. The time could be different for Wolbachia infected mosquitoes. These approaches result in models with time delays.
3.1 Local stability of the WNvfree equilibria
Equilibria of system (3.46) may exist in which WNv is absent. Such WNvfree equilibria include the equilibrium \((M^*,F^*,0,0)\) considered in Theorem 2.1, in which both WNv and Wolbachia are absent, and equilibria in which WNv is absent but Wolbachia are present. We show that multiple WNvfree equilibria may coexist that have both Wolbachia uninfected and Wolbachia infected mosquitoes, we present a necessary and sufficient condition for any particular WNvfree equilibrium to be locally stable, and we show that the most likely scenario for eradication of WNv is to have large numbers of Wolbachia infected mosquitoes, with solutions of system (3.46) evolving to a WNvfree equilibrium that has large numbers of Wolbachia infected mosquitoes and relatively few uninfected ones. Theorem 3.1 applies to any WNvfree equilibrium, of which there may be several. Of course, we may have \(R_0<1\) at one WNvfree equilibrium and \(R_0>1\) at another. It depends on the values of \(F_s^*\), \(F_{ws}^*\) and \(B_s^*\) for the particular WNvfree equilibrium under consideration. For clarity of exposition, we include as a hypothesis that the equilibrium be stable to the subset of perturbations in which WNv is absent (i.e. stable as a solution of the subsystem (2.2)–(2.5)), rather than including explicit conditions for stability of an equilibrium as a solution of that subsystem. The latter stability problem is a tedious one in its own right and is under consideration elsewhere in this paper. Theorem 3.1, in the form presented below, highlights clearly the particular role played by \(R_0\).
Theorem 3.1
Proof
4 Numerical simulations
Definition of parameters
Symbol  Definition  Value 

\(\mu _m\)  Percapita mortality rate of male mosquitoes  1/20 
\(\mu _f\)  Percapita mortality rate of female mosquitoes  1/20 
\(\mu _{wm}\)  Percapita mortality rate of Winfected male mosquitoes  1/20 
\(\mu _{wf}\)  Per capita mortality rate of Winfected female mosquitoes  1/20 
r  Maximum percapita mosquito egglaying rate  30 
k  Competition coefficient for mosquitoes  5000 
\(\beta \)  Fitness cost of Winfection on reproduction  \(\in [0,1]\) 
\(\tau \)  Maternal transmission rate of Wolbachia  \(\in [0,1]\) 
q  Strength of CI due to Winfection  \(\in [0,1]\) 
\(\gamma \)  Male killing rate due to Winfection  \(\in [0,1]\) 
\(\alpha _f \)  Percapita Wfree mosquito biting rate  0.09 
\(\alpha _{fw} \)  Percapita Winfected mosquito biting rate  0.09 
\(p_{bf}\)  WNv transmission coefficient from birds to mosquitoes  0.16 
\(p_{fb} \)  WNv transmission coefficient from mosquitoes to birds  0.88 
\(\mu _b\)  Natural percapita mortality rate of birds  \(1/(365\times 3)\) 
\(\mu _{bi}\)  WNvinduced percapita death rate of birds  0.1 
\(\Pi (B)\)  Birth rate of birds  100/365 
\(\nu _f\)  Percapita rate at which Wfree mosquitoes  
complete WNvlatency and become WNvinfectious  \(\in [0,1]\)  
\(\varepsilon \)  \(\varepsilon \, \nu _f\) is the percapita rate at which Winfected mosquitoes  
complete WNvlatency and become WNvinfectious  
\(\nu _b\)  Percapita rate at which exposed birds become infectious  0.2 
\(\nu _i\)  Per capita rate at which infectious birds recover  0.2 
5 Conclusion
In this paper we have derived a detailed sexstructured model for a mosquito population infected with Wolbachia . The model captures many of the wellknown key effects of Wolbachia infection, including cytoplasmic incompatibility, male killing, reduction in reproductive output and incomplete maternal transmission of the Wolbachia infection. Our analysis shows that the mosquito population can stabilise at a Wolbachia free equilibrium under certain circumstances, which include situations when inequality (2.11) holds. Such circumstances include, for example, if Wolbachia infection significantly reduces reproductive output, and/or Wolbachia infection significantly lowers female life expectancy. We also showed that if \(\tau =1\), i.e. maternal transmission of Wolbachia is complete, then the mosquito population can stabilise at an equilibrium in which all mosquitoes are infected with Wolbachia. This happens in the case of sufficiently high cytoplasmic incompatibility. In the case of \(\tau \) close to 1 we have shown that Wolbachia infected mosquitoes can coexist with small numbers of uninfected mosquitoes. We have also shown that under some additional assumptions our model has multiple coexistence steady states.
