1
|
Wan H, Wu Y, Fan G, Li D. Wolbachia invasion dynamics of a random mosquito population model with imperfect maternal transmission and incomplete CI. J Math Biol 2024; 88:72. [PMID: 38678110 DOI: 10.1007/s00285-024-02094-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/30/2021] [Revised: 04/10/2023] [Accepted: 04/11/2024] [Indexed: 04/29/2024]
Abstract
In this work, we formulate a random Wolbachia invasion model incorporating the effects of imperfect maternal transmission and incomplete cytoplasmic incompatibility (CI). Under constant environments, we obtain the following results: Firstly, the complete invasion equilibrium of Wolbachia does not exist, and thus the population replacement is not achievable in the case of imperfect maternal transmission; Secondly, imperfect maternal transmission or incomplete CI may obliterate bistability and backward bifurcation, which leads to the failure of Wolbachia invasion, no matter how many infected mosquitoes would be released; Thirdly, the threshold number of the infected mosquitoes to be released would increase with the decrease of the maternal transmission rate or the intensity of CI effect. In random environments, we investigate in detail the Wolbachia invasion dynamics of the random mosquito population model and establish the initial release threshold of infected mosquitoes for successful invasion of Wolbachia into the wild mosquito population. In particular, the existence and stability of invariant probability measures for the establishment and extinction of Wolbachia are determined.
Collapse
Affiliation(s)
- Hui Wan
- Ministry of Education Key Laboratory of NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China
| | - Yin Wu
- Ministry of Education Key Laboratory of NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China
| | - Guihong Fan
- Department of Mathematics, Columbus State University, Columbus, GA, 31907, USA
| | - Dan Li
- School of Mathematical Sciences, Anhui University, Hefei, 230601, China.
| |
Collapse
|
2
|
Zhu Z, Feng X, He X, Guo H. Mirrored dynamics of a wild mosquito population suppression model with Ricker-type survival probability and time delay. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2024; 21:1884-1898. [PMID: 38454666 DOI: 10.3934/mbe.2024083] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 03/09/2024]
Abstract
Here, we formulated a delayed mosquito population suppression model including two switching sub-equations, in which we assumed that the growth of the wild mosquito population obeys the Ricker-type density-dependent survival function and the release period of sterile males equals the maturation period of wild mosquitoes. For the time-switched delay model, to tackle with the difficulties brought by the non-monotonicity of its growth term to its dynamical analysis, we employed an essential transformation, derived an auxiliary function and obtained some expected analytical results. Finally, we proved that under certain conditions, the number of periodic solutions and their global attractivities for the delay model mirror that of the corresponding delay-free model. The findings can boost a better understanding of the impact of the time delay on the creation/suppression of oscillations harbored by the mosquito population dynamics and enhance the success of real-world mosquito control programs.
Collapse
Affiliation(s)
- Zhongcai Zhu
- Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China
- College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China
| | - Xiaomei Feng
- College of Science, Xi'an University of Science and Technology, Xi'an 710054, China
- School of Mathematics and Informational Technology, Yuncheng University, Yuncheng 044000, China
| | - Xue He
- College of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China
| | - Hongpeng Guo
- Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China
- College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China
| |
Collapse
|
3
|
Yu J, Li J. Mathematical modelling and release thresholds of transgenic sterile mosquitoes. JOURNAL OF BIOLOGICAL DYNAMICS 2023; 17:2285749. [PMID: 38015718 DOI: 10.1080/17513758.2023.2285749] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/11/2023] [Accepted: 11/11/2023] [Indexed: 11/30/2023]
Abstract
We formulate simple differential equation models to study the impact of releases of transgenic sterile mosquitoes carrying a dominant lethal on mosquito control based on the modified sterile insects technique. The early acting bisex, late acting bisex, early acting female-killing, and late acting female-killing lethality strategies are all considered. We determine release thresholds of the transgenic sterile mosquitoes, respectively, for these models by investigating the existence of positive equilibria and their stability. We compare the model dynamics, in particular, the thresholds of the models numerically. The late acting lethality strategies are generally more effective than their corresponding early acting lethality strategies, but the comparison between the late acting bisex and early acting female-killing lethality strategies depends on different parameter settings.
