scholarly journals Modeling the Control of Zika Virus Vector Population Using the Sterile Insect Technology

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
William Atokolo ◽  
Godwin Mbah Christopher Ezike
Keyword(s):  
Equilibrium Point ◽  
Equilibrium Points ◽  
Mosquito Population ◽  
Sterile Male ◽  
Wild Female ◽  
Globally Asymptotically Stable ◽  
Offspring Number ◽  
Vector Population ◽  
Control Intervention ◽  
Male Mosquitoes

This work is aimed at formulating a mathematical model for the control of mosquito population using sterile insect technology (SIT). SIT is an environmental friendly method, which depends on the release of sterile male mosquitoes that compete with wild male mosquitoes and mate with wild female mosquitoes, which leads to the production of no offspring. The basic offspring number of the mosquitoes’ population was computed, after which we investigated the existence of two equilibrium points of the model. When the basic offspring number of the model M0, is less than or equal to 1, a mosquito extinction equilibrium point E2, which is often biologically unattainable, was shown to exits. On the other hand, if M0>1, we have the nonnegative equilibrium point E1 which is shown to be both locally and globally asymptotically stable whenever M0>1. Local sensitivity analysis was then performed to know the parameters that should be targeted by control intervention strategies and result shows that female mating probability to be with the sterile male mosquitoes ρS, mating rate of the sterile mosquito β2, and natural death rates of both aquatic and female mosquitoesμA+μF have greater impacts on the reduction and elimination of mosquitoes from a population. Simulation of the model shows that enough release of sterile male mosquitoes into the population of the wild mosquitoes controls the mosquito population and as such can reduce the spread of mosquito borne disease such as Zika.

MATHEMATICAL MODELING OF THE IMPACT OF PERIODIC RELEASE OF STERILE MALE MOSQUITOES AND SEASONALITY ON THE POPULATION ABUNDANCE OF MALARIA MOSQUITOES

2020 ◽  
Vol 28 (02) ◽  
pp. 277-310 ◽  
Author(s):  
ENAHORO A. IBOI ◽  
ABBA B. GUMEL ◽  
JESSE E. TAYLOR
Keyword(s):  
Seasonal Variation ◽  
Mosquito Population ◽  
Local Temperature ◽  
Effective Control ◽  
Population Abundance ◽  
Sterile Male ◽  
Density Dependent ◽  
Malaria Mosquitoes ◽  
The Impact ◽  
Male Mosquitoes

This study presents a new mathematical model for assessing the impact of sterile insect technology (SIT) and seasonal variation in local temperature on the population abundance of malaria mosquitoes in an endemic setting. Simulations of the model, using temperature data from Kipsamoite area of Kenya, show that a peak abundance of the mosquito population is attained in the Kipsamoite area when the mean monthly temperature reaches [Formula: see text]. Furthermore, in the absence of seasonal variation in local temperature, our results show that releasing more sterile male mosquitoes (e.g., 100,000) over a one year period with relatively short duration between releases (e.g., weekly, bi-weekly or even monthly) is more effective than releasing smaller numbers of the sterile male mosquitoes (e.g., 10,000) over the same implementation period and frequency of release. It is also shown that density-dependent larval mortality plays an important role in determining the threshold number of sterile male mosquitoes that need to be released in order to achieve effective control (or elimination) of the mosquito population in the community. In particular, low(high) density-dependent mortality requires high(low) numbers of sterile male mosquitoes to be released to achieve such control. In the presence of seasonal variation in local temperature, effective control of the mosquito population using SIT is only feasible if a large number of the sterile male mosquitoes (e.g., 100,000) is periodically released within a very short time interval (at most weekly). In other words, seasonal variation in temperature necessitates more frequent releases (of a large number) of sterile male mosquitoes to ensure the effectiveness of the SIT intervention in curtailing the targeted mosquito population.

scholarly journals An SIRS Epidemic Model Supervised by a Control System for Vaccination and Treatment Actions Which Involve First-Order Dynamics and Vaccination of Newborns

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 36
Author(s):  
Santiago Alonso-Quesada ◽  
Manuel De la Sen ◽  
Raúl Nistal
Keyword(s):  
Control System ◽  
Equilibrium Point ◽  
Epidemic Model ◽  
Equilibrium Points ◽  
Threshold Value ◽  
Design Parameters ◽  
Globally Asymptotically Stable ◽  
Sirs Epidemic Model ◽  
Existence And Stability ◽  
Disease Free

