Simple discrete-time malarial models

We investigate dynamics of mosquito population models under two assumptions, respectively, and then formulate simple discrete-time compartmental susceptible-exposed-infective-recovered models for the malaria transmission based on the mosquito population models. We show that the mosquito population models either have robust dynamics or exhibit period-doubling bifurcation depending on the model assumptions. We derive a formula for the reproductive number of infection for the malaria model, which determines the stability of the infection-free fixed point. We then determine the existence of endemic fixed points for the malaria models. Using numerical simulations, we demonstrate that the dynamical characteristics of the mosquito populations, such as the global stability of the endemic fixed point and the appearance of a period-doubling bifurcation, are reflected in the dynamics of the malaria transmission.     

Seasonality on the life cycle of Aedes aegypti mosquito and its statistical relation with dengue outbreaks

Aedes aegypti is the most common and important vector that transmits dengue disease. It has been observed that its abundance is one of the main factors influencing dengue incidence and hence the importance of studying its population dynamics.By means of a dynamical model, in this work, we study the effects temperature and precipitation in the abundance of the mosquito Aedes aegypti. We also analyze the correlation between mosquito and dengue outbreaks using historical data of 8 Mexican regions and the role of diapause in seasonality outbreaks.We found that the joint action of different mechanisms can enhance the mosquito abundance, but also inhibit it when they act in an asynchronous way. For the studied regions, the seasonality of the low temperature rather than mean temperature is the main driving force promoting mosquito outbreaks and hence dengue outbreaks.We conclude that this simpler model rather than more complex dengue models can also be used to predict the risk of dengue outbreaks with good accuracy. In addition to this, the model can be used to discover the underlying mechanism of dengue outbreaks in different regions and thus help to apply targeted control measures.     

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.     

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.

     

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.     

How does within‐host dynamics affect population‐level dynamics? Insights from an immuno‐epidemiological model of malaria

Malaria is one of the most common mosquito‐borne diseases widespread in the tropical and subtropical regions. Few models coupling the within‐host malaria dynamics with the between‐host mosquito‐human dynamics have been developed. In this paper, by adopting the nested approach, a malaria transmission model with immune response of the host is formulated. Applying age‐structured partial differential equations for the between‐host dynamics, we describe the asymptomatic and symptomatic infectious host population for malaria transmission. The basic reproduction numbers for the within‐host model and for the coupled system are derived, respectively. The existence and stability of the equilibria of the coupled model are analyzed. We show numerically that the within‐host model can exhibit complex dynamical behavior, possibly even chaos. In contrast, equilibria in the immuno‐epidemiological model are globally stable and their stabilities are determined by the reproduction number. Increasing the activation rate of the within‐host immune response “dampens” the sensitivity of the population level reproduction number and prevalence to the increase of the within‐host reproduction of the pathogen. From public health perspective this means that treatment in a population with higher immunity has less impact on the population‐level reproduction number and prevalence than in a population with less immunity.     

Reducing mosquito-borne disease outbreak size: The relative importance of contact and transmissibility in a network model

The complex biological and environmental factors involved in the transmission of mosquito-borne diseases in humans have made their control elusive in many instances. Conceptual models contribute to gain insight and help to reduce the risk of taking poor managerial decisions. The focus of this paper is to compare, using a contact network model, the impact that perturbation of the number infectious contacts and of transmissibility have on the size of an outbreak. We illustrate the analysis on a contact network parametrized with data that associates humans and the mosquito Culex quinquefasciatus, a vector for lymphatic filariasis. The model suggests that, if the values corresponding to transmissibility and number of infectious contacts is relatively large, variations in the size of an outbreak are significantly in favor of control measures to reduce infectious contacts.     

Analysis of a malaria model with mosquito-dependent transmission coefficient for humans

In this paper, we discuss an ordinary differential equation mathematical model for the spread of malaria in human and mosquito population. We suppose the human population to act as a reservoir. Both the species follow a logistic population model. The transmission coefficient or the interaction coefficient of humans is considered to be dependent on the mosquito population. It is seen that as the factors governing the transmission coefficient of humans increase, so does the number of infected humans. Further, it is observed that as the immigration constant increases, it leads to a rise in infected humans, giving an endemic shape to the disease.     

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.     

On the dynamics of a deterministic and stochastic model for mosquito control

In this work, we analyze a system of nonlinear difference equations describing community intervention in mosquito control. More specifically, we extend the model given in [M. Predescu, R. Levins, T. Awerbuch, Analysis of a nonlinear system for community intervention in mosquito control, Discrete Contin. Dyn. Syst. Ser. B 6 (3) (2006) 605–622] to allow for consciousness to be created in an ongoing way by educational efforts that are independent of the presence of mosquito breeding sites. In order to quantify the effect of random external events, such as weather or public concerns, we consider a stochastic version of the model. Numerical simulations show that the stochastic model is consistent with the deterministic one.     

