Mathematical analysis of an influenza epidemic model, formulation of different controlling strategies using optimal control and estimation of basic reproduction number

<abstract><p>A generalized "SVEIR" epidemic model with general nonlinear incidence rate has been proposed as a candidate model for measles virus dynamics. The basic reproduction number $ \mathcal{R} $, an important epidemiologic index, was calculated using the next generation matrix method. The existence and uniqueness of the steady states, namely, disease-free equilibrium ($ \mathcal{E}_0 $) and endemic equilibrium ($ \mathcal{E}_1 $) was studied. Therefore, the local and global stability analysis are carried out. It is proved that $ \mathcal{E}_0 $ is locally asymptotically stable once $ \mathcal{R} $ is less than. However, if $ \mathcal{R} > 1 $ then $ \mathcal{E}_0 $ is unstable. We proved also that $ \mathcal{E}_1 $ is locally asymptotically stable once $ \mathcal{R} > 1 $. The global stability of both equilibrium $ \mathcal{E}_0 $ and $ \mathcal{E}_1 $ is discussed where we proved that $ \mathcal{E}_0 $ is globally asymptotically stable once $ \mathcal{R}\leq 1 $, and $ \mathcal{E}_1 $ is globally asymptotically stable once $ \mathcal{R} > 1 $. The sensitivity analysis of the basic reproduction number $ \mathcal{R} $ with respect to the model parameters is carried out. In a second step, a vaccination strategy related to this model will be considered to optimise the infected and exposed individuals. We formulated a nonlinear optimal control problem and the existence, uniqueness and the characterisation of the optimal solution was discussed. An algorithm inspired from the Gauss-Seidel method was used to resolve the optimal control problem. Some numerical tests was given confirming the obtained theoretical results.</p></abstract>

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Dynamical Behavior of a Malaria Model with Discrete Delay and Optimal Insecticide Control

Biophysical Reviews and Letters ◽

10.1142/s1793048017500023 ◽

2017 ◽

Vol 12 (01) ◽

pp. 19-38 ◽

Cited By ~ 3

Author(s):

Tuhin Kumar Kar ◽

Soovoojeet Jana

Keyword(s):

Optimal Control ◽

Optimal Control Problem ◽

Control Problem ◽

Basic Reproduction Number ◽

Human Population ◽

Reproduction Number ◽

Dynamical Behavior ◽

Control Variable ◽

Discrete Delay ◽

Insecticide Control

In this paper we have proposed and analyzed a simple three-dimensional mathematical model related to malaria disease. We consider three state variables associated with susceptible human population, infected human population and infected mosquitoes, respectively. A discrete delay parameter has been incorporated to take account of the time of incubation period with infected mosquitoes. We consider the effect of insecticide control, which is applied to the mosquitoes. Basic reproduction number is figured out for the proposed model and it is shown that when this threshold is less than unity then the system moves to the disease-free state whereas for higher values other than unity, the system would tend to an endemic state. On the other hand if we consider the system with delay, then there may exist some cases where the endemic equilibrium would be unstable although the numerical value of basic reproduction number may be greater than one. We formulate and solve the optimal control problem by considering insecticide as the control variable. Optimal control problem assures to obtain better result than the noncontrol situation. Numerical illustrations are provided in support of the theoretical results.

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Spatial dynamics of a diffusive SIRI model with distinct dispersal rates and heterogeneous environment

Communications on Pure & Applied Analysis ◽

10.3934/cpaa.2021120 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Lian Duan ◽

Lihong Huang ◽

Chuangxia Huang

Keyword(s):