This paper analyses an SIRS epidemic model with the vaccination of susceptible individuals and treatment of infectious ones. Both actions are governed by a designed control system whose inputs are the subpopulations of the epidemic model. In addition, the vaccination of a proportion of newborns is considered. The control reproduction number Rc of the controlled epidemic model is calculated, and its influence in the existence and stability of equilibrium points is studied. If such a number is smaller than a threshold value Rc, then the model has a unique equilibrium point: the so-called disease-free equilibrium point at which there are not infectious individuals. Furthermore, such an equilibrium point is locally and globally asymptotically stable. On the contrary, if Rc>Rc, then the model has two equilibrium points: the referred disease-free one, which is unstable, and an endemic one at which there are infectious individuals. The proposed control strategy provides several free-design parameters that influence both values Rc and Rc. Then, such parameters can be appropriately adjusted for guaranteeing the non-existence of the endemic equilibrium point and, in this way, eradicating the persistence of the infectious disease.

scholarly journals Dynamics of COVID-19 Epidemic Model with Asymptomatic Infection, Quarantine, Protection and Vaccination

2021 ◽  
Vol 4 (2) ◽  
pp. 106-124
Author(s):  
Raqqasyi Rahmatullah Musafir ◽  
Agus Suryanto ◽  
Isnani Darti
Keyword(s):  
Equilibrium Point ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Asymptomatic Infection ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Proposed Model ◽  
Disease Free ◽  
The Basic Reproduction Number

We discuss the dynamics of new COVID-19 epidemic model by considering asymptomatic infections and the policies such as quarantine, protection (adherence to health protocols), and vaccination. The proposed model contains nine subpopulations: susceptible (S), exposed (E), symptomatic infected (I), asymptomatic infected (A), recovered (R), death (D), protected (P), quarantined (Q), and vaccinated (V ). We first show the non-negativity and boundedness of solutions. The equilibrium points, basic reproduction number, and stability of equilibrium points, both locally and globally, are also investigated analytically. The proposed model has disease-free equilibrium point and endemic equilibrium point. The disease-free equilibrium point always exists and is globally asymptotically stable if basic reproduction number is less than one. The endemic equilibrium point exists uniquely and is globally asymptotically stable if the basic reproduction number is greater than one. These properties have been confirmed by numerical simulations using the fourth order Runge-Kutta method. Numerical simulations show that the disease transmission rate of asymptomatic infection, quarantine rates, protection rate, and vaccination rates affect the basic reproduction number and hence also influence the stability of equilibrium points.

scholarly journals Mathematical Modeling and Analysis of Transmission Dynamics and Control of Schistosomiasis

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
Ebrima Kanyi ◽  
Ayodeji Sunday Afolabi ◽  
Nelson Owuor Onyango
Keyword(s):  
Equilibrium Point ◽  
Control Strategies ◽  
Equilibrium Points ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Transmission Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Modeling And Analysis ◽  
Disease Free

This paper presents a mathematical model that describes the transmission dynamics of schistosomiasis for humans, snails, and the free living miracidia and cercariae. The model incorporates the treated compartment and a preventive factor due to water sanitation and hygiene (WASH) for the human subpopulation. A qualitative analysis was performed to examine the invariant regions, positivity of solutions, and disease equilibrium points together with their stabilities. The basic reproduction number, R 0 , is computed and used as a threshold value to determine the existence and stability of the equilibrium points. It is established that, under a specific condition, the disease-free equilibrium exists and there is a unique endemic equilibrium when R 0 > 1 . It is shown that the disease-free equilibrium point is both locally and globally asymptotically stable provided R 0 < 1 , and the unique endemic equilibrium point is locally asymptotically stable whenever R 0 > 1 using the concept of the Center Manifold Theory. A numerical simulation carried out showed that at R 0 = 1 , the model exhibits a forward bifurcation which, thus, validates the analytic results. Numerical analyses of the control strategies were performed and discussed. Further, a sensitivity analysis of R 0 was carried out to determine the contribution of the main parameters towards the die out of the disease. Finally, the effects that these parameters have on the infected humans were numerically examined, and the results indicated that combined application of treatment and WASH will be effective in eradicating schistosomiasis.

scholarly journals Mathematical Modeling of Typhoid Fever Disease Incorporating Unprotected Humans in the Spread Dynamics

2019 ◽  
pp. 1-11
Author(s):  
Julia Wanjiku Karunditu ◽  
George Kimathi ◽  
Shaibu Osman
Keyword(s):  
Typhoid Fever ◽  
Global Stability ◽  
Equilibrium Point ◽  
Local Stability ◽  
Jacobian Matrix ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Spread Dynamics ◽  
Disease Free

A deterministic mathematical model of typhoid fever incorporating unprotected humans is formulated in this study and employed to study local and global stability of equilibrium points. The model incorporating Susceptible, unprotected, Infectious and Recovered humans which are analyzed mathematically and also result into a system of ordinary differential equations which are used for interpretations and comparison to the qualitative solutions in studying the spread dynamics of typhoid fever. Jacobian matrix was considered in the study of local stability of disease free equilibrium point and Castillo-Chavez approach used to determine global stability of disease free equilibrium point. Lyapunov function was used to study global stability of endemic equilibrium point. Both equilibrium points (DFE and EE) were found to be local and globally asymptotically stable. This means that the disease will be dependent on numbers of unprotected humans and other factors who contributes positively to the transmission dynamics.

