Discrete Dynamics in Nature and Society
Volume 2020 (2020), Article ID 8987490, 11 pages
https://doi.org/10.1155/2020/8987490

Research Article

Modeling Wolbachia Diffusion in Mosquito Populations by Discrete Competition Model

Yijie Li, Zhiming Guo, and Yanyuan Xing

School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China

Correspondence should be addressed to Zhiming Guo; guozm@gzhu.edu.cn

Received 11 November 2019; Accepted 20 January 2020; Published 27 February 2020

Academic Editor: Guang Zhang

Copyright © 2020 Yijie Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Dengue fever is caused by dengue virus and transmitted by Aedes mosquitoes. A promising avenue to control this disease is to infect the wild Aedes population with the bacterium Wolbachia driven by cytoplasmic incompatibility (CI). To study the invasion of Wolbachia into wild mosquito population, we formulate a discrete competition model and analyze the competition between released mosquitoes and wild mosquitoes. We show the global asymptotic properties of the trivial equilibrium, boundary equilibrium, and positive equilibrium and give the conditions for the successful invasion of Wolbachia. Finally, we verify our findings by numerical simulations.

1. Introduction

Dengue fever is a mosquito-borne infectious disease caused by dengue virus, mainly transmitted by Aedes aegypti [1]. More than 2.5 billion people in more than 100 countries are currently at risk from this disease, making mosquito control a major public health priority. Conventional measures, such as mass spraying of insecticides, are limited and unsafe, and worse, no effective vaccine that against dengue has yet been developed [2].

An innovative and effective method of mosquito control is using specific endosymbiotic bacterium Wolbachia to block the transmission of dengue fever. With infection, Wolbachia can reduce the mosquito’s dengue transmission potential. Besides, Wolbachia often induces cytoplasmic incompatibility (CI) which leads to early embryonic death when Wolbachia-infected males mate with uninfected females, so the release of infected males helps to reduce mosquito populations [3]. This bacterium infects many arthropods, including some mosquito species in nature, which was first identified in 1924 [4]. However, Wolbachia is not present in some important mosquito vectors. In 2005, Xi et al. established stable mosquito strains carrying Wolbachia for the first time by injecting Wolbachia into Aedes aegypti [5, 6], which is the basis of using Wolbachia to control mosquito-borne diseases. The strategies of using Wolbachia for mosquito vector control mainly include population suppression and population replacement. The former only releases infected males, and produces CI effect after mating with female mosquitoes in the field to suppress the number of mosquito vectors in the field [7, 8]; the latter releases both male and female mosquitoes infected with Wolbachia using vertical maternal transmission of Wolbachia and certain advantages to let Wolbachia spread in the mosquito population to stop spreading disease [9].

In recent years, the spreading dynamics of Wolbachia has become a hot research topic in academia. In 2014, Zheng et al. have established a delay differential equation model to study Wolbachia infection dynamics [10]. They gave a precise threshold for the infection frequency, and their numerical simulation is well fitting with the experimental data; then, Huang et al. proposed the corresponding reaction diffusion equations with homogeneous Neumann boundary condition and obtained similar results [11, 12]. In 2019, Zheng et al. have carried out a mathematical model to study combining incompatible and sterile insect techniques (IIT-SIT) to eliminate the population of Aedes mosquitoes in the wild, and they proved the feasibility of regional control of mosquito vector population by combining IIT-SIT through field experiments [13]. Besides, some interesting mathematical models have also been developed, such as the time-delay continuous model [8, 14–16], the discrete model [17, 18], and the stochastic systems of differential equations [19].

Motivated by those valuable research studies and since the experimental data of the detection of mosquito population in the field are discrete, we managed to establish a discrete model and make it as reasonable as possible. In this paper, a discrete model was established to study the spread of Wolbachia in mosquito populations from the perspective of competition, which is rarely mentioned in the existing literature.

The most frequently used equation for studying the discrete model is the Beverton-Holt equation:

where denotes the birth rate and denotes the intraspecific competition coefficient. If , then all solutions of the equation with tend monotonically to the equilibrium . If is considered as the number of a species at time t, it is a discrete model similar to the classical logistic continuous model [20]. If we consider the discrete model of competition between two species, the Beverton-Holt equation needs to be modified, which leads to the Leslie-Gower discrete competition model:

where denote the birth rate of , respectively. denote the intraspecific competition coefficient and denote the interspecific competition coefficient, respectively. Model (2) has rich dynamic properties and satisfies the competitive exclusion principle [21, 22]. Based on (2), we establish a discrete competition model describing Wolbachia spread in mosquito population. We assume that represents the number of mosquito population infected with Wolbachia at t and represents the number of uninfected mosquito population at t, and their birth rates are and , respectively. Since the mosquitoes are overlapping generation populations, we introduce as the mortality rates of and , respectively. Then, their survival rates are, respectively, and . Because Wolbachia often induces fitness cost [10], and it is reasonable to assume that . Furthermore, considering that both and denote the numbers of Aedes aegypti, for simplicity, we ignore the difference between intraspecific competition and interspecific competition and focus on the effect of competitions on their birth rates, so we assume and denote the competition coefficient within (or between) species, respectively. Finally, considering the infection Wolbachia mechanism of mosquito population and vertical maternal transmission of Wolbachia [23], the birth rate of is replaced by ; hence, the discrete model of competition between two species is as follows:

where all the coefficients are positive, and we mainly discuss this model.

The structure of this paper is arranged as follows. In Section 2, firstly, we prove the nonnegativity and boundedness of the solution of model (3) and obtain the positive invariant set. Secondly, we give four equilibrium points, including the trivial equilibrium , boundary equilibrium , and positive equilibrium as well as their existence conditions, and we also introduce the global attraction conditions for each equilibrium by stability analysis. In Section 3, some numerical simulations are presented to illustrate the behavior of the model and we interpret out main results biologically. Finally, the conclusion and discussion will be given in Section 4.

2. Model Analysis

We first consider the positivity and boundedness of the solution of model (3), positivity is clear, and the boundedness can be obtained as follows. From the first equation of model (3), we find

Let , then we have . Furthermore, we can obtain a series of inequalities

Add them up, and we get

As , it is easy to obtain the following inequality for any given initial value :

Similarly, for , we have

Therefore,

is a positive invariant.

Next, we analyze the existence and stability of the equilibria to (3). Note that (3) is not defined at the origin. In order to study the dynamical behavior of solutions to (3) near the origin, we define an auxiliary function:

Thus, (3) can be modified as

Thus, can be regarded as an equilibrium of (3). The other equilibria of (3) satisfy the following algebraic equations:

There are two nonnegative boundary equilibria of model (12), and , provided and . Besides, a positive equilibrium

exists if , and .

Theorem 1. For model (3), we have the following conclusions.

(a)If , then is globally asymptotically stable.(b)If , then is unstable and is globally asymptotically stable.(c)If , then is unstable and is globally asymptotically stable.

Proof. The Jacobian of model (3) is

The Jacobians evaluated at read as

respectively. Their eigenvalues appear along the diagonals. Recall that an equilibrium is globally asymptotically stable if it is locally asymptotically stable and all orbits tend to the equilibrium as .

(a)If , there is no nonnegative equilibrium except for . When , from (3), we have

 the system is monotonically decreasing, and if , then , so that is globally asymptotically stable. Furthermore, when , from (3), we have

 Obviously, , if and only if for all . Otherwise, we claim that and for all . It follows that and exist and must be zero. Therefore, is globally asymptotically stable. For the remaining cases, i.e., and , the conclusion can be verified similarly.(b)If , there are two equilibria, and . When , from the first equation of model (3), we have

 Clearly, if and only if . It follows that the solution emanating from the X-axis with will be away from along the X-axis, so is unstable. The eigenvalues of belong to the interval , and hence is locally asymptotically stable. In addition, according to a theorem due to Liu and Elaydi [24], all bounded orbits will eventually converge to an equilibrium; thus, all orbits will converge to . Therefore, is globally asymptotically stable.(c)If , there are two equilibria, and . In this case, we can prove the results in a similar way to case (b), so we omit it.

From now on, we assume that . By a similar argument, we know that is unstable. Therefore, in the sequel, we will mainly study the boundary equilibria , and positive equilibrium .
We claim that exists if and only if

We analyze it graphically, from the first equation of (12), we can obtain the perpendicular isoclines

denoted by , which is a segment of lines with slope and intercept . From the second equation of (12), we can obtain the horizontal isoclines

denoted by , which is a parabola, and it goes through three fixed points , and . We distinguish two cases depending on the position of and as shown in Figure 1. The inequalities satisfied by the coefficients that correspond to these cases can be listed as follows:

 In this case, there exist three equilibria , and , but does not exist. 

In this case, there exist four equilibria , and , the positive equilibrium is the intersection of and .
For the above two cases, a feasible biological interpretation of the conditions is that if , then the environment is more favorable for infected mosquitoes, and we say that Wolbachia infection has a fitness benefit. On the contrary, if , then the environment is less favorable for infected mosquitoes, and we claim that the infection has a fitness cost [10]. The stability analysis of the equilibria in both cases will be shown in the following theorem.