Steady State ◽

Basic Reproduction Number ◽

Epidemic Model ◽

Reproduction Number ◽

Spatial Dynamics ◽

Heterogeneous Environment ◽

Threshold Parameter ◽

Dispersal Rate ◽

Siri Model ◽

The Basic Reproduction Number

<p style='text-indent:20px;'>In this paper, we are concerned with the dynamics of a diffusive SIRI epidemic model with heterogeneous parameters and distinct dispersal rates for the susceptible and infected individuals. We first establish the basic properties of solutions to the model, and then identify the basic reproduction number <inline-formula><tex-math id="M1">\begin{document}$ \mathscr{R}_{0} $\end{document}</tex-math></inline-formula> which serves as a threshold parameter that predicts whether epidemics will persist or become globally extinct. Moreover, we study the asymptotic profiles of the positive steady state as the dispersal rate of the susceptible or infected individuals approaches zero. Our analytical results reveal that the epidemics can be extinct by limiting the movement of the susceptible individuals, and the infected individuals concentrate on certain points in some circumstances when limiting their mobility.</p>

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Optimal Control of a Dengue-Dengvaxia Model: Comparison Between Vaccination and Vector Control

Computational and Mathematical Biophysics ◽

10.1515/cmb-2020-0124 ◽

2021 ◽

Vol 9 (1) ◽

pp. 198-212

Author(s):

Cheryl Q. Mentuda

Keyword(s):

Optimal Control ◽

Vector Control ◽

Basic Reproduction Number ◽

Human Population ◽

Model Comparison ◽

Reproduction Number ◽

Control Strategies ◽

Minimum Principle ◽

Transmitted Disease ◽

The Basic Reproduction Number

Abstract Dengue is the most common mosquito-borne viral infection transmitted disease. It is due to the four types of viruses (DENV-1, DENV-2, DENV-3, DENV-4), which transmit through the bite of infected Aedes aegypti and Aedes albopictus female mosquitoes during the daytime. The first globally commercialized vaccine is Dengvaxia, also known as the CYD-TDV vaccine, manufactured by Sanofi Pasteur. This paper presents a Ross-type epidemic model to describe the vaccine interaction between humans and mosquitoes using an entomological mosquito growth population and constant human population. After establishing the basic reproduction number ℛ0, we present three control strategies: vaccination, vector control, and the combination of vaccination and vector control. We use Pontryagin’s minimum principle to characterize optimal control and apply numerical simulations to determine which strategies best suit each compartment. Results show that vector control requires shorter time applications in minimizing mosquito populations. Whereas vaccinating the primary susceptible human population requires a shorter time compared to the secondary susceptible human.

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PENGEMBANGAN MODEL EPIDEMIK SIRA UNTUK PENYEBARAN VIRUS PADA JARINGAN KOMPUTER

Journal of Fundamental Mathematics and Applications (JFMA) ◽

10.14710/jfma.v2i1.26 ◽

2019 ◽

Vol 2 (1) ◽

pp. 13

Author(s):

Panca Putra Pemungkas ◽

Sutrisno Sutrisno ◽

Sunarsih Sunarsih

Keyword(s):

Equilibrium Point ◽

Basic Reproduction Number ◽

Epidemic Model ◽

Computer Network ◽

Reproduction Number ◽

Equilibrium Points ◽

Virus Spread ◽

Computer Based ◽

Stability Issues ◽

Spread Analysis

This paper is addressed to discuss the development of epidemic model of SIRA (Susceptible-Infected-Removed-Antidotal) for virus spread analysis purposes on a computer network. We have developed the existing model by adding a possibility of antidotal computer returned to susceptible computer. Based on the results, there are two virus-free equilibrium points and one endemic equilibrium point. These equilibrium points were analyzed for stability issues using basic reproduction number and Routh-Hurwitz Method.

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On an SEIR Epidemic Model with Vaccination of Newborns and Periodic Impulsive Vaccination with Eventual On-Line Adapted Vaccination Strategies to the Varying Levels of the Susceptible Subpopulation

Applied Sciences ◽

10.3390/app10228296 ◽

2020 ◽

Vol 10 (22) ◽

pp. 8296 ◽

Cited By ~ 1

Author(s):

Malen Etxeberria-Etxaniz ◽

Santiago Alonso-Quesada ◽

Manuel De la Sen

Keyword(s):

