- Research
- Open Access
The roles of maturation delay and vaccination on the spread of Dengue virus and optimal control
- Lin-Fei Nie^{1}Email authorView ORCID ID profile and
- Ya-Nan Xue^{1}
https://doi.org/10.1186/s13662-017-1323-y
© The Author(s) 2017
Received: 7 June 2017
Accepted: 19 August 2017
Published: 12 September 2017
Abstract
A mathematical model of Dengue virus transmission between mosquitoes and humans, incorporating a control strategy of imperfect vaccination and vector maturation delay, is proposed in this paper. By using some analytical skills, we obtain the threshold conditions for the global attractiveness of two disease-free equilibria and prove the existence of a positive equilibrium for this model. Further, we investigate the sensitivity analysis of threshold conditions. Additionally, using the Pontryagin maximum principle, we obtain the optimal control strategy for the disease. Finally, numerical simulations are delivered to verify the correctness of the theoretical results, the feasibility of a vaccination control strategy, and the influences of the controlling parameters on the control and elimination of this disease. Theoretical results and numerical simulations show that the vaccination rate and effectiveness of vaccines are two key factors for the control of Dengue spread, and the manufacture of the Dengue vaccine is also architecturally significant.
Keywords
- Dengue vaccination
- maturation delay
- disease-free equilibrium and endemic equilibrium
- attractiveness and bifurcation
- sensitivity and optimal control
1 Introduction
Dengue is a vector-borne disease which transcends international borders as the most important arbovirus disease currently threatening human populations. In the light of evolution, at least approximately 50-100 million people are affected by the Dengue virus each year [1]. The Dengue virus is transmitted to humans by mosquitos, mostly the Aedes aegypti and Aedes albopictus. As far back as 1981, Jousset [2] published geographic locations of Aedes aegypti strains and the Dengue virus. To better describe the influences of the Dengue virus, many scholars have investigated Dengue transmission in mathematical models (see [3–5] and the references therein). Particularly, Esteva et al. [6] proposed a Dengue virus transmission model and analyzed the global stability of equilibria, and the control measures of the vector population are also discussed in terms of threshold conditions. Further, Wang et al. [7] proposed a nonlocal and time-delayed reaction-diffusion model of the Dengue virus, and established threshold dynamics in terms of the basic reproduction number. In addition, Garba et al. [8] proposed a deterministic model for the transmission dynamics of a strain of Dengue, which allows for transmission by exposed humans and mosquitoes. They proved the existence and local asymptotical stability of the disease-free equilibrium if the basic reproduction number is less than unity. The authors also examined the phenomenon of backward bifurcation.
How to control and eliminate the Dengue virus has always been a hot topic. Until now, the available strategy that controls the spread of Dengue virus only controls the vector. Despite combined community participation and vector control, together with active disease surveillance and insecticides, the examples of successful Dengue prevention and control on a national scale are few [1]. Besides, with the increase of vector resistance, the intervals between treatments are shorter. Moreover, as a result of the high costs of development and registration and low gains, only few insecticide products are offered on the market [9]. Considering these realities, vaccination could be more effective to protect against the Dengue virus [10].
It is a well-known fact that vaccination has already been successfully applied to control and eliminate various infectious diseases. Particularly, in 1760, the Swiss mathematician Daniel Bernoulli published an investigation on the impact of immunization with cowpox. Then, the means of protecting people from infection through immunization began to be widely used. In addition, the method has already successfully decreased both mortality and morbidity [11–13]. In fact, during the 1940s, Dengue vaccines were under development. In recent years, however, with the increase in Dengue infections and a serious need for faster development of a vaccine [14], the progress in Dengue vaccines development has amazingly accelerated. To guide public support for vaccine development in both industrialized and developing countries, economic analysis has been conducted, including previous cost-effectiveness studies of Dengue [15–17]. The cost of the disease burden is compared with the possibility of making a vaccination campaign, by the authors of this analytical work; finally, they consider that Dengue vaccines, as a means of intervention, have a potential economic benefit.
