Discrete dynamical models on Wolbachia infection frequency in mosquito populations with biased release ratios

We develop two discrete models to study how supplemental releases affect the Wolbachia spreading dynamics in cage mosquito populations. The first model focuses on the case when only infected males are released at each generation. This release strategy has been proved to be capable of speeding up the Wolbachia persistence by suppressing the compatible matings between uninfected individuals. The second model targets the case when only infected females are released at each generation. For both models, detailed model formulation, enumeration of the positive equilibria and their stability analysis are provided. Theoretical results show that the two models can generate bistable dynamics when there are three positive equilibrium points, semi-stable dynamics for the case of two positive equilibrium points. And when the positive equilibrium point is unique, it is globally asymptotically stable. Some numerical simulations are offered to get helpful implications on the design of the release strategy.


Introduction
Dengue is a mosquito-borne viral disease which is mainly endemic in tropical and subtropical areas, and has spread rapidly to temperate regions in recent years. About 100-400 million people infect dengue each year and nowadays, almost half of the world's population is at risk of dengue. As a mosquito-borne disease, dengue is transmitted by the bites of female Aedes aegypti and Aedes albopictus which are also vectors of Zika, Chikungunya and yellow fever [11]. The most direct and traditional way to prevent mosquito-borne disease transmission is to kill mosquitoes by spraying insecticides and removing breeding sites, which only has a short-term effect because of the emergence and enhancement of insecticide resistance of mosquitoes and the continual creation of ubiquitous larval sources in the warm and humid seasons [4,26,31]. Although the history of dengue vaccines can be traced back to 1993, dengue vaccine was first applied for use until 2015. However, experiments in [7,8] have proved the phenomenon of antibody dependent enhancement (ADE for short) in dengue serotypes, and further report [3] shows that 130 among 830,000 vaccinated children have died, 19 of those have dengue, meaning that ADE does play a role.
An innovative biological method involves an intracellular bacterium, named Wolbachia, which was first identified by Hertig and Wolbach in 1924 [12]. Wolbachia, which exists in up to 75% of insects, gained widespread attention of scholars in 1956 when Laven [23] revealed its role in cytoplasmic incompatibility (CI for short) in Culex pipiens. Unfortunately, Wolbachia does not exist in Aedes aegypti. Although Aedes albopictus naturally carries two Wolbachia strains, these two strains could not block the replication of the dengue viruses in mosquito. The groundbreaking work is credited to Xi, who established a stable Wolbachia infection in Aedes aegypti for the first time [32].
As a maternally transmitted bacterium, Wolbachia can induce CI when Wolbachiainfected males mate with uninfected females, resulting in an early embryonic death [13,24] and no offspring can be produced from these mated females. Based on these two mechanisms, two release strategies targeting controlling mosquito populations emerge as promising methods to reduce the occurrence of diseases transmitted by mosquitoes. The first one is usually termed as population suppression [46], when a large number of Wolbachia-infected males are released into the wild to suppress, or even eradicate, the wild female mosquitoes through CI. Population replacement, as an alternative release strategy, release both Wolbachia-infected males and females to replace wild mosquito population with infected one, among which females lose their ability in transmitting dengue viruses owing to Wolbachia infection. With promising results to reduce the occurrence of diseases transmitted by mosquitoes, the dynamics of Wolbachia in mosquito population has attracted a lot of attention, and various mathematical models have been developed, including ordinary differential models [16,34,36,38,42,44], delay differential models [18][19][20][21][22]33,35,41], stochastic models [15], reaction-diffusion models [17] and discrete models [6,10,13,14,25,[27][28][29][30]37,39,40,43,45].
Non-overlapping cage mosquito populations whose dynamics can be monitored by infection frequency rather than number, where the discrete model becomes the first choice for its easy mathematical tractability. The first discrete model was developed by Caspari and Watson [6] to characterize the evolutionary importance of CI sterility in mosquitoes, which reads as where x n is the frequency of Wolbachia infection at the nth generation, s f ∈ (0, 1) is the fitness cost of Wolbachia-infected mosquitoes to wild ones, and s h ∈ (0, 1] is the proportion of unhatched eggs produced from incompatible cross [28,29]. Experimental observations show that Wolbachia can be stably maintained with strong CI and a mild fitness cost [5,24,32]. Hence, infections with s f < s h is widely accepted. Later in 1978, observing that the maternal transmission of Wolbachia is not perfect, Fine [10] introduced the maternal leakage rate μ ∈ [0, 1) and generalized model (1) to which has also been used [14,[27][28][29][30] to characterize Wolbachia spreading dynamics in Drosophila simulans during 1990s. Recently, model (2) was revisited in [37]. By introducing the threshold on the maternal leakage rate a complete description for the dynamics of model (2) was obtained.
which is semi-stable: stable from the right side but unstable from the left side, and x * 0 is locally asymptotically stable.
The value μ * 1 is interpreted as the maximal maternal leakage rate in [37], above which Wolbachia persistence is impossible. For the case when μ < μ * 1 , both models (1) and (2) generate bistable dynamics, with the existence of an unstable equilibrium, which is s f /s h for μ = 0, or x * 1 (μ) for μ ∈ (0, μ * 1 ). When the initial infection frequency x 0 is larger than the unstable equilibrium, Wolbachia infection in mosquito population is guaranteed to be persistent. When x 0 lies below the unstable equilibrium, wild mosquito populations outcompete the Wolbachia-infected ones. To change the fate of Wolbachia, supplemental releases are needed to guarantee the success of Wolbachia persistence until at some generation n, x n surpasses the unstable equilibrium.
Assume that a proportional release strategy is implemented where both infected females and infected males are released simultaneously at the same ratio r. The next model was developed in [37] to characterize how supplemental releases affect the Wolbachia infection frequency threshold in [6,10], where r is the constant ratio of infected females/males to the total number of wild females/males at each generation. A release ratio threshold r * was found in [37]: for r ∈ (0, r * ), the Wolbachia infection frequency threshold is reduced, and for r ≥ r * , the threshold is further lowered to 0 which implies that Wolbachia persistence is always successful for any initial infection frequency above 0.
In this paper, we continue to study how supplemental releases affect the Wolbachia spreading dynamics in mosquito populations. Section 2 focuses on the case when only infected males are supplementally released at each generation. This release strategy has been proved to be capable of speeding up the Wolbachia infection by suppressing the compatible matings between uninfected mosquitoes in lab experiments [5]. Detailed model formulation, enumeration of the positive equilibria and their stability analysis are provided. Section 3 studies the case when only infected females are released at each generation. Similar to Section 2, we propose the corresponding discrete model, enumerate the possible equilibria, and analyse their stability. Finally, in Section 4, some numerical simulations are offered to get helpful implications on the design of the release strategy.

