Analysis of SIS-SI Stochastic Model with CTMC on the Spread of Malaria Disease

Various mathematical models have been developed to describe the transmission of malaria disease. The purpose of this study was to modify an existing mathematical model of malaria disease by using a CTMC stochastic model. The investigation focused on the transition probability, the basic reproduction number (R0), the outbreak probability, the expected time required to reach a disease-free equilibrium, and the quasi-stationary probability distribution. The population system will experience disease outbreak if R0>1, whereas an outbreak will not occur in the population system if R0≤1. The probability that a mosquito bites an infectious human is denoted as k, while θ is associated with human immunity. Based on the numerical analysis conducted, k and θ have high a contribution to the distribution of malaria disease. This conclusion is based on their impact on the outbreak probability and the expected time required to reach a disease-free equilibrium.

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Stochastic model of the transmission dynamics of COVID-19 pandemic

Advances in Difference Equations ◽

10.1186/s13662-021-03597-1 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Aychew Wondyfraw Tesfaye ◽

Tesfaye Sama Satana

Keyword(s):

Stochastic Model ◽

Global Stability ◽

Equilibrium Point ◽

Basic Reproduction Number ◽

Deterministic Model ◽

Reproduction Number ◽

Index Formula ◽

Sensitivity Index ◽

Simulation Results ◽

Disease Free

AbstractIn this paper, we formulate an SVITR deterministic model and extend it to a stochastic model by introducing intensity of stochastic factors and Brownian motion. Our basic qualitative analysis of both models includes the positivity of the solution, invariant region, disease-free equilibrium point, basic reproduction number, local and global stability of disease-free equilibrium point, endemic equilibrium point, and sensitivity. We obtain the stochastic reproduction number and local stability by using twice differentiable Itô’s formula. We prove the global stability of the disease-free equilibrium point by using a Lyapunov function. We determine the sensitivity of the effect of each parameter on basic reproduction number of the model by using a normalized sensitivity index formula. On the other hand, we demonstrate numerical simulation results of deterministic and stochastic models of COVID-19 by using Maple 18 and MATLAB software. Our simulation results indicate that reducing the contact between infected and susceptible individuals and improvement of treatment play a vital role in COVID-19 pandemic control.

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Analysis of stability and sensitivity of deterministic and stochastic models for the spread of the new corona virus SARS-CоV-2

Filomat ◽

10.2298/fil2103045j ◽

2021 ◽

Vol 35 (3) ◽

pp. 1045-1063

Author(s):

Bojana Jovanovic ◽

Jasmina Ðordjevic ◽

Jelena Manojlovic ◽

Nenad Suvak

Keyword(s):

Sensitivity Analysis ◽

Stochastic Model ◽

Basic Reproduction Number ◽

Stochastic Models ◽

Reproduction Number ◽

Endemic Equilibrium ◽

Stochastic Version ◽

Analysis Of Stability ◽

Parameter Values ◽

Disease Free

Basic reproduction number for deterministic SEIPHAR model and its stochastic counterpart for the spread of SARS-CoV-2 virus are analyzed and compared. For deterministic version of the model, conditions for stability of the disease-free equilibrium are derived and, in addition, conditions for existence of bifurcation state related to endemic equilibrium are established. For stochastic model, conditions for extinction and persistence in mean of the disease are derived. Complete sensitivity analysis of thresholds between the extinction and mean-persistence are performed for both the deterministic and the stochastic version of the model. Influence of variation in parameter values is illustrated for epidemics in Wuhan in early 2020.