On the other hand, there are three successive aquatic juvenile phases (egg, larva and pupa) and one adult pupa of the life cycle of mosquitoes [18]. The duration of the development from egg to adult (1-2 weeks) is often compared to the average life span of an adult mosquito (about 3 weeks). The size of the mosquito population is strongly affected by temperature, and the number of female mosquitoes changes accordingly due to seasonal variations [19, 20]. Therefore, it is vital to consider the maturation time of mosquitoes [21], the length of the larval phase from egg to adult mosquitoes, and the impact on the spread of the Dengue virus.
Based on the above-mentioned information and the immature Dengue vaccine, a delayed mathematical model of dynamical Dengue transmission between mosquitoes and humans, incorporating a control strategy of imperfect vaccination, is proposed in this paper, aiming to discuss the influences of vaccination and a maturation delay for controlling and eliminating the Dengue virus. The rest of the paper is structured as follows. Section 2 describes an imperfect vaccination model with the maturation time of mosquitoes, and the basic properties of this model are presented in Section 3. In Section 4, the threshold conditions and the existence and attractiveness of equilibria of the model are discussed. In Section 5, we will investigate the sensitivity of our threshold conditions. In Section 6, we discuss the optimal control strategy for the disease. Finally, we give numerical simulations in Section 7, and present some concluding comments in Section 8.
2 Model formulation
In this section, we present a mathematical model to study the transmission dynamics of the Dengue virus. The model is based on a susceptible, infectious, recovered and vaccinated structure and explains the transmission process of humans and mosquitoes. Let \(S_{h}(t)\), \(I_{h}(t)\), \(R_{h}(t)\) and \(V_{h}(t)\) denote the numbers of susceptible (individuals who can contract the disease), infectious (individuals who are capable of transmitting the disease), resistant (individuals who have recovered and acquired immunity) and vaccinated (individuals who were vaccinated and are now immune) individuals at time t, respectively. Similarly, \(S_{m}(t)\) and \(I_{m}(t)\) represent the numbers of susceptible (mosquitoes able to contract the disease) and infectious (mosquitoes capable of transmitting the disease to humans) adult female mosquitoes at time t. Here the total numbers of humans and mosquitoes are denoted by \(N_{h}(t)=S_{h}(t)+I_{h}(t)+V_{h}(t)+R_{h}(t)\) and \(N_{m}(t)=S_{m}(t)+I_{m}(t)\), respectively.
Parameter interpretations, value ranges and sources of model ( 1 )
Param. | Description | Value | Source |
---|---|---|---|
b | Average number of bites by infectious mosquitoes (day^{−1}) | [0,1] | [10] |
\(\beta_{hm}\) | Transmission probability from infectious individuals to mosquitoes | [0,1] | [10] |
\(\beta_{mh}\) | Transmission probability from infectious mosquitoes to humans | [0,1] | [10] |
\({1}/{\mu_{h}}\) | Average human life expectancy (day) | [18250,27375] | [23] |
\(\eta_{h}\) | Dengue recovery rate in humans (day^{−1}) | [0.1,0.6] | [23] |
1/α | Size of mosquitoes at which egg laying is maximized without delay | - | - |
\(r_{m}\) | Maximum per capita daily mosquito egg production rate (day^{−1}) | [0.036,42.5] | [22] |
τ | Maturation time of the mosquito (day) | [5,30] | [22] |
\(d_{j}\) | Death rate of juvenile mosquitoes (day^{−1}) | [0.28,0.46] | [22] |
\(d_{m}\) | Natural death rate of adult female mosquitoes (day^{−1}) | [0.016,0.25] | [23] |
q | Vertical transmission probability of virus in mosquitoes | [0,1] | - |
ψ | Fraction of susceptible class that has been vaccinated | [0,1] | - |
θ | Waning rate of immunity | [0,1] | - |
σ | Infection rate of vaccinated members | [0,1] | - |
3 Basic properties
The following theorem describes the global dynamical behavior of model (3).