(a)

(b)

Figure 1: The two possible cases of and . (a) Case A. (b) Case B.

Theorem 2. Suppose . Then, we have the following conclusions.

(a)In the fitness benefit case , is a saddle and is globally asymptotically stable.(b)In the fitness cost case , both and are locally asymptotically stable while is a saddle.

Proof.

(a): from (see (16)), we know that is a saddle and is unstable and the stable manifold is the Y-axis. From the stable manifold theory and Hartman-Grobman theorems [25, 26], it follows that no solution can approach . By Liu and Elaydi’s theorem, all solutions must approach . Besides, the eigenvalues of (see (15)) are positive and less than 1, so is locally asymptotically stable. Therefore, is globally asymptotically stable.(b): all of the eigenvalues of (see (15)) and (see (16)) are positive and less than 1, so both and are locally asymptotically stable.

The stability of is discussed below. The Jacobian at takes the form

According to the Jury criteria [27], the equilibrium is locally asymptotically stable if the Jury condition

holds, where is the trace and is the determinant of . If at least one of these inequalities is reversed, then the equilibrium is unstable. The inequalities are equivalent to the following three inequalities.

(1)(2)(3)

By a direct calculation, it shows that

Using these formulas, we find that inequality is equivalent to

and the reverse inequality holds in , so that is unstable.
In what follows, we prove that is a saddle.
Inequality is equivalent to

We will prove that it holds in .
Let’s analyze the two complicated parts of the above equation. Obviously, the inequality

holds in . Also, we have

(i)If , then (ii)If , then (iii)If , then

If or , then

Consequently, we only need to prove holds when . By using this condition, we have

The last inequality employs the condition .
Now, we consider the characteristic quadratic polynomial of , namely,

Clearly, because the inequality is reversed in . Moreover, because the inequality holds, and . Thus, has a positive root and a root between and 1. Therefore, is a saddle.
The first part of Theorem 2 predicts that if the environment favors Wolbachia-infected populations, then successful invasion is ensured for any initial release size [28]. Biologically, the other case is more feasible when Wolbachia-infected populations have a fitness cost. In this case, model (3) possesses a saddle point , whose stable curves separate the attraction regions of and . We illustrate our results by numerical simulations in the next section.

3. Numerical Results

In order to show the dynamic stability of the model (3), we use MATLAB technical computing software for numerical simulations. The simulations for and in Theorem 2 is mainly conducted by using different parameters. Some results are shown as follows.

From Figure 2 we see that in , namely, the fitness benefit case, is globally asymptotically stable. The result of in Theorem 2 is verified. Whatever the initial value of Wolbachia-infected mosquito population and uninfected mosquito population are, will extinct and will persist, this means that the result of the competition is that Wolbachia successfully spreads throughout the mosquito population. In addition, it can be shown from the conditions that increasing the birth rate and reducing the competition coefficient α and the death rate will be beneficial to this result.

(a)

(b)

(c)

Figure 2: The red solid and the black dotted curves represent the numbers of infected and uninfected mosquitoes in Wolbachia, respectively. The parameter values are . The initial conditions are given by in (a), in (b), and in (c), and all of them show is globally asymptotically stable in .

From Figure 3, we see that in , namely, the fitness cost case, both and are locally asymptotically stable, which is consistent with the result of in Theorem 2. Each equilibrium has its own attractive regions and the initial values of Wolbachia-infected mosquitoes and uninfected mosquitoes will affect the trend of the solution; if the solution approaches , then will become extinct and will persist, that is to say the result of the competition is that Wolbachia successfully spreads throughout the mosquito population. On the contrary, if the solution approaches , then will go to extinct and will persist, namely, Wolbachia diffusion fails.

(a)

(b)

(c)

(d)

Figure 3: The red solid and the black dotted curves represent the numbers of infected and uninfected mosquitoes in Wolbachia, respectively. The parameter values are . It satisfies the condition of , both and are locally asymptotically stable, and they have their own attractive regions. The solutions with different initial values will approach to different equilibria. The initial value and will lead to the solution approach as , shown in (a) and (b), while the initial value and will lead to the solution approach as , shown in (c) and (d).

From above, we find the factors that affect the competition results are the competition coefficient within (or between) species, the birth rate, the death rate, and the initial value of the population. Reducing the competition coefficient and the death rate of the population and increasing the birth rate and the initial value of the population will be beneficial to the lasting survival of the population. It also makes sense in biological terms.