Collapse
Affiliation(s)
- Jianshe Yu
- Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou, People's Republic of China
| | - Jia Li
- Department of Mathematical Sciences, The University of Alabama in Huntsville, Huntsville, AL, USA
| |
Collapse
|
4
|
Xue L, Jin X, Zhu H. Assessing the impact of serostatus-dependent immunization on mitigating the spread of dengue virus. J Math Biol 2023; 87:5. [PMID: 37301798 DOI: 10.1007/s00285-023-01944-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/30/2021] [Revised: 05/29/2023] [Accepted: 05/29/2023] [Indexed: 06/12/2023]
Abstract
Dengue is the most rapidly spreading mosquito-borne disease that poses great threats to public health. We propose a compartmental model with primary and secondary infection and targeted vaccination to assess the impact of serostatus-dependent immunization on mitigating the spread of dengue virus. We derive the basic reproduction number and investigate the stability and bifurcations of the disease-free equilibrium and endemic equilibria. The existence of a backward bifurcation is proved and is used to explain the threshold dynamics of the transmission. We also carry out numerical simulations and present bifurcation diagrams to reveal rich dynamics of the model such as bi-stability of the equilibria, limit cycles, and chaos. We prove the uniform persistence and global stability of the model. Sensitivity analysis suggests that mosquito control and protection from mosquito bites are still the key measures of controlling the spread of dengue virus, though serostatus-dependent immunization is implemented. Our findings provide insightful information for public health in mitigating dengue epidemics through vaccination.
Collapse
Affiliation(s)
- Ling Xue
- College of Mathematical Sciences, Harbin Engineering University, Harbin, 150001, Heilongjiang, China.
| | - Xiulei Jin
- College of Mathematical Sciences, Harbin Engineering University, Harbin, 150001, Heilongjiang, China
| | - Huaiping Zhu
- Laboratory of Mathematical Parallel Systems (LAMPS), Department of Mathematics and Statistics, Centre for Diseases Modelling (CDM), York University, Toronto, Canada.
| |
Collapse
|
5
|
Joseph D, Ramachandran R, Alzabut J, Jose SA, Khan H. A Fractional-Order Density-Dependent Mathematical Model to Find the Better Strain of Wolbachia. Symmetry (Basel) 2023. [DOI: 10.3390/sym15040845] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/05/2023] Open
Abstract
The primary objective of the current study was to create a mathematical model utilizing fractional-order calculus for the purpose of analyzing the symmetrical characteristics of Wolbachia dissemination among Aedesaegypti mosquitoes. We investigated various strains of Wolbachia to determine the most sustainable one through predicting their dynamics. Wolbachia is an effective tool for controlling mosquito-borne diseases, and several strains have been tested in laboratories and released into outbreak locations. This study aimed to determine the symmetrical features of the most efficient strain from a mathematical perspective. This was accomplished by integrating a density-dependent death rate and the rate of cytoplasmic incompatibility (CI) into the model to examine the spread of Wolbachia and non-Wolbachia mosquitoes. The fractional-order mathematical model developed here is physically meaningful and was assessed for equilibrium points in the presence and absence of disease. Eight equilibrium points were determined, and their local and global stability were determined using the Routh–Hurwitz criterion and linear matrix inequality theory. The basic reproduction number was calculated using the next-generation matrix method. The research also involved conducting numerical simulations to evaluate the behavior of the basic reproduction number for different equilibrium points and identify the optimal CI value for reducing disease spread.