Theorem 1
- (i)
if \(\mathcal{R}_{01}\leq1\), then solution \(N_{m}(t)\) is bounded and the trivial equilibrium \(N_{m}=0\) is globally asymptotically stable;
- (ii)if \(\mathcal{R}_{01}>1\), then \(h< N_{m}(t)< H\) for any \(t\geq0\), whereMoreover, model (3) has a unique positive equilibrium \(N_{m}^{*}\) which is globally asymptotically stable.$$ h=\frac{1}{2}\min \Bigl\{ \min_{\theta\in[-\tau, 0]} \bigl\{ \phi _{s}(\theta)+\phi_{i}(\theta) \bigr\} , N_{m}^{*} \Bigr\} ,\quad\quad H=1+\max \Bigl\{ N_{m}^{*}, \max _{\theta\in[-\tau, 0]} \bigl\{ \phi _{s}(\theta)+\phi_{i}( \theta) \bigr\} \Bigr\} . $$
Proof
In order to prove that the global stability of equilibria \(N_{m}=0\) and \(N_{m}^{*}\), we denote the right hand side of (3) as functions \(f(N_{m}(t)\text{ and }N_{m}(t-\tau))\). Since \(\partial f(x, y)/\partial y>0\), it follows that (3) generates an eventually strongly monotone semiflow on the space \(\mathcal{C}\) of a continuous function on \([-\tau, 0]\) with the usual pointwise ordering (see Smith [24]). If \(\mathcal{R}_{01}\leq1\), there is only a single trivial equilibrium \(N_{m}=0\). By Theorem 2.3.1 in [24], the equilibrium \(N_{m}=0\) is globally asymptotically stable. If \(\mathcal{R}_{01}>1\), there are two equilibria \(N_{m}=0\) and \(N_{m}^{*}\). By Theorem 2.3.2 in [24], solutions of (3) converge to one of two equilibria. To eliminate the possibility of \(N_{m}=0\) as an attractor, we linearize the system about \(N_{m}=0\) and use Theorem A2 in [25] to conclude that it is unstable when \(\mathcal{R}_{01}>1\). Hence \(N_{m}(t)\rightarrow N_{m}^{*}\) as \(t\rightarrow\infty\). □
4 Existence and attractiveness of equilibria
Theorem 2
Noting that \(C\leq0\) if and only if \(\mathcal{R}_{02}\geq1\). It is clear from Theorem 2 (Case (i)) that the model has a unique endemic equilibrium if \(\mathcal{R}_{01}\geq1\) and \(\mathcal {R}_{02}>1\). Further, Case (ii) indicates the possibility of backward bifurcation (where a local asymptotically stable disease-free equilibrium co-exists with a locally asymptotically stable endemic equilibrium) in model (1) for \(\mathcal{R}_{01}\geq1\) and \(\mathcal{R}_{02}<1\). To check for this, the discriminant \(B^{2}-4AC\) is set to zero and solved for the critical value of \(\mathcal{R}_{02}\), denoted by \(\mathcal{R}_{02}^{c}\). Thus, backward bifurcation would occur for values of \(\mathcal{R}_{02}\) such that \(\mathcal{R}_{01}\geq 1\) and \(\mathcal{R}_{02}^{c}<\mathcal{R}_{02}<1\).
Now, on the globally asymptotically stable disease-free equilibrium without mosquitoes \(E_{01}\) of model (1), we have Theorem 3.
Theorem 3
If \(\mathcal{R}_{01}<1\), then model (1) has a unique disease-free equilibrium without mosquitoes \(E_{01}\), which is globally asymptotically stable.
Proof
Finally, we give a conclusion on the global attractiveness of the disease-free equilibrium with mosquitoes \(E_{02}\) of model (1).
Theorem 4
Proof
According to the above discussion and the comparison theorem of differential equations, we know that \(\lim_{t\rightarrow\infty} I_{m}(t)=0\) and \(\lim_{t\rightarrow\infty} I_{h}(t)=0\) for \(\mathcal {R}_{01}>1\) and \(\mathcal{R}_{02}^{*}<1\). Finally, in the light of Theorem 1, we get \(\lim_{t\rightarrow\infty }(S_{h}(t),I_{h}(t),V_{h}(t),R_{h}(t),S_{m}(t),I_{m}(t))=E_{02}\). This completes the proof. □
Remark 1
Obviously, \(qe^{d_{m}\tau}\approx q\) due to the small vertical transmission probability q according to existing literature, therefore \(\mathcal{R}_{02}\approx\mathcal{R}_{02}^{*}\).
