Wolbachia-based biocontrol for dengue reduction using dynamic optimization approach

Aedes aegypti females mosquitoes are the principal transmitters of dengue and other arboviral infections. In recent years, it was disclosed that, when deliberately infected with Wolbachia symbiont, this mosquito species loses its vectorial competence and becomes less capable of transmitting the virus to human hosts. Thanks to this important discovery, Wolbachia-based biocontrol is now accepted as an ecologically friendly and potentially cost-effective method for prevention and control of dengue and other arboviral infections. In this paper, we propose a dengue transmission model that accounts for the presence of wild Aedes aegypti females and those deliberately infected with wMelPop Wolbachia strain, which is regarded as the best blocker of dengue and other arboviral infections. However, wMelPop strain of Wolbachia considerably reduces the individual fitness of mosquitoes, what makes rather challenging to achieve the gradual extrusion of wild mosquitoes and ensure their posterior replacement by Wolbachia-carriers. Nonetheless, this obstacle have been overcome by employing the optimal control approach for design of specific intervention programs based on daily releases of Wolbachia-carrying mosquitoes. The resulting optimal release programs ensure the population replacement and eventual local extinction of wild mosquitoes in the finite time and also entail a significant reduction in the number of expected dengue infections among human hosts under the long-term settings.     

The impact of Wolbachia on dengue transmission dynamics in an SEI–SIS model

Dengue has grown dramatically in recent decades globally. In order to investigate the spread of dengue with vector control, especially, the impact of Wolbachia on dengue transmission, a mathematical model is established and analyzed to study dengue transmission between humans and mosquitoes. Firstly, model qualitative analysis including the existence and local asymptotic stability of dengue-free equilibria and endemic equilibria is done. It is found that dengue will disappear when the basic reproduction number is less than one, and dengue will prevail when the basic reproduction number is larger than one. More important finding is that the persistence of Wolbachia is determined by its fitness effect on mosquitoes, and Wolbachia can drastically reduce dengue fever transmission. All the results are verified by numerical simulation. Secondly, sensitivity analysis is done to explore the relative importance of different parameters on the system. It is obtained that parameters with strong sensitivity and controllability are the biting rate, the probability of dengue infection between mosquitoes and humans and the recovery rate of infectious humans. Finally, the control methods are discussed.     

The threshold infection level for Wolbachia invasion in random environments

Dengue fever and Zika are mosquito-borne diseases threatening human health. A novel strategy for mosquito-borne disease control uses the bacterium Wolbachia to block virus transmission. It requires releasing Wolbachia infected mosquitoes to exceed a threshold level. Since an accurate forecast for temperature and rainfall, the major environmental conditions regulating the mosquito dynamics, is often not available over a long time period, it is important to explore how the threshold releasing level changes in random environments. In this work, we estimate the threshold level in a stochastic system of differential equations where the reproduction rates of mosquitoes change randomly. We prove that the threshold level is, surprisingly, defined by a deterministic curve that does not fluctuate with environmental conditions. The major difficulty in the proof is to construct various auxiliary curves to limit the dynamic behaviors of the whole family of innumerable solutions satisfying a given initial condition.     

An impulsive model for Wolbachia infection control of mosquito-borne diseases with general birth and death rate functions

Mosquito-borne diseases are global health problems, which mainly affect low-income populations in tropics and subtropics. In order to prevent the transmission of mosquito-borne diseases, the intracellular symbiotic bacteria named as Wolbachia is becoming a promising candidate to interrupt the virus transmission. In this paper, an impulsive mosquito population model with general birth and death rate functions is established to study the cytoplasmic incompatibility (CI) effect caused by mating of Wolbachia-infected males and uninfected females. The dynamics of the spread of Wolbachia in mosquito population are studied, and the strategies of mosquito extinction or replacing Wolbachia-uninfected mosquitoes with Wolbachia-infected mosquitoes are analyzed. Moreover, the results are applied to models with specific birth and death rate functions. It is shown that strategies may be different due to different birth and death rate functions, the type of Wolbachia strains and the initial number of Wolbachia-infected mosquitoes. Furthermore, numerical simulations are conducted to illustrate our conclusions.     