Releasing infected males with a constant ratio α
Continuous supplemental releases of infected male mosquitos at each generation can promote Wolbachia persistence by suppressing the effective matings between uninfected individuals [5]. In the following, we introduce our first discrete model and give a complete analysis of its dynamics.

Model formulation
Let I F n , I M n , U F n and U M n be the numbers of infected females, infected males, uninfected females and uninfected males at the nth generation, respectively. Under the assumption of equal sex determination [2], we have Set I n = I F n + I M n and U n = U F n + U M n . Then We assume that infected male mosquitoes are released at a ratio α to the female/male mosquito population size T n = I F n + U F n (= I M n + U M n ), which means that the number of released Wolbachia-infected males at the nth generation is αT n . Supplemental releases of infected males do not change the infection frequency of females, which is still x n . While the infection frequency of male mosquitos goes from x n to Let P I n+1 and P U n+1 be the proportions of infected and uninfected offspring at the (n + 1)th generation, respectively. Then the proportion of infected offspring is Under the assumptions of random mating [6] and incomplete CI, the proportion of uninfected offspring P U n+1 contains μ(1 − s f )x n produced by infected females, from matings between uninfected individuals. Hence, we have Therefore, a direct computation gives the first discrete model in this paper Model (4) contains (2) as a special case when α = 0. The number of nonnegative equilibria of model (4) and their stability are determined by different combinations of μ and α. In Section 2.2, we divide the parameter region {(μ, α) : 0 ≤ μ < 1, α > 0} into six subregions to study the existence of nonnegative equilibria, respectively. In Section 2.3, we give a complete analysis of the stability of nonnegative equilibria for each case.