5 Description of sensitivity analysis
Sensitivity indices enable us to measure the relative change in a state variable when a model parameter changes. The normalized forward sensitivity index of a variable to a model parameter is the ratio of the relative change in the variable to the relative change in the parameter. When the variable is a differentiable function of the parameter, the sensitivity index may be alternatively defined using partial derivatives.
Definition 1
Sensitivity index [27]
Sensitivity indices of \(\pmb{\mathcal{R}_{02}}\) and \(\pmb{\mathcal {R}_{02}^{*}}\) to the parameter values for model ( 1 )
Variable | Parameter | Sensitivity index |
---|---|---|
\(\mathcal{R}_{02}\) | θ | 0.06051 |
σ | 0.92280 | |
ψ | −0.06075 | |
q | 0.01010 | |
τ | - | |
b | 2 | |
\(\beta_{mh}\) | 1 | |
\(\beta_{hm}\) | 1 | |
\(\mathcal{R}_{02}^{*}\) | θ | - |
σ | - | |
ψ | - | |
q | 0.01237 | |
τ | 0.0025 | |
b | 2 | |
\(\beta_{mh}\) | 1 | |
\(\beta_{hm}\) | 1 |
Note from Table 2 that \(\mathcal{R}_{02}\) and \(\mathcal {R}_{02}^{*}\) all show the greatest sensitivities to the biting rate b, followed by the transmission probabilities \(\beta_{mh}\) and \(\beta_{hm}\). Accordingly, a reduction of 1% in the biting rate b decreases \(\mathcal{R}_{02}\) by 2%, which equals the decrease in \(\mathcal{R}_{02}^{*}\) when identically varying the biting rate parameter b; further, a reduction of 1% in the transmitting rate \(\beta_{mh}\) or \(\beta_{hm}\) decreases \(\mathcal{R}_{02}\) and \(\mathcal{R}_{02}^{*}\) both by 1%. Next, a reduction of 1% in the waning rate θ decreases \(\mathcal{R}_{02}\) by 0.06051%, a reduction of 1% in the infection rate of vaccinated members σ decreases \(\mathcal{R}_{02}\) by 0.92280%, and a reduction of 1% of the vaccinated fraction of the susceptible class ψ increases \(\mathcal{R}_{02}\) by 0.06075%. Lastly, a reduction of 1% in the vertical transmission probability q decreases \(\mathcal{R}_{02}\) and \(\mathcal{R}_{02}^{*}\) by 0.01010% and 0.01237%, respectively; a reduction of 1% in the maturation time of the mosquito τ decreases \(\mathcal{R}_{02}^{*}\) by 0.0025%.
Obviously, the sensitivity index of the infection rate of vaccinated members σ exceeds that of the fraction ψ of the susceptible class that was vaccinated, though the value of σ (\(\sigma=0.2\)) is smaller than the value of ψ (\(\psi=0.6\)). The sensitivity index of the fraction ψ of the susceptible class that was vaccinated is substantial near the sensitivity index of the waning rate θ for the values above. Then the sensitivity index of the vertical transmission probability q is very small. This is perhaps related to the small value of q (\(q=0.01\)). The sensitivity level of τ is the smallest, that is, the maturation time of the mosquito has less effect on the variation of \(\mathcal{R}_{02}^{*}\).
6 Analysis of optimal vaccination
Theorem 5
There is an optimal control \(\psi^{*}(t)\) such that \(J(\psi ^{*}(t))=\min J(\psi(t))\), subject to the control system (12) with the initial condition (2).
Now, let us derive a necessary condition for the optimal control strategy by means of the Pontryagin maximum principle [32]. Similar proof methods can also be found in [33–35] and the references therein.