Collapse
Affiliation(s)
- Dianavinnarasi Joseph
- Centre for Nonlinear Systems, Chennai Institute of Technology, Chennai 600069, Tamil Nadu, India
| | - Raja Ramachandran
- Ramanujan Centre for Higher Mathematics, Alagappa University, Karaikudi 630004, India
- Department of Computer Science and Mathematics, Lebanese American University, Beirut 1102-2801, Lebanon
| | - Jehad Alzabut
- Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
- Department of Industrial Engineering, OSTIM Technical University, Ankara 06374, Turkey
| | - Sayooj Aby Jose
- Department of Mathematics, Alagappa University, Karaikudi 630004, India
- School of Mathematics & Statistics, Mahatma Gandhi University, Kottayam 686560, Kerala, India
| | - Hasib Khan
- Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
- Department of Mathematics, Shaheed Benazir Bhutto University Sheringal Dir Upper, Khyber Pakhtunkhwa 18000, Pakistan
| |
Collapse
|
6
|
Yan R, Sun Q. Uniqueness and stability of periodic solutions for an interactive wild and Wolbachia-infected male mosquito model. JOURNAL OF BIOLOGICAL DYNAMICS 2022; 16:254-276. [PMID: 35166193 DOI: 10.1080/17513758.2022.2037760] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/30/2021] [Accepted: 01/26/2022] [Indexed: 06/14/2023]
Abstract
We investigate a mosquito population suppression model, which includes the release of Wolbachia-infected males causing incomplete cytoplasmic incompatibility (CI). The model consists of two sub-equations by considering the density-dependent birth rate of wild mosquitoes. By assuming the release waiting period T is larger than the sexual lifespan T¯ of Wolbachia-infected males, we derive four thresholds: the CI intensity threshold sh∗, the release amount thresholds g∗ and c∗, and the waiting period threshold T∗. From a biological view, we assume sh>sh∗ throughout the paper. When g∗<c<c∗, we prove the origin E0 is locally asymptotically stable iff T<T∗, and the model admits a unique T-periodic solution iff T≥T∗, which is globally asymptotically stable. When c≥c∗, we show the origin E0 is globally asymptotically stable iff T≤T∗, and the model has a unique T-periodic solution iff T>T∗, which is globally asymptotically stable. Our theoretical results are confirmed by numerical simulations.
Collapse
Affiliation(s)
- Rong Yan
- College of Mathematics and Information Sciences, Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou, People's Republic of China
| | - Qiwen Sun
- College of Mathematics and Information Sciences, Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou, People's Republic of China
| |
Collapse
|
7
|
Zhu Z, Yan R, Feng X. Existence and stability of two periodic solutions for an interactive wild and sterile mosquitoes model. JOURNAL OF BIOLOGICAL DYNAMICS 2022; 16:277-293. [PMID: 35007472 DOI: 10.1080/17513758.2021.2023666] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/27/2021] [Accepted: 11/28/2021] [Indexed: 06/14/2023]
Abstract
In this paper, we study the periodic and stable dynamics of an interactive wild and sterile mosquito population model with density-dependent survival probability. We find a release amount upper bound G∗, depending on the release waiting period T, such that the model has exactly two periodic solutions, with one stable and another unstable, provided that the release amount does not exceed G∗. A numerical example is also given to illustrate our results.
Collapse
Affiliation(s)
- Zhongcai Zhu
- Center for Applied Mathematics, Guangzhou University, Guangzhou, China
| | - Rong Yan
- Center for Applied Mathematics, Guangzhou University, Guangzhou, China
| | - Xiaomei Feng
- School of Mathematics and Informational Technology, Yuncheng University, Yuncheng, China
| |
Collapse
|
8
|
Dynamic Behavior of an Interactive Mosquito Model under Stochastic Interference. MATHEMATICS 2022. [DOI: 10.3390/math10132284] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
For decades, mosquito-borne diseases such as dengue fever and Zika have posed serious threats to human health. Diverse mosquito vector control strategies with different advantages have been proposed by the researchers to solve the problem. However, due to the extremely complex living environment of mosquitoes, environmental changes bring significant differences to the mortality of mosquitoes. This dynamic behavior requires stochastic differential equations to characterize the fate of mosquitoes, which has rarely been considered before. Therefore, in this article, we establish a stochastic interactive wild and sterile mosquito model by introducing the white noise to represent the interference of the environment on the survival of mosquitoes. After obtaining the existence and uniqueness of the global positive solution and the stochastically ultimate boundedness of the stochastic system, we study the dynamic behavior of the stochastic model by constructing a series of suitable Lyapunov functions. Our results show that appropriate stochastic environmental fluctuations can effectively inhibit the reproduction of wild mosquitoes. Numerical simulations are provided to numerically verify our conclusions: the intensity of the white noise has an effect on the extinction and persistence of both wild and sterile mosquitoes.