Optimal bed net use for a dengue disease model with mosquito seasonal pattern

Controversial results concerning the effectiveness of bed net in reducing dengue fever transmission make further research necessary in this direction. At this aim, we consider a mathematical model of dengue transmission where the use by individuals of insecticide‐treated bed nets is taken into account, combined or not with insecticide spraying. Furthermore, as climatic factors play a key role in mosquito‐borne diseases, we model the effect of seasonality through a periodic mosquito birth rate. We numerically investigate some specific scenarios according to different rainfall and mean temperature values. We set an optimal control problem to minimize the number of human infections and the cost of efforts placed into bed net adoption and maintenance and insecticide spraying. To assess the most appropriate strategy to eliminate dengue with minimum costs, we perform a comparative cost‐effectiveness analysis, which also shows how the cost‐benefit of intervention efforts is affected by changes in the amplitude of seasonal variation. One general result is that in any case the combination of bed net use and insecticide spraying produces the highest ratio of infections averted, whereas in terms of cost‐benefit only spraying campaigns should be implemented in control programs for regions with no large seasonality.     

A note on immersions of domains of fractional powers of certain sectorial operators in Sobolev spaces

In this note we give a proof of a result on immersions of domains of fractional powers of certain sectorial operators associated to strongly elliptic operators in Sobolev spaces; such immersions preserve information on fractional derivatives. We also briefly comment on the application of this result to a problem of optimal control of mosquito populations.     

Modelling the $Wolbachia$ Strains for Dengue Fever Virus Control in the Presence of Seasonal Fluctuation

Consider that infection with $Wolbachiacan$ limit a mosquito''s ability to transmit Dengue fever virus through its saliva, a mathematical model describing the transmission of Dengue fever between vector mosquitoes and human, incorporating $Wolbachia$-carrying mosquito population and seasonal fluctuation, is proposed. Firstly, the stability and bifurcation of this model are investigated exactly in the case where seasonality can be neglected. Further, the basic reproductive number $\mathcal{R}_0^s$ for this model with seasonal variation is obtained, that is, if $\mathcal{R}_0^s$ is less than unity the disease is extinct and $\mathcal{R}_0^s$ is greater than unity the disease is uniformly persistent. Finally, numerical simulations verify the theoretical results. Theoretical results suggest that, compared with the mosquito reduction strategies (such as the elimination of mosquito breeding sites, killing of adult mosquitoes by spraying), introducing $Wolbachia$ strains is as effectual to fight against the transmission of Dengue virus.     

A free boundary problem for Aedes aegypti mosquito invasion

An advection–reaction–diffusion model with free boundary is proposed to investigate the invasive process of Aedes aegypti mosquitoes. By analyzing the free boundary problem, we show that there are two main scenarios of invasive regime: vanishing regime or spreading regime, depending on a threshold in terms of model parameters. Once the mortality rate of the mosquito becomes large with a small specific rate of maturation, the invasive mosquito will go extinct. By introducing the definition of asymptotic spreading speed to describe the spreading front, we provide an estimate to show that the boundary moving speed cannot be faster than the minimal traveling wave speed. By numerical simulations, we consider that the mosquitoes invasive ability and wind driven advection effect on the boundary moving speed. The greater the mosquito invasive ability or advection, the larger the boundary moving speed. Our results indicate that the mosquitoes asymptotic spreading speed can be controlled by modulating the invasive ability of winged mosquitoes.     

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.     

Control strategies for a population dynamics model of Aedes aegypti with seasonal variability and their effects on dengue incidence

Aedes aegypti female mosquitoes are the principal transmitters of dengue and other vector-borne infections. This species is closely associated with human habitation, due to its blood-feeding habits and the presence of breeding sites widely available around households. In this paper, we introduce a mathematical model for the life cycle of Aedes aegypti mosquitoes comprising two stages, aerial and aquatic, that reflects seasonal changes in the mosquito abundance. This model is further amended by three season-dependent control actions. Two coercive actions are introduced during the hot seasons characterized by higher abundance and enhanced growth rates of mosquitoes. They consist in the application of two chemical substances, insecticide and larvicide, acting upon the aerial and aquatic mosquito stages, respectively. During the cool seasons, characterized by the slower growth rates of mosquitoes and abundance of quiescent unhatched eggs, we introduce a preventive vector control measure consisting in mechanical elimination of mosquito breeding sites. Using the framework of optimal control in combination with the cost-benefit approach and epidemiological assessment, we identify the most efficient strategy capable of essentially reducing the population of adult and immature mosquitoes during both seasons and provide a sketch for its modus operandi.     

Steady states in a structured epidemic model with Wentzell boundary condition

We introduce a non-linear structured population model with diffusion in the state space. Individuals are structured with respect to a continuous variable which represents a pathogen load. The class of uninfected individuals constitutes a special compartment that carries mass; hence the model is equipped with generalized Wentzell (or dynamic) boundary conditions. Our model is intended to describe the spread of infection of a vertically transmitted disease, for e.g., Wolbachia in a mosquito population. Therefore, the (infinite dimensional) non-linearity arises in the recruitment term. First, we establish global existence of solutions and the principle of linearised stability for our model. Then, in our main result, we formulate simple conditions which guarantee the existence of non-trivial steady states of the model. Our method utilises an operator theoretic framework combined with a fixed-point approach. Finally in the last section, we establish a sufficient condition for the local asymptotic stability of the positive steady state.     

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.     

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.     

Convexity preserving jump-diffusion models for option pricing

We investigate which jump-diffusion models are convexity preserving. The study of convexity preserving models is motivated by monotonicity results for such models in the volatility and in the jump parameters. We give a necessary condition for convexity to be preserved in several-dimensional jump-diffusion models. This necessary condition is then used to show that, within a large class of possible models, the only convexity preserving models are the ones with linear coefficients.