Theorem 6
Proof
7 Numerical simulation and discussion
The parameter values for model ( 1 )
Parameter | b | \(\beta_{mh}\) | \(\beta_{hm}\) | ψ | σ | θ | τ |
Value | 0.8 | 0.375 | 0.375 | 0.6 | 0.05 | 1/365 | 12 |
Parameter | q | \(\eta_{h}\) | \(\mu_{h}\) | α | \(r_{m}\) | \(d_{j}\) | \(d_{m}\) |
Value | 0.007 | 1/3 | 1/(71 × 365) | 0.0000025 | 15 | 0.37 | 0.05 |
8 Concluding remarks
In this paper, we propose a mathematical model to describe Dengue virus transmission between mosquitoes and humans, where imperfect vaccination and vector maturation delay are introduced. The notation used in our mathematical model includes the compartment \(V_{h}\), which represents the group of human population that is vaccinated, in order to distinguish the resistance obtained through vaccination and the one achieved by disease recovery. By using some analytical skills, the dynamical behavior of this model is discussed. This includes the global attractiveness of two disease-free equilibria, the existence of a positive equilibrium, the sensitivity analysis of threshold conditions, and the optimal control strategy for the disease. In addition, numerical simulations are also carried out to verify the correctness of the theoretical results and the feasibility of the vaccination control strategy. Theoretical results and numerical simulations show that the vaccination rate and effectiveness of the vaccine are two key factors for control of the spread of Dengue.
It is well known that there are four distinct serotypes of Dengue virus (DEN1, DEN2, DEN3 and DEN4), according to clinical data collected during the past years. Therefore, one person in an endemic area can suffer from four Dengue infections during his lifetime, one with each serotype. Epidemiological studies [36] support the hypothesis that recovered0 people can be re-infected with a different serotype, and face an increased risk of developing Dengue hemorrhagic fever and Dengue shock syndrome. In recent publications, some multi-strain Dengue fever transmission models have been discussed (see [37–40] and the references therein). However, all individuals who are capable of transmitting the disease are in one class in our model for the purpose of mathematical analysis. Therefore, for a more detailed understanding of the transmission of four Dengue virus strains between mosquitoes and humans, we intend to study the influences of vaccination and maturation delay for a multi-strain Dengue model in the future.
Dengue is a tropical vector-borne disease, difficult to prevent and manage. Researchers agree that the development of a vaccine for Dengue is a question of high priority. Recently, a novel method to fight mosquitoes is using a bacterium called Wolbachia, which exists in spiders and up to 75% of the insects, including ticks and mites. Stable Wolbachia strains in Aedes aegypti have also been established. And subsequent studies have shown that, very importantly, Wolbachia blocks the replication of Dengue viruses in mosquitoes. Thence, an increasing number of people realize that replacing the wild mosquitoes with Wolbachia infected mosquitoes is safer, and more feasible than vaccination to some extent. Based on this, there are many mathematical models (including discrete-time and continuous-time models) that are used to investigated the spread of Wolbachia infection (see [41–46] and the references therein). As future work we intend to compare the advantages and disadvantages of the two control strategies (Wolbachia and vaccination). It would also be interesting to investigate what happens if two control strategies are taken at the same time.