Finally, we give an explanation on the stable manifold of . Theorem 2 shows that is a saddle whose stable manifold separates the attractive regions of and . However, the specific position of the stable manifold cannot be determined. Here, we use numerical simulation to roughly estimate the position of the stable curve by using different initial values, as shown in Figure 4, and observing which equilibrium that the solution will tend to. Through figure analysis, we claim that the stable curve can be expressed as a smooth function that starts at , strictly increasing in the positive direction, and its image biased towards the X-axis. The stable manifold is composed of two separation curves with as the dividing point. Therefore, the attraction region of is larger than that of .

(a)

(b)

(c)

(d)

Figure 4: The parameters and some conclusions are given in Figure 3. is the saddle, and its stable manifolds separate the attractive regions of and . The solutions with different initial values will approach to different equilibria. The initial value and will lead to the solution approaches as , shown in (a) and (c), while the initial value and will lead to the solution approaches as , shown in (b) and (d).

4. Conclusion and Discussion

Dengue fever is a mosquito-borne disease, which has great harm to human society. Due to the lack of vaccines and efficient clinical cures, we need to control mosquito population to block the spread of the disease. An innovative and effective method to control mosquitoes is to employ Wolbachia, which has led to a growing number of researchers building models to study the dynamics of Wolbachia transmission. Considering that the collection data of mosquitoes in the wild are discrete, we established a discrete competition model to study the conditions for Wolbachia to successful spread in mosquitoes.

Model (3) is a discrete competition model of overlapping generations of Wolbachia-infected mosquitoes population and noninfected mosquitoes population . We showed the global asymptotic properties of the four equilibria of the model through elaborate analysis. First of all, the trivial equilibrium is globally asymptotically stable when and , mosquito populations will go to extinct, otherwise is unstable. Then, we obtain a complete result of and . If , then is globally asymptotically stable, which means that Wolbachia-infected mosquitoes population will persist and noninfected mosquitoes will become extinct, namely, Wolbachia successfully spreads. On the contrary, if , then is globally asymptotically stable, namely, Wolbachia fails to spread. Furthermore, we study the case when . In the fitness benefit case, namely, , and then is a saddle while is globally asymptotically stable and is not exist; that is, Wolbachia successfully spreads. In the fitness cost case, namely, , both and are locally asymptotically stable while is a saddle, and have their own attractive regions, and the stable manifold of is separated from their attracting domain. The size of the initial population value will affect its own persistence.

Numerical simulations are also provided to demonstrate these theoretical results. We mainly showed the simulation under condition and we found that the simulation results are consistent with the theoretical results. In particular, since we cannot determine the exact position of the stable manifold of , we showed its approximate position through simulations.

There are some limitations of the model presented in this paper. When we built the model, we only took into account the competitive factors affecting the birth rate of the population, not the death rate. In fact, both intraspecific and interspecific competition can affect mortality in populations. Also, we assume and to obtain a simpler model. It is obvious that if , the model will be more reasonable and can better describe the actual situation of competition between two mosquito populations. However, it presents great challenges in equilibrium analysis.

Data Availability

No data were used to support the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (No. 11771104) and Program for Changjiang Scholars and Innovative Research Team in University (IRT16R16).

References

J. L. Kyle and E. Harris, “Global spread and persistence of dengue,” Annual Review of Microbiology, vol. 62, no. 1, pp. 71–92, 2008.

S. V. Bardina, P. Bunduc, S. Tripathi et al., “Enhancement of Zika virus pathogenesis by preexisting antiflavivirus immunity,” Science, vol. 356, no. 6334, pp. 175–180, 2017.

H. Laven, “Cytoplasmic inheritance in Culex,” Nature, vol. 177, no. 4499, pp. 141-142, 1956.

M. Hertig and S. B. Wolbach, “Studies on Rickettsia-like micro-organisms in insects,” The Journal of Medical Research, vol. 44, no. 3, pp. 329–374, 1924.

Z. Xi, C. C. Khoo, S. L. Dobson et al., “Wolbachia establishment and invasion in an Aedes aegypti laboratory population,” Science, vol. 310, no. 5746, pp. 326–328, 2005.

Z. Xi, J. L. Dean, C. Khoo, and S. L. Dobson, “Generation of a novel Wolbachia infection in Aedes albopictus (Asian tiger mosquito) via embryonic microinjection,” Insect Biochemistry and Molecular Biology, vol. 35, no. 8, pp. 903–910, 2005.

M. G. Huang, J. W. Luo, L. C. Hu et al., “Assessing the efficiency of, Wolbachia, driven, Aedes, mosquito suppression by delay differential equations,” Journal of Theoretical Biology, vol. 440, pp. 1–11, 2017.