Comparing the efficiency of Wolbachia driven Aedes mosquito suppression strategies

Wolbachia is an endosymbiotic bacterium which manipulates host reproduction by cytoplasmic incompatibility, and restrains the transmission of dengue virus in Aedes mosquitoes. A novel strategy for dengue control involves releasing Wolbachia infected males into nature to suppress wild Aedes mosquito population. We develop a model of delay differential equations, integrating larval density-dependent competition and diapausing eggs, to compare the efficiency of different suppression strategies. The global asymptotical stability of the complete suppression state identifies the releasing amount threshold ascertaining suppression. Based on the experimental data for Aedes albopictus population in Guangzhou, our simulations show that the mosquito density in the highest peak season can be reduced by more than $95\%$ when the number of released males is above the releasing threshold. The best time to initiate the suppression is in early March, lasting until the end of June, followed by the parallel releasing policy from July to November. However, the egg bank has neglectable effects on the control of dengue vector in Guangzhou.     

One discrete dynamical model on the Wolbachia infection frequency in mosquito populations

How to prevent and control the outbreak of mosquito-borne diseases, such as malaria, dengue fever and Zika, is an urgent worldwide public health problem. The most conventional method for the control of these diseases is to directly kill mosquitoes by spraying insecticides or removing their breeding sites. However, the traditional method is not effective enough to keep the mosquito density below the epidemic risk threshold. With promising results international, the World Mosquito Program’s Wolbachia method is helping to reduce the occurrence of diseases transmitted by mosquitoes. In this paper, we introduce a generalized discrete model to study the dynamics of the Wolbachia infection frequency in mosquito populations where infected mosquitoes are impulsively released. This generalized model covers all the relevant existing models since 1959 as some special cases. After summarizing known results of discrete models deduced from the generalized one, we put forward some interesting open questions to be further investigated for the periodic impulsive releases.

     

Controlling Aedes aegypti populations by limited Wolbachia‐based strategies in a seasonal environment

In this work, the linear feedback limited control strategy is proposed to indicate how the Wolbachia‐infected mosquitoes should be introduced in the seasonal environment to reduce the non‐Wolbachia mosquito population. The numerical simulations show that the proposed strategy reduces the population level of non‐Wolbachia mosquitos, avoiding mosquito spread and, consequently, reducing the number of cases of vector‐borne diseases.     

Wolbachia infection dynamics by reaction-diffusion equations

Dengue fever is caused by the dengue virus and transmitted by Aedes mosquitoes.A promising avenue for eradicating the disease is to infect the wild aedes population with the bacterium Wolbachia driven by cytoplasmic incompatibility(CI).When releasing Wolbachia infected mosquitoes for population replacement,it is essential to not ignore the spatial inhomogeneity of wild mosquito distribution.In this paper,we develop a model of reaction-diffusion system to investigate the infection dynamics in natural areas,under the assumptions supported by recent experiments such as perfect maternal transmission and complete CI.We prove non-existence of inhomogeneous steady-states when one of the diffusion coefficients is sufficiently large,and classify local stability for constant steady states.It is seen that diffusion does not change the criteria for the local stabilities.Our major concern is to determine the minimum infection frequency above which Wolbachia can spread into the whole population of mosquitoes.We find that diffusion drives the minimum frequency slightly higher in general.However,the minimum remains zero when Wolbachia infection brings overwhelming fitness benefit.In the special case when the infection does not alter the longevity of mosquitoes but reduces the birth rate by half,diffusion has no impact on the minimum frequency.     

Spatial-temporal risk index and transmission of a nonlocal dengue model

The nonlocal incidence and free boundaries are introduced into a classic SIR-SI model describing the transmission dynamics of dengue fever, where the nonlocal incidence allows for interactions of susceptible population at a given location with infected mosquitoes in the whole area, and free boundaries represent the expanding front of the area contaminated by dengue virus. We derive a spatial–temporal risk index in terms of the basic reproduction number, which depends on the nonlocal incidence and time variable. More importantly, we explore the relationships between different model variants regarding these risk indices. We additionally find sufficient conditions to ensure the vanishing and spreading of dengue fever, and demonstrate, for a special case, the asymptotic behavior of its solution when spreading occurs. Finally, we carry out numerical simulations to demonstrate our analytical findings and further provide their epidemiological explanations.     