Collapse
|
9
|
Periodic Orbits of a Mosquito Suppression Model Based on Sterile Mosquitoes. MATHEMATICS 2022. [DOI: 10.3390/math10030462] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/10/2022]
Abstract
In this work, we investigate the existence and stability of periodic orbits of a mosquito population suppression model based on sterile mosquitoes. The model switches between two sub-equations as the actual number of sterile mosquitoes in the wild is assumed to take two constant values alternately. Employing the Poincaré map method, we show that the model has at most two T-periodic solutions when the release amount is not sufficient to eradicate the wild mosquitoes, and then obtain some sufficient conditions for the model to admit a unique or exactly two T-periodic solutions. In particular, we observe that the model displays bistability when it admits exactly two T-periodic solutions: the origin and the larger periodic solution are asymptotically stable, and the smaller periodic solution is unstable. Finally, we give two numerical examples to support our lemmas and theorems.
Collapse
|
10
|
Huang M, Liu S, Song X. Study of the sterile insect release technique for a two-sex mosquito population model. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2021; 18:1314-1339. [PMID: 33757187 DOI: 10.3934/mbe.2021069] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
In this paper, to study the large-scale time control and limited-time control of mosquito population in a field, a two-sex mosquito population model with stage structure and impulsive releases of sterile males is proposed. For the large-scale time control, a wild mosquito-free periodic solution is given and conditions under which it is globally stable are obtained by the use of the monotone system theory. Besides, based on the stability analysis, threshold conditions under which the wild mosquito population is eliminated or not are obtained. Then we study three different optimal release strategies for the limited-time control, which takes into account both of the population control level of wild mosquitoes and the economic input. To solve technical problems in optimal impulsive control, a time rescaling technique is applied and the gradients of cost function with respect to all control parameters are obtained. In addition, by the aid of numerical simulation, we get the optimal release amounts and release timings for each release strategy. Our study indicates that the optimal release timing control is superior to the optimal release amount control. However, simultaneous optimal selection of release amount and release timing leads to the best control performance.
Collapse
Affiliation(s)
- Mingzhan Huang
- College of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China
| | - Shouzong Liu
- College of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China
| | - Xinyu Song
- College of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China
- College of Mathematics and Statistics, Huanghuai University, Zhumadian 463000, China
| |
Collapse
|
11
|
Lin G, Hui Y. Stability analysis in a mosquito population suppression model. JOURNAL OF BIOLOGICAL DYNAMICS 2020; 14:578-589. [PMID: 32672096 DOI: 10.1080/17513758.2020.1792565] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/18/2020] [Accepted: 06/26/2020] [Indexed: 06/11/2023]
Abstract
In this work, we study a non-autonomous differential equation model for the interaction of wild and sterile mosquitoes. Suppose that the number of sterile mosquitoes released in the field is a given nonnegative continuous function. We determine a threshold [Formula: see text] for the number of sterile mosquitoes and provide a sufficient condition for the origin [Formula: see text] to be globally asymptotically stable based on the threshold [Formula: see text]. For the case when the number of sterile mosquitoes keeps at a constant level, we find that the origin [Formula: see text] is globally asymptotically stable if and only if the constant number [Formula: see text] of sterile mosquitoes released in the field is above [Formula: see text].