Equilibrium Point ◽

Basic Reproduction Number ◽

Epidemic Model ◽

Disease Transmission ◽

Reproduction Number ◽

Equilibrium Points ◽

Endemic Equilibrium ◽

Impulsive Vaccination ◽

Disease Free ◽

The Basic Reproduction Number

This paper investigates a susceptible-exposed-infectious-recovered (SEIR) epidemic model with demography under two vaccination effort strategies. Firstly, the model is investigated under vaccination of newborns, which is fact in a direct action on the recruitment level of the model. Secondly, it is investigated under a periodic impulsive vaccination on the susceptible in the sense that the vaccination impulses are concentrated in practice in very short time intervals around a set of impulsive time instants subject to constant inter-vaccination periods. Both strategies can be adapted, if desired, to the time-varying levels of susceptible in the sense that the control efforts be increased as those susceptible levels increase. The model is discussed in terms of suitable properties like the positivity of the solutions, the existence and allocation of equilibrium points, and stability concerns related to the values of the basic reproduction number. It is proven that the basic reproduction number lies below unity, so that the disease-free equilibrium point is asymptotically stable for larger values of the disease transmission rates under vaccination controls compared to the case of absence of vaccination. It is also proven that the endemic equilibrium point is not reachable if the disease-free one is stable and that the disease-free equilibrium point is unstable if the reproduction number exceeds unity while the endemic equilibrium point is stable. Several numerical results are investigated for both vaccination rules with the option of adapting through ime the corresponding efforts to the levels of susceptibility. Such simulation examples are performed under parameterizations related to the current SARS-COVID 19 pandemic.

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A TWO-STRAIN EPIDEMIC MODEL WITH UNCERTAINTY IN THE INTERACTION

The ANZIAM Journal ◽

10.1017/s1446181112000326 ◽

2012 ◽

Vol 54 (1-2) ◽

pp. 108-115 ◽

Cited By ~ 1

Author(s):

M. G. ROBERTS

Keyword(s):

Differential Equations ◽

Basic Reproduction Number ◽

Epidemic Model ◽

Influenza A ◽

Reproduction Number ◽

The Other ◽

Orthogonal Functions ◽

Dependent Variables ◽

Independent Variable ◽

Cross Immunity

AbstractAnnual epidemics of influenza A typically involve two subtypes, with a degree of cross-immunity. We present a model of an epidemic of two interacting viruses, where the degree of cross-immunity may be unknown. We treat the unknown as a second independent variable, and expand the dependent variables in orthogonal functions of this variable. The resulting set of differential equations is solved numerically. We show that if the population is initially more susceptible to one variant, if that variant invades earlier, or if it has a higher basic reproduction number than the other variant, then its dynamics are largely unaffected by cross-immunity. In contrast, the dynamics of the other variant may be considerably restricted.

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Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies

Mathematics ◽

10.3390/math6120328 ◽

2018 ◽

Vol 6 (12) ◽

pp. 328 ◽

Cited By ~ 3

Author(s):

Yanli Ma ◽

Jia-Bao Liu ◽

Haixia Li

Keyword(s):

Lyapunov Function ◽

Basic Reproduction Number ◽

Epidemic Model ◽

Reproduction Number ◽

Endemic Equilibrium ◽

Global Dynamics ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Disease Free ◽

The Basic Reproduction Number

In this paper, an SIQR (Susceptible, Infected, Quarantined, Recovered) epidemic model with vaccination, elimination, and quarantine hybrid strategies is proposed, and the dynamics of this model are analyzed by both theoretical and numerical means. Firstly, the basic reproduction number R 0 , which determines whether the disease is extinct or not, is derived. Secondly, by LaSalles invariance principle, it is proved that the disease-free equilibrium is globally asymptotically stable when R 0 < 1 , and the disease dies out. By Routh-Hurwitz criterion theory, we also prove that the disease-free equilibrium is unstable and the unique endemic equilibrium is locally asymptotically stable when R 0 > 1 . Thirdly, by constructing a suitable Lyapunov function, we obtain that the unique endemic equilibrium is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists when R 0 > 1 . Finally, some numerical simulations are presented to illustrate the analysis results.

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