scholarly journals On a Discrete SEIR Epidemic Model with Exposed Infectivity, Feedback Vaccination and Partial Delayed Re-Susceptibility

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 520
Author(s):  
Manuel De la Sen ◽  
Santiago Alonso-Quesada ◽  
Asier Ibeas
Keyword(s):  
Equilibrium Point ◽  
Epidemic Model ◽  
Transmission Rate ◽  
Equilibrium Points ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Control Gain ◽  
Disease Free ◽  
Partial Immunity

A new discrete Susceptible-Exposed-Infectious-Recovered (SEIR) epidemic model is proposed, and its properties of non-negativity and (both local and global) asymptotic stability of the solution sequence vector on the first orthant of the state-space are discussed. The calculation of the disease-free and the endemic equilibrium points is also performed. The model has the following main characteristics: (a) the exposed subpopulation is infective, as it is the infectious one, but their respective transmission rates may be distinct; (b) a feedback vaccination control law on the Susceptible is incorporated; and (c) the model is subject to delayed partial re-susceptibility in the sense that a partial immunity loss in the recovered individuals happens after a certain delay. In this way, a portion of formerly recovered individuals along a range of previous samples is incorporated again to the susceptible subpopulation. The rate of loss of partial immunity of the considered range of previous samples may be, in general, distinct for the various samples. It is found that the endemic equilibrium point is not reachable in the transmission rate range of values, which makes the disease-free one to be globally asymptotically stable. The critical transmission rate which confers to only one of the equilibrium points the property of being asymptotically stable (respectively below or beyond its value) is linked to the unity basic reproduction number and makes both equilibrium points to be coincident. In parallel, the endemic equilibrium point is reachable and globally asymptotically stable in the range for which the disease-free equilibrium point is unstable. It is also discussed the relevance of both the vaccination effort and the re-susceptibility level in the modification of the disease-free equilibrium point compared to its reached component values in their absence. The influences of the limit control gain and equilibrium re-susceptibility level in the reached endemic state are also explicitly made viewable for their interpretation from the endemic equilibrium components. Some simulation examples are tested and discussed by using disease parameterizations of COVID-19.

scholarly journals Successful suppression of a field population of Ae. aegypti mosquitoes using a novel biological vector control strategy is associated with significantly lower incidence of dengue

2019 ◽  
Author(s):  
Lisiane C Poncio ◽  
Filipe A dos Anjos ◽  
Deborah A de Oliveira ◽  
Débora Rebechi ◽  
Rodrigo N de Oliveira ◽  
...  
Keyword(s):  
Vector Control ◽  
Lower Incidence ◽  
Large Scale ◽  
Control Area ◽  
Environmentally Benign ◽  
Mosquito Population ◽  
Mosquito Vector ◽  
Sterile Male ◽  
Treated Area ◽  
Male Mosquitoes

AbstractBackgroundDespite extensive efforts to prevent recurrent Aedes-borne arbovirus epidemics, there is a steady rise in their global incidence. Vaccines/treatments show very limited efficacy and together with the emergence of mosquito resistance to insecticides, it has become urgent to develop alternative solutions for efficient, sustainable and environmentally benign mosquito vector control. Here we present a new Sterile Insect Technology (SIT)-based program that uses large-scale releases of sterile male mosquitoes produced by a highly effective, safe and environmentally benign method.Methods and findingsTo test the efficacy of this approach, a field trial was conducted in a Brazilian city (Jacarezinho), which presented a history of 3 epidemics of dengue in the past decade. Sterile male mosquitoes were produced from a locally acquired Aedes aegypti colony, and releases were carried out on a weekly basis for seven months in a predefined area. This treated area was matched to a control area, in terms of size, layout, historic mosquito infestation index, socioeconomic patterns and comparable prevalence of dengue cases in past outbreaks. Releases of sterile male mosquitoes resulted in up to 91.4% reduction of live progeny of field Ae. aegypti mosquitoes recorded over time. The reduction in the mosquito population was corroborated by the standard monitoring system (LIRAa index) as determined by the local municipality, which found that our treated neighborhoods were almost free of Ae. aegypti mosquitoes after 5 months of release, whereas neighborhoods adjacent to the treated area and the control neighborhoods were highly infested. Importantly, when a dengue outbreak started in Jacarezinho in March 2019, the effective mosquito population suppression was shown to be associated with a far lower incidence of dengue in the treated area (16 cases corresponding to 264 cases per 100,000 inhabitants) almost 16 times lower than the dengue incidence in the control area (198 cases corresponding to 4,360 dengue cases per 100,000 inhabitants).ConclusionsOur data present the first demonstration that a SIT-based intervention has the potential to prevent the spread of dengue, opening exciting new opportunities for preventing mosquito-borne disease.