We extended the sexstructured mosquito population model (2.2)–(2.5) to include West Nile virus, which is spread by birds and mosquitoes, treating WNv as an SEI infection for mosquitoes, and as an SEIR infection for birds. We were motivated by results recently reported in Hussain (2013), which suggest that a particular strain of Wolbachia substantially reduces WNv replication in the mosquito species Aedes aegypti. We modelled this crucial phenomenon by incorporating a small parameter \(\varepsilon \), the reciprocal of which is proportional to the time spent in the WNv exposed class for Wolbachia infected mosquitoes. This enabled us to assess the potential of Wolbachia infection to eradicate WNv via its effect on WNv replication in Wolbachia infected mosquitoes. Notably the expression we obtained for the basic reproduction number \(R_0\) suggests that WNv will be eradicated if at the steady state the overwhelming majority of mosquitoes are infected with Wolbachia, and the Wolbachia infection substantially reduces WNv replication in mosquitoes. The first of these hypotheses is in fact shown to hold for a number of Wolbachia strains and mosquito species, see e.g. Engelstädter and Telschow (2009).
Notes
Acknowledgements
We thank the American Institute of Mathematics for financial support through the SQuaRE program. We also thank the reviewers for their helpful comments.
References
 Bergsman LD, Hyman JM, Manore C (2016) A mathematical model for the spread of West Nile virus in migratory and resident birds. Math Biosci Eng 13:401–424MathSciNetMATHGoogle Scholar
 Blagrove MSC et al (2012) Wolbachia strain wMel induces cytoplasmic incompatibility and blocks dengue transmission in Aedes albopictus. PNAS 109:255–260CrossRefGoogle Scholar
 Blayneh KW, Gumel AB, Lenhart S, Clayton T (2010) Backward bifurcation and optimal control in transmission dynamics of West Nile virus. Bull Math Biol 72:1006–1028MathSciNetCrossRefMATHGoogle Scholar
 Bowman C, Gumel AB, van den Driessche P, Wu J, Zhu H (2005) A mathematical model for assessing control strategies against West Nile virus. Bull Math Biol 67:1107–1133MathSciNetCrossRefMATHGoogle Scholar
 Caspari E, Watson GS (1959) On the evolutionary importance of cytoplasmic sterility in mosquitoes. Evolution 13:568–570CrossRefGoogle Scholar
 Dodson BL, Hughes GL, Paul O, Matacchiero AC, Kramer LD, Rasgon JL (2014) Wolbachia enhances West Nile Virus (WNV) infection in the mosquito Culex tarsalis. PLoS Negl Trop Dis 8(7):e2965CrossRefGoogle Scholar
 Dufourd C, Dumont Y (2012) Modeling and simulations of mosquito dispersal. The case of Aedes albopictus. BIOMATH 1:1209262, 7MathSciNetCrossRefMATHGoogle Scholar
 Engelstädter J, Telschow A (2009) Cytoplasmic incompatibility and host population structure. Heredity 103:196–207CrossRefGoogle Scholar
 Engelstädter J, Telschow A, Hammerstein P (2004) Infection dynamics of different Wolbachiatypes within one host population. J Theor Biol 231:345–355MathSciNetCrossRefGoogle Scholar
 Farkas JZ, Hinow P (2010) Structured and unstructured continuous models for Wolbachia infections. Bull Math Biol 72:2067–2088MathSciNetCrossRefMATHGoogle Scholar
 Gourley SA, Liu R, Wu J (2006/07) Some vector borne diseases with structured host populations: extinction and spatial spread. SIAM J Appl Math 67:408–433Google Scholar
 Hadeler KP (2012) Pair formation. J Math Biol 64:613–645MathSciNetCrossRefMATHGoogle Scholar
 Hadeler KP, Waldstätter R, WörzBusekros A (1988) Models for pair formation in bisexual populations. J Math Biol 26:635–649MathSciNetCrossRefMATHGoogle Scholar
 Hancock PA, Sinkins SP, Godfray HCJ (2011a) Population dynamic models of the spread of Wolbachia. Am Nat 177:323–333CrossRefGoogle Scholar
 Hancock PA, Sinkins SP, Godfray HCJ (2011b) Strategies for introducing Wolbachia to reduce transmission of mosquitoborne diseases. PLoS Negl Trop Dis 5:e1024CrossRefGoogle Scholar