Effect of partial immunity on transmission dynamics of dengue disease with optimal control

Dengue fever is one of the most dangerous vector‐borne diseases in the world in terms of death and economic cost. Hence, the modeling of dengue fever is of great significance to understand the dynamics of dengue. In this paper, we extend dengue disease transmission models by including transmit vaccinated class, in which a portion of recovered individual loses immunity and moves to the susceptibles with limited immunity and hence a less transmission probability. We obtain the threshold dynamics governed by the basic reproduction number R₀; it is shown that the disease‐free equilibrium is locally asymptotically stable if R₀ ≤ 1, and the system is uniformly persistence if R₀ > 1. We do sensitivity analysis in order to identify the key factors that greatly affect the dengue infection, and the partial rank correlation coefficient (PRCC) values for R₀ shows that the bitting rate is the most effective in lowering dengue new infections, and moreover, control of mosquito size plays an essential role in reducing equilibrium level of dengue infection. Hence, the public are highly suggested to control population size of mosquitoes and to use mosquito nets. By formulating the control objective, associated with the low infection and costs, we propose an optimal control question. By the application of optimal control theory, we analyze the existence of optimal control and obtain necessary conditions for optimal controls. Numerical simulations are carried out to show the effectiveness of control strategies; these simulations recommended that control measures such as protection from mosquito bites and mosquito eradication strategies effectively control and eradicate the dengue infections during the whole epidemic.     

The asymptotic profile of a dengue model on a growing domain driven by climate change

Global warming results in a slow expansion of habitat range of mosquitoes, an important vector of dengue virus. To understand the impact of this changing environment on the transmission of dengue virus, we develop a dengue model on a growing domain under the framework of reaction diffusion equations. By overcoming some difficulties of dynamical behaviors caused by diffusion terms with variable-dependent coefficients, we investigate the stabilities of the disease-free and endemic equilibria in terms of the associated basic reproduction number. Comparing our dengue model on a growing domain to the model on a fixed domain in terms of the basic reproduction number, we conclude that habitat expansion resulting from global warming catalyzes the spread of dengue fever, and it is negative to the control of dengue fever. Finally, numerical simulations are performed and show a good agreement with our analytical results.     

HIV low viral load persistence under treatment: Insights from a model of cell-to-cell viral transmission

HIV infection persists despite long-term administration of antiretroviral therapy. The mechanisms underlying HIV persistence are not fully understood. Direct viral transmission from infected to uninfected cells (cell-to-cell transmission) may be one of them. During cell-to-cell transmission, multiple virions are delivered to an uninfected cell, making it possible that at least one virion can escape HIV drugs and establish infection. In this paper, we develop a mathematical model that includes cell-to-cell viral transmission to study HIV persistence. During cell-to-cell transmission, it is assumed that various number of virus particles are transmitted with different probabilities and antiretroviral therapy has different effectiveness in blocking their infection. We analyze the model by deriving the basic reproduction number and investigating the stability of equilibria. Sensitivity analysis and numerical simulation show that the viral load is still sensitive to the change of the treatment effectiveness in blocking cell-free virus infection. To reduce this sensitivity, we modify the model by including density-dependent infected cell death or HIV latent infection. The model results suggest that although cell-to-cell transmission may have reduced susceptibility to HIV drugs, HIV latency represents a major reason for HIV persistence in patients on suppressive treatment.     

Understanding epidemics from mathematical models: Details of the 2010 dengue epidemic in Bello (Antioquia,Colombia)

Dengue is the most threatening vector-borne viral disease in Colombia. At the moment, there is no treatment or vaccine available for its control or prevention; therefore, the main measure is to exert control over mosquito population. To reduce the economic impact of control measures, it is important to focus on specific characteristics related to local dengue epidemiology at the local level, and know the main factors involved in an epidemic. To this end, we used a mathematical model based on ordinary differential equations and experimental data regarding mosquito populations from Bello (Antioquia, Colombia) to simulate the epidemic occurred in 2010. The results showed that the parameters to which the incidence of dengue cases are most sensitive are the biting and mortality rates of adult mosquitoes as well as the virus transmission probabilities. Finally, we found that the Basic Reproductive Number (R₀) of this epidemic was between 1.5 and 2.7, with an infection force (Λ) of 0.061, meaning that R₀ values slightly above one are sufficient to result in a significant dengue outbreak in this region.     

Global properties for virus dynamics model with mitotic transmission and intracellular delay

In this paper we study the basic model of viral infections with mitotic transmission and intracellular delay discrete. The delay corresponds to the time between infection of uninfected cells and the emission of virus on a cellular level. By means of Volterra-type Lyapunov functionals, we provide the global stability for this model. Let η be the number of virus produced per infected cell. If ηcrit, the critical number, satisfies η?ηcrit, then the virus-free steady state is globally asymptotically stable. On the contrary if η>ηcrit, then the infected steady state is globally asymptotically stable if a sufficient condition is satisfied.     