scholarly journals Dynamical Analysis of a Fractional Order HIV/AIDS Model

2021 ◽  
Vol 5 (1) ◽  
pp. 14
Author(s):  
Septiangga Van Nyek Perdana Putra ◽  
Agus Suryanto ◽  
Nur Shofianah
Keyword(s):  
Equilibrium Point ◽  
Fractional Order ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Dynamical Analysis ◽  
Order Model ◽  
Globally Asymptotically Stable ◽  
Fractional Order Model ◽  
Disease Free ◽  
Hiv Aids

This article discusses a dynamical analysis of the fractional-order model of HIV/AIDS. Biologically, the rate of subpopulation growth also depends on all previous conditions/memory effects. The dependency of the growth of subpopulations on the past conditions is considered by applying fractional derivatives. The model is assumed to consist of susceptible, HIV infected, HIV infected with treatment, resistance, and AIDS. The fractional-order model of HIV/AIDS with Caputo fractional-order derivative operators is constructed and then, the dynamical analysis is performed to determine the equilibrium points, local stability and global stability of the equilibrium points. The dynamical analysis results show that the model has two equilibrium points, namely the disease-free equilibrium point and endemic equilibrium point. The disease-free equilibrium point always exists and is globally asymptotically stable when the basic reproduction number is less than one. The endemic equilibrium point exists if the basic reproduction number is more than one and is globally asymptotically stable unconditionally. To illustrate the dynamical analysis, we perform some numerical simulation using the Predictor-Corrector method. Numerical simulation results support the analytical results.

scholarly journals Analysis of a virus-resistant HIV-1 model with behavior change in non-progressors

BIOMATH ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 2006143
Author(s):  
Musa Rabiu ◽  
Robert Willie ◽  
Nabendra Parumasur
Keyword(s):  
Equilibrium Point ◽  
Equilibrium Points ◽  
Behavioural Change ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Reproduction Numbers ◽  
Special Case ◽  
Disease Free ◽  
Hiv 1

We develop a virus-resistant HIV-1 mathematical model with behavioural change in HIV-1 resistant non-progressors. The model has both disease-free and endemic equilibrium points that are proved to be locally asymptotically stable depending on the value of the associated reproduction numbers. In both models, a non-linear Goh{Volterra Lyapunov function was used to prove that the endemic equilibrium point is globally asymptotically stable for special case while the method of Castillo-Chavez was used to prove the global asymptotic stability of the disease-free equilibrium point. In both the analytic and numerical results, this study shows that in the context of resistance to HIV/AIDS, total abstinence can also play an important role in protection against this notorious infectious disease.

scholarly journals Dynamics of an Eco-Epidemic Predator–Prey Model Involving Fractional Derivatives with Power-Law and Mittag–Leffler Kernel

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 785
Author(s):  
Hasan S. Panigoro ◽  
Agus Suryanto ◽  
Wuryansari Muharini Kusumawinahyu ◽  
Isnani Darti
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Derivative ◽  
Equilibrium Point ◽  
Power Law ◽  
Equilibrium Points ◽  
Essential Difference ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Infected Prey

In this paper, we consider a fractional-order eco-epidemic model based on the Rosenzweig–MacArthur predator–prey model. The model is derived by assuming that the prey may be infected by a disease. In order to take the memory effect into account, we apply two fractional differential operators, namely the Caputo fractional derivative (operator with power-law kernel) and the Atangana–Baleanu fractional derivative in the Caputo (ABC) sense (operator with Mittag–Leffler kernel). We take the same order of the fractional derivative in all equations for both senses to maintain the symmetry aspect. The existence and uniqueness of solutions of both eco-epidemic models (i.e., in the Caputo sense and in ABC sense) are established. Both models have the same equilibrium points, namely the trivial (origin) equilibrium point, the extinction of infected prey and predator point, the infected prey free point, the predator-free point and the co-existence point. For a model in the Caputo sense, we also show the non-negativity and boundedness of solution, perform the local and global stability analysis and establish the conditions for the existence of Hopf bifurcation. It is found that the trivial equilibrium point is a saddle point while other equilibrium points are conditionally asymptotically stable. The numerical simulations show that the solutions of the model in the Caputo sense strongly agree with analytical results. Furthermore, it is indicated numerically that the model in the ABC sense has quite similar dynamics as the model in the Caputo sense. The essential difference between the two models is the convergence rate to reach the stable equilibrium point. When a Hopf bifurcation occurs, the bifurcation points and the diameter of the limit cycles of both models are different. Moreover, we also observe a bistability phenomenon which disappears via Hopf bifurcation.

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