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Dynamical balance between the transmission, intervention of COVID-19 and economic development

10.1101/2020.08.13.20174755 ◽

2020 ◽

Author(s):

Zhaowang Zhang ◽

Hualiang Lin ◽

Guanghu Zhu

Keyword(s):

Disease Transmission ◽

Reproduction Number ◽

Economic Loss ◽

Disease Outbreak ◽

Intervention Strategies ◽

Poverty Trap ◽

Coupling Mechanism ◽

The Stability ◽

The Cost ◽

Disease Free

The current explosive outbreak of coronavirus (COVID-19) is posing serious threats to public health and economy around the world. To clarify the coupling mechanism between this disease and economy, a new dynamical system is established. It is theoretically proved that the basic reproduction number is a nonlinear combination of parameters regarding disease transmission, intervention and economy effect, which totally determines the stability of the disease-free and endemic equilibria. Further results indicate the existence of interaction and mutual restraint among the transmission, intervention and economy, in which strong coupling of COVID-19 and economy would trigger disease outbreak and form poverty trap, while adaptive isolation of at-risk population could effectively reduce morbidity at the cost of least economic loss. Our findings can offer new insights to improve the intervention strategies against COVID-19.

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Stochastic Modeling for HIV and AIDS Epidemics With Viral Load Detectability

International Journal of Statistics and Probability ◽

10.5539/ijsp.v9n4p33 ◽

2020 ◽

Vol 9 (4) ◽

pp. 33

Author(s):

Peter Emeda Tengaa ◽

Samuel Mwalili ◽

George Otieno Orwa

Keyword(s):

Viral Load ◽

Stochastic Model ◽

Existence And Uniqueness ◽

Reproduction Number ◽

Control Strategies ◽

Random Perturbation ◽

Disease Outbreak ◽

Hiv And Aids ◽

Persistence Properties ◽

Stochastic Epidemic

In this paper, we investigated on the stochastic epidemic model by incorporating viral load detectability. We derived HIV and AIDS stochastic model from the deterministic counterpart model and presented a stochastic threshold in terms of stochastic basic reproduction number. We showed that R_s^0 < 1 then the disease dies out exponentially and when R_s^0 > 1 the disease persists in the population. We further derived the existence and uniqueness, extinction and persistence properties of the stochastic model models, then the numerical simulation is done by using Milsten’s numerical scheme. The finding shows that the random perturbation introduced in the stochastic model suppresses disease outbreak as compared to its deterministic counterparts which provide some useful control strategies to eradicate the disease

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Threshold Dynamics in a Model for Zika Virus Disease with Seasonality

Bulletin of Mathematical Biology ◽

10.1007/s11538-020-00844-6 ◽

2021 ◽

Vol 83 (4) ◽

Cited By ~ 1

Author(s):

Mahmoud A. Ibrahim ◽

Attila Dénes

Keyword(s):

Zika Virus ◽

Reproduction Number ◽

Virus Disease ◽

Threshold Dynamics ◽

Global Dynamics ◽

Threshold Parameter ◽

Globally Asymptotically Stable ◽

Linear Integral ◽

The Impact ◽

Disease Free

AbstractWe present a compartmental population model for the spread of Zika virus disease including sexual and vectorial transmission as well as asymptomatic carriers. We apply a non-autonomous model with time-dependent mosquito birth, death and biting rates to integrate the impact of the periodicity of weather on the spread of Zika. We define the basic reproduction number $${\mathscr {R}}_{0}$$ R 0 as the spectral radius of a linear integral operator and show that the global dynamics is determined by this threshold parameter: If $${\mathscr {R}}_0 < 1,$$ R 0 < 1 , then the disease-free periodic solution is globally asymptotically stable, while if $${\mathscr {R}}_0 > 1,$$ R 0 > 1 , then the disease persists. We show numerical examples to study what kind of parameter changes might lead to a periodic recurrence of Zika.

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A Stochastic TB Model for a Crowded Environment

Journal of Applied Mathematics ◽

10.1155/2018/3420528 ◽

2018 ◽

Vol 2018 ◽

pp. 1-8 ◽

Cited By ~ 3

Author(s):

Sibaliwe Maku Vyambwera ◽

Peter Witbooi

Keyword(s):

Compartmental Model ◽

Deterministic Model ◽

Reproduction Number ◽

Stochastic Perturbation ◽

Equation Model ◽

Ordinary Differential Equation