 Hoffmann AA, Turelli M (1997) Cytoplasmic incompatibility in insects. In: Hoffmann AA, O’Neill SL, Werren JH (eds) Influential Passengers. Oxford University Press, Oxford, pp 42–80Google Scholar
 Hoffmann AA et al (2011) Successful establishment of Wolbachia in Aedes populations to suppress dengue transmission. Nature 476:454–457CrossRefGoogle Scholar
 Hussain M et al (2013) Effect of Wolbachia on replication of West Nile virus in a mosquito cell line and adult mosquitoes. J Virol 87:851–858CrossRefGoogle Scholar
 Hughes H, Britton NF (2013) Modelling the use of Wolbachia to control dengue fever transmission. Bull Math Biol 75:796–818MathSciNetCrossRefMATHGoogle Scholar
 Keeling MJ, Jiggins FM, Read JM (2003) The invasion and coexistence of competing Wolbachia strains. Heredity 91:382–388CrossRefGoogle Scholar
 Kendall DG (1949) Stochastic processes and population growth. J R Stat Soc Ser B 11:230–264MathSciNetMATHGoogle Scholar
 Keyfitz N (1972) The mathematics of sex and marriage. In: Proceedings of the sixth Berkeley symposium on mathematical statistics and probability, vol IV: biology and health. Univ. California Press, pp 89–108Google Scholar
 Lewis M, Renclawowicz J, van den Driessche P (2006) Traveling waves and spread rates for a West Nile virus model. Bull Math Biol 68:3–23MathSciNetCrossRefMATHGoogle Scholar
 Li J (2011) Modelling of transgenic mosquitoes and impact on malaria transmission. J Biol Dyn 5:474–494MathSciNetCrossRefMATHGoogle Scholar
 Li J, Yuan Z (2015) Modelling releases of sterile mosquitoes with different strategies. J Biol Dyn 9:1–14MathSciNetCrossRefGoogle Scholar
 McMeniman CJ et al (2009) Stable introduction of a lifeshortening Wolbachia infection into the mosquito Aedes aegypti. Science 323:141–144CrossRefGoogle Scholar
 O’Neill SL, Hoffmann AA, Werren JH (eds) (1997) Influential passengers. Oxford University Press, OxfordGoogle Scholar
 Rasgon JL, Scott TW (2004) Impact of population age structure on Wolbachia transgene driver efficacy: ecologically complex factors and release of genetically modified mosquitoes. Insect Biochem Mol Biol 34:707–713CrossRefGoogle Scholar
 Smith HL (1995) Monotone dynamical systems. An introduction to the theory of competitive and cooperative systems. Mathematical Surveys and Monographs, 41. American Mathematical Society, ProvidenceGoogle Scholar
 Stouthamer R (1997) Wolbachia induced parthenogenesis. In: Hoffmann AA, O’Neill SL, Werren H (eds) Influential passengers. Oxford University Press, Oxford, pp 102–124Google Scholar
 Telschow A, Hammerstein P, Werren JH (2005a) The effect of Wolbachia versus genetic incompatibilities on reinforcement and speciation. Evolution 59:1607–1619CrossRefGoogle Scholar
 Telschow A, Yamamura N, Werren JH (2005b) Bidirectional cytoplasmic incompatibility and the stable coexistence of two Wolbachia strains in parapatric host populations. J Theor Biol 235:265–274MathSciNetCrossRefGoogle Scholar
 Turelli M (1994) Evolution of incompatibilityinducing microbes and their hosts. Evolution 48:1500–1513CrossRefGoogle Scholar
 Vautrin E et al (2007) Evolution and invasion dynamics of multiple infections with Wolbachia investigated using matrix based models. J Theor Biol 245:197–209MathSciNetCrossRefGoogle Scholar
 Walker T et al (2011) The wMel Wolbachia strain blocks dengue and invades caged Aedes aegypti populations. Nature 476:450–453CrossRefGoogle Scholar
 Werren JH (1997) Biology of Wolbachia. Annu Rev Entomol 42:587–609CrossRefGoogle Scholar
 Wonham MJ, Lewis MA (2008) A comparative analysis of models for West Nile Virus. Mathematical epidemiology. Lecture Notes in Math., 1945. Springer, Berlin, pp 365–390Google Scholar
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