Existence of equilibria
It is easy to see that the origin, denoted by x * 0 , is a boundary equilibrium of (4). For a nontrivial equilibrium of model (4), it satisfies (4). Now, we are going to determine the positive roots of f (x, α) = 0 lying in (0, 1). where We have the following result on the sign of D(μ, α).

Lemma 2.1:
The following three statements hold: Meanwhile, the x-coordinate of the minimum of y = f (x, α) determine the position and the number of positive solutions of f (x, α) = 0 lying in (0, 1). Set Then f (0, α) and x (α) can be rewritten as This leads to the following two lemmas on the signs of f (0, α) and x (α).

Lemma 2.2:
The following three statements hold: It's easy to prove that both α * 1 (μ) and α * 2 (μ) are strictly increasing functions, and α * Figure 1 divides the μα-plane into six subregions according to the signs of D(μ, α), f (0, α) and x (α), from which we can enumerate the positive equilibria of (4) as follows.

Theorem 2.3:
The following two statements are true.

Releasing infected females with a constant ratio β
Population replacement aims to replace a local mosquito population with Wolbachiainfected ones so that their capacity in transmitting disease is reduced, whose implementation requires the release of infected females. For this purpose, we formulate the second discrete model and then analyse its dynamics.

Model formulation
When supplemental infected females are released with a constant ratio β to the total number of male/female mosquitoes T n = I M n + U M n (= I F n + U F n ), the infection frequency of males is still x n , while the infection frequency of females increases from x n to I F n + βT n T n + βT n = x n + β 1 + β .
The proportion of infected mosquitoes at the (n + 1)th generation is since P I n+1 does not depend on the parental infection status. On P U n+1 , taking the imperfect maternal transmission and incomplete CI into consideration, we have where μ(1 − s f )(x n + β)/(1 + β) counts the proportion from infected females owing to maternal leakage, (1 − s h )x n (1 − x n )/(1 + β) is the proportion survived from CI, and (1 − x n ) 2 /(1 + β) represents the proportion from uninfected matings. Therefore, the second discrete model in this paper is expressed as
Particularly, since there are three possible cases to consider.

Comparisons on three release strategies introduced in (3), (4) and (16)
The above observation that model (16) performs better than model (4) for μ = 0.15 and α = β = 0.01 is not a special case, but a general one. To see this, we plot the infection frequency thresholds driven by model (3) with r = 0.0005, model (16) with β = 0.0005 and model (4) with α = 0.0005 for μ ∈ (0, μ * 1 ), s f = 0.1 and s h = 0.9 in Figure 6(A), the infection frequency threshold generated from model (3) is the smallest, while the release strategy with only infected males released requires the largest threshold for Wolbachia fixation. Meanwhile, Figure 6(B) plots the polymorphic states (the largest positive equilibria) for μ under the three release strategies driven by (3), (16) and (4), respectively. It shows that releasing both infected females and males brings the Wolbachia to fix at the highest infection level. And when only infected males are supplementally released, the Wolbachia infection frequency will fix at the lowest one. For the three release strategies, the increase of μ pulls the infection frequency thresholds higher, and drags down the Wolbachia fixation frequencies.
Following this procedure, we plot the curves of N by randomly selecting initial values in (0.18, 0.92) in Figure 7(A), which shows that among these three release strategies driven by (3), (16) and (4), the fastest to reach persistence is the release of both infected females and males, followed by the release of only infected females, and the lowest is the release of only infected males.
We end the whole manuscript with numerical trials for answering the second question, i.e. how supplemental releases of infected mosquitoes lower the infection frequency threshold to make the persistence achievable. To this end, still letting s f = 0.1, s h = 0.9, μ = 0.05, we plot the infection frequency thresholds for the release ratios lying in (0, 0.007) in Figure  7(B). Here we take 0.007 to guarantee the existence of the thresholds under three release strategies. And numerical observation agree with our theoretical results that higher release ratios lead to lower infection frequency thresholds to guarantee the success of Wolbachia persistence.

Disclosure statement
No potential conflict of interest was reported by the author(s).