Collapse
Affiliation(s)
- Genghong Lin
- Center for Applied Mathematics, Guangzhou University, Guangzhou, People's Republic of China
| | - Yuanxian Hui
- Center for Applied Mathematics, Guangzhou University, Guangzhou, People's Republic of China
- School of Mathematics and Statistics, Pu'er University, Pu'er, People's Republic of China
| |
Collapse
|
12
|
The Threshold Infection Level for [Formula: see text] Invasion in a Two-Sex Mosquito Population Model. Bull Math Biol 2019; 81:2596-2624. [PMID: 31161558 DOI: 10.1007/s11538-019-00620-1] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/18/2018] [Accepted: 05/25/2019] [Indexed: 10/26/2022]
Abstract
In this paper, we formulate a new [Formula: see text] infection model in a two-sex mosquito population with stage structure. Some key factors of [Formula: see text] infection, including cytoplasmic incompatibility (CI), male killing (MK) effect, maternal transmission, fecundity cost due to fitness effect and different mortality rates for infected individuals, are captured. Dynamical analysis has been carried out, and the basic reproduction number [Formula: see text] for [Formula: see text] infection has been calculated. Our analysis shows that [Formula: see text] can establish in a mosquito population if [Formula: see text] is greater than unity. If [Formula: see text] is less than unity, [Formula: see text] establishment still can be achieved if backward bifurcation occurs. Under this circumstance, the initial values lying in the basin of attraction of the stable [Formula: see text]-established equilibrium are essential to guarantee [Formula: see text] establishment. In particular, the method to find the basin of attraction and evaluate the threshold initial values is given. Besides, according to a comparison of different releasing strategies, it is shown that, from the perspective of economy and disease control, keeping the number of infected female mosquitoes to a necessary minimum by relying on higher number of male mosquitoes released is a desirable strategy. Moreover, global and local sensitivity analysis and numerical simulation have been performed to explore the impact of model parameters to the success of population establishment. Our results suggest that low levels of MK effect and fitness costs as well as high levels of CI and maternal inheritance are in favor of [Formula: see text] establishment. Moreover, not considering MK effect and incomplete CI effect may result in the underestimation of the number of infected mosquitoes needed to be released.
Collapse
|
13
|
Zheng B, Yu J. Characterization of Wolbachia enhancing domain in mosquitoes with imperfect maternal transmission. JOURNAL OF BIOLOGICAL DYNAMICS 2018; 12:596-610. [PMID: 30025503 DOI: 10.1080/17513758.2018.1499969] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/12/2017] [Accepted: 07/03/2018] [Indexed: 06/08/2023]
Abstract
A novel method to reduce the burden of dengue is to seed wild mosquitoes with Wolbachia-infected mosquitoes in dengue-endemic areas. Concerns in current mathematical models are to locate the Wolbachia introduction threshold. Our recent findings manifest that the threshold is highly dependent on the initial population size once Wolbachia infection alters the logistic control death rate of infected females. However, counting mosquitoes is beyond the realms of possibility. A plausible method is to monitor the infection frequency. We propose the concept of Wolbachia enhancing domain in which the infection frequency keeps increasing. A detailed description of the domain is presented. Our results suggest that both the initial population size and the infection frequency should be taken into account for optimal release strategies. Both Wolbachia fixation and extinction permit the oscillation of the infection frequency.
Collapse
Affiliation(s)
- Bo Zheng
- a College of Mathematics and Information Sciences , Guangzhou University , Guangzhou , People's Republic of China
- b Center for Applied Mathematics , Guangzhou University , Guangzhou , People's Republic of China
| | - Jianshe Yu
- a College of Mathematics and Information Sciences , Guangzhou University , Guangzhou , People's Republic of China
- b Center for Applied Mathematics , Guangzhou University , Guangzhou , People's Republic of China
| |
Collapse
|
14
|
Li J. New revised simple models for interactive wild and sterile mosquito populations and their dynamics. JOURNAL OF BIOLOGICAL DYNAMICS 2017; 11:316-333. [PMID: 27686737 DOI: 10.1080/17513758.2016.1216613] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
Based on previous research, we formulate revised, new, simple models for interactive wild and sterile mosquitoes which are better approximations to real biological situations but mathematically more tractable. We give basic investigations of the dynamical features of these simple models such as the existence of equilibria and their stability. Numerical examples to demonstrate our findings and brief discussions are also provided.
Collapse
Affiliation(s)
- Jia Li
- a Department of Mathematical Sciences , University of Alabama in Huntsville , Huntsville , AL , USA
| |
Collapse
|