Declarations
Acknowledgements
This research is supported by the Natural Science Foundation of Xinjiang (Grant No. 2016D01C046).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Dengue and dengue haemorrhagic fever. WHO factsheets No. 117 (2009) Google Scholar
- Jousset, FX: Geographic Aedes aegypti strains and dengue-2 virus: susceptibility, ability to transmit to vertebrate and transovarial transmission. Ann. Inst. Pasteur., Virol. 132, 357-370 (1981) View ArticleGoogle Scholar
- Aldila, D, Götz, T, Soewono, E: An optimal control problem arising from a dengue disease transmission model. Math. Biosci. 242, 9-16 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Kooi, BW, Maira, A, Nico, S: Analysis of an asymmetric two-strain dengue model. Math. Biosci. 248, 128-139 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Coutinho, FAB, Burattini, MN, Lopez, LF, Massad, E: Threshold conditions for a non-autonomous epidemic system describing the population dynamics of dengue. Bull. Math. Biol. 68, 2263-2282 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Esteva, L, Vargas, C: Analysis of a dengue disease transmission model. Math. Biosci. 150, 131-151 (1998) View ArticleMATHGoogle Scholar
- Wang, WD, Zhao, XQ: A nonlocal and time-delayed reaction-diffusion model of dengue transmission. SIAM J. Appl. Math. 71, 147-168 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Garba, SM, Gumel, AB, Bakar, MRA: Backward bifurcations in dengue transmission dynamics. Math. Biosci. 215, 11-25 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Keeling, MJ, Rohani, P: Modeling Infectious Diseases in Humans and Animals p. 415. Princeton University Press, Princeton (2008) MATHGoogle Scholar
- Rodrigues, HS, Monteiro, MTT, Torres, DFM: Vaccination models and optimal control strategies to dengue. Math. Biosci. 247, 1-12 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Scherer, A, McLean, A: Mathematical models of vaccination. Br. Med. Bull. 62, 187-199 (2002) View ArticleGoogle Scholar
- Kar, TK, Jana, S: Application of three controls optimally in a vector-borne disease: a mathematical study. Commun. Nonlinear Sci. Numer. Simul. 18, 2868-2884 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Nelson, OO: Determination of optimal vaccination strategies using an orbital stability threshold from periodically driven systems. J. Math. Biol. 68, 763-784 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Murrel, S, Butler, SCW: Review of dengue virus and the development of a vaccine. Biotechnol. Adv. 29, 239-247 (2011) View ArticleGoogle Scholar
- Clark, DV, Mammen, MPJ, Nisalak, A, Puthimethee, V, Endy, TP: Economic impact of dengue fever/dengue hemorrhagic fever in Thailand at the family and population levels. Am. J. Trop. Med. Hyg. 72, 786-791 (2005) Google Scholar
- Shepard, DS, et al.: Cost-effectiveness of a pediatric dengue vaccine. Vaccine 22, 1275-1280 (2004) View ArticleGoogle Scholar
- Suaya, JA, et al.: Cost of dengue cases in eight countries in the Americas and Asia: a prospective study. Am. J. Trop. Med. Hyg. 80, 846-855 (2009) Google Scholar
- http://www.cdc.gov/ncidod/dvbid/westnile/
- Bayoh, MN, Lindsay, SW: Effect of temperature on the development of the aquatic stages of Anopheles gambiae sensu stricto (Diptera: Culicidae). Bull. Entomol. Res. 93, 375-381 (2003) View ArticleGoogle Scholar
- Shaman, J, Spiegelman, M, Cane, M, Stieglitz, M: A hydrologically driven model of swamp water mosquito population dynamics. Ecol. Model. 194, 395-404 (2006) View ArticleGoogle Scholar
- Zhao, XQ, Zou, XF: Threshold dynamics in a delayed SIS epidemic model. J. Math. Anal. Appl. 257, 282-291 (2001) MathSciNetView ArticleMATHGoogle Scholar
- Fan, GH, Wu, JH, Zhu, HP: The impact of maturation delay of mosquitoes on the transmission of West Nile virus. Math. Biosci. 228, 119-126 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Tridip, S, Sourav, R, Joydev, C: A mathematical model of dengue transmission with memory. Commun. Nonlinear Sci. Numer. Simul. 22, 511-525 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Smith, HJ: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Mathematical Surveys and Monographs, vol. 41. Am. Math. Soc., Providence (1995) MATHGoogle Scholar
- Cooke, KL, Driessche, P: Analysis of an SEIRS epidemic model with two delays. J. Math. Biol. 32, 240-260 (1996) MathSciNetView ArticleMATHGoogle Scholar
- Smith, HL: An Introduction to Delay Differential Equations with Applications to the Life Sciences, vol. 57. Springer, New York (2011) MATHGoogle Scholar
- Nakul, C, James, MH, Jim, MC: Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bull. Math. Biol. 70, 1272-1296 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Cesari, L: Optimization-Theory and Applications, Problems with Ordinary Differential Equations. Applications and Mathematics, vol. 17. Springer, New York (1983) MATHGoogle Scholar
- Kamien, MI, Schwartz, NL: Dynamics Optimization: The Calculus of Variations and Optimal Control in Economics and Management. Elsevier, Amsterdam (2000) MATHGoogle Scholar
- Nababan, S: A Filippov-type lemma for functions involving delays and its application to time-delayed optimal control problems. J. Optim. Theory Appl. 27(3), 357-376 (1979) MathSciNetView ArticleMATHGoogle Scholar
- Seierstad, A, Sydsaeter, K: Optimal Control Theory with Economic Applications. Elsevier, Amsterdam (1975) MATHGoogle Scholar
- Fleming, WH, Rishel, RW: Deterministic and Stochastic Optimal Control. Springer, New York (1975) View ArticleMATHGoogle Scholar
- Bashier, EBM, Patidar, KC: Optimal control of an epidemiological model with multiple time delays. Appl. Math. Comput. 292, 47-56 (2017) MathSciNetGoogle Scholar
- Chen, LJ, Hattaf, K, Sun, JT: Optimal control of a delayed SLBS computer virus model. Physica A 427, 244-250 (2015) MathSciNetView ArticleGoogle Scholar
- Zhu, QY, Yang, XF, Yang, LX, Zhang, CM: Optimal control of computer virus under a delayed model. Appl. Math. Comput. 218, 11613-11619 (2012) MathSciNetMATHGoogle Scholar
- Stech, H, Williams, M: Alternate hypothesis on the pathogenesis of dengue hemorrhagic fever (DHF)/dengue shock syndrome (DSS) in dengue virus infection. Exp. Biol. Med. 233(4), 401-408 (2008) View ArticleGoogle Scholar
- Chung, KW, Lui, R: Dynamics of two-strain influenza model with cross-immunity and no quarantine class. J. Math. Biol. 73(6), 1-23 (2016) MathSciNetMATHGoogle Scholar
- Esteva, L, Vargas, C: Coexistence of different serotypes of dengue virus. J. Math. Biol. 46(1), 31-47 (2003) MathSciNetView ArticleMATHGoogle Scholar
- Feng, Z, Velasco-Hernández, JX: Competitive exclusion in a vector-host model for the dengue fever. J. Math. Biol. 35(5), 523-544 (1997) MathSciNetView ArticleMATHGoogle Scholar
- Hartley, LM, Donnelly, CA, Garnett, GP: The seasonal pattern of dengue in endemic areas: mathematical models of mechanisms. Trans. R. Soc. Trop. Med. Hyg. 96, 387-397 (2002) View ArticleGoogle Scholar
- Zheng, B, Tang, MX, Yu, JS: Modeling Wolbachia spread in mosquitoes through delay differential equations. SIAM J. Appl. Math. 74(3), 743-770 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Zheng, B, Tang, MX, Yu, JS, Qiu, JX: Wolbachia spreading dynamics in mosquitoes with imperfect maternal transmission. J. Math. Biol. 2017 (2017, in press). doi:10.1007/s00285-017-1142-5
- Huang, MG, Tang, MX, Yu, JS: Wolbachia infection dynamics by reaction-diffusion equations. Sci. China Math. 58(1), 77-96 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Huang, MG, Yu, JS, Hu, LC, Zhang, B: Qualitative analysis for a Wolbachia infection model with diffusion. Sci. China Math. 59(7), 1249-1266 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Hu, LC, Huang, MG, Tang, MX, Yu, JS, Zheng, B: Wolbachia spread dynamics in stochastic environments. Theor. Popul. Biol. 106, 32-34 (2015) View ArticleMATHGoogle Scholar
- Turelli, M, Bartom, NH: Deploying dengue-suppressing Wolbachia: robust models predict slow but effective spatial spread in Aedes aegypti. Theor. Popul. Biol. 115, 45-60 (2017) View ArticleGoogle Scholar