Dynamical behaviors of a discrete HIV‐1 virus model with bilinear infective rate

We establish a discrete virus dynamic model by discretizing a continuous HIV‐1 virus model with bilinear infective rate using ‘hybrid’ Euler method. We discuss not only the existence and global stability of the uninfected equilibrium but also the existence and local stability of the infected equilibrium. We prove that there exists a crucial value similar to that of the continuous HIV‐1 virus dynamics, which is called the basic reproductive ratio of the virus. If the basic reproductive ratio of the virus is less than one, the uninfected equilibrium is globally asymptotically stable. If the basic reproductive ratio of the virus is larger than one, the infected equilibrium exists and is locally stable. Moreover, we consider the permanence for such a system by constructing a Lyapunov function v_n. Copyright © 2013 John Wiley & Sons, Ltd.     

Comparative analysis of dengue versus chikungunya outbreaks in Costa Rica

For decades, dengue virus has been a cause of major public health concern in Costa Rica, due to its landscape and climatic conditions that favor the circumstances in which the vector, Aedes aegypti, thrives. The emergence and introduction throughout tropical and subtropical countries of the chikungunya virus, as of 2014, challenged Costa Rican health authorities to provide a correct diagnosis since it is also transmitted by the same vector and infected hosts may share similar symptoms. We study the 2015–2016 dengue and chikungunya outbreaks in Costa Rica while establishing how point estimates of epidemic parameters for both diseases compare to one another. Longitudinal weekly incidence reports of these outbreaks signal likely misdiagnosis of infected individuals: underreporting of chikungunya cases, while overreporting cases of dengue. Our comparative analysis is formulated with a single-outbreak deterministic model that features an undiagnosed class. Additionally, we also used a genetic algorithm in the context of weighted least squares to calculate point estimates of key model parameters and initial conditions, while formally quantifying misdiagnosis.     

Optimal control approach to dengue reduction and prevention in Cali,Colombia

The aim of this paper is to propose optimal strategies for dengue reduction and prevention in Cali, Colombia. For this purpose, we consider two variants of a simple dengue transmission model, epidemic and endemic, each of which is amended with two control variables. These variables express feasible control actions to be taken by an external decision‐maker. First control variable stands for the insecticide spraying and thus targets to suppress the vector population. The second one expresses the protective measures (such as use of repellents, mosquito nets, and insecticide‐treated clothes) that are destined to reduce the number of contacts (bites) between female mosquitoes (principal dengue transmitters) and human individuals. We use the Pontryagin's maximum principle in order to derive the optimal strategies for dengue control and then perform the cost‐effectiveness analysis of these strategies in order to choose the most sustainable one in terms of cost–benefit relationship. Copyright © 2016 John Wiley & Sons, Ltd.     

Global analysis of a vector-host epidemic model with nonlinear incidences

In this paper, an epidemic model with nonlinear incidences is proposed to describe the dynamics of diseases spread by vectors (mosquitoes), such as malaria, yellow fever, dengue and so on. The constant human recruitment rate and exponential natural death, as well as vector population with asymptotically constant population, are incorporated into the model. The stability of the system is analyzed for the disease-free and endemic equilibria. The stability of the system can be controlled by the threshold number R₀. It is shown that if R₀ is less than one, the disease free equilibrium is globally asymptotically stable and in such a case the endemic equilibrium does not exist; if R₀ is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Our results imply that the threshold condition of the system provides important guidelines for accessing control of the vector diseases, and the spread of vector epidemic in an efficient way can be prevented. The contribution of the nonlinear saturating incidence to the basic reproduction number and the level of the endemic equilibrium are also analyzed, respectively.     

Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response

In this paper, we study the global dynamics of a viral infection model with a latent period. The model has a nonlinear function which denotes the incidence rate of the virus infection in vivo. The basic reproduction number of the virus is identified and it is shown that the uninfected equilibrium is globally asymptotically stable if the basic reproduction number is equal to or less than unity. Moreover, the virus and infected cells eventually persist and there exists a unique infected equilibrium which is globally asymptotically stable if the basic reproduction number is greater than unity. The basic reproduction number determines the equilibrium that is globally asymptotically stable, even if there is a time delay in the infection.