J. Yu, “Modeling mosquito population suppression based on delay differential equations,” SIAM Journal on Applied Mathematics, vol. 78, no. 6, pp. 3168–3187, 2018.

J. K. Axford, A. G. Callahan, A. A. Hoffmann, H. L. Yeap, and P. A. Ross, “Fitness of wAlbB Wolbachia infection in Aedes aegypti: parameter estimates in an outcrossed background and potential for population invasion,” The American Journal of Tropical Medicine and Hygiene, vol. 94, no. 3, pp. 507–516, 2016.

B. Zheng, M. X. Tang, and J. S. Yu, “Modeling Wolbachia spread in mosquitoes through delay differential equations,” SIAM Journal of Applied Mathematics, vol. 74, pp. 734–770, 2014.

M. Huang, J. Yu, L. Hu, and B. Zheng, “Qualitative analysis for a Wolbachia infection model with diffusion,” Science China Mathematics, vol. 59, no. 7, pp. 1249–1266, 2016.

M. Huang, M. Tang, and J. Yu, “Wolbachia infection dynamics by reaction-diffusion equations,” Science China Mathematics, vol. 58, no. 1, pp. 77–96, 2015.

X. Zheng, D. Zhang, Y. Li et al., “Incompatible and sterile insect techniques combined eliminate mosquitoes,” Nature, vol. 572, no. 7767, pp. 56–61, 2019.

M. Huang, M. Tang, J. S. Yu, and B. Zheng, “A stage structured model of delay differential equations for Aedes mosquito population suppression,” Discrete & Continuous Dynamical Systems—A, 2019.

M. G. Huang, L. C. Hu, and B. Zheng, “Comparing the efficiency of Wolbachia driven Aedes mosquito suppression strategies,” Journal of Applied Analysis and Computation, vol. 9, no. 1, pp. 211–230, 2019.

M. Huang, M. Tang, J. S. Yu, and B. Zheng, “The impact of mating competitiveness and incomplete cytoplasmic incompatibility on Wolbachia-driven mosquito population suppression,” Mathematical Biosciences and Engineering, vol. 16, no. 5, pp. 4741–4757, 2019.

J. Yu and B. Zheng, “Modeling Wolbachia infection in mosquito population via discrete dynamical models,” Journal of Difference Equations and Applications, vol. 25, no. 11, pp. 1549–1567, 2019.

B. Zheng, X. P. Liu, M. X. Tang et al., “Use of age-stage structural models to seek optimal Wolbachia-infected male mosquito releases for mosquito-borne disease control,” Journal of Theoretical Biology, vol. 472, pp. 95–109, 2019.

L. C. Hu, M. X. Tang, Z. D. Wu et al., “The threshold infection level for Wolbachia invasion in random environments,” Journal of Differential Equations, vol. 266, pp. 4377–4393, 2019.

J. M. Cushing, R. F. Costantino, B. Dennis et al., Chaos in Ecology Theoretical Ecology Series, Academic Press/Elsevier, San Diego, CA, USA, 2003.

J. M. Cushing, S. Levarge, N. Chitnis et al., “Some discrete competition models and the competitive exclusion principle,” Journal of Difference Equations and Applications, vol. 10, no. 13–15, pp. 1139–1151, 2004.

P. H. Leslie and J. C. Gower, “The properties of a stochastic model for two competing species,” Biometrika, vol. 45, no. 3-4, pp. 316–330, 1958.

J. H. Werren, L. Baldo, and M. E. Clark, “Wolbachia: master manipulators of invertebrate biology,” Nature Reviews Microbiology, vol. 6, no. 10, pp. 741–751, 2008.

P. Liu and S. N. Elaydi, “Discrete competitive and cooperative models of Lotka-Volterra type,” Journal of Computational Analysis and Applications, vol. 3, no. 1, pp. 53–73, 2001.

S. N. Elaydi, An Introduction to Difference Equations, Springer, New York, NY, USA, Second edition, 1999.

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, Berlin, Germany, 1983.

H. Caswell, Matrix Population Models: Construction, Analysis and Interpretation, Sinauer Associates, Inc. Publisher, Sunderland, MA, USA, Second edition, 2001.

M. Turelli, “Cytoplasmic incompatibility in populations with overlapping generations,” Evolution, vol. 64, no. 1, pp. 232–241, 2010.

D. Xiao and S. Ruan, “Global dynamics of a ratio-dependent predator-prey system,” Journal of Mathematical Biology, vol. 43, no. 3, pp. 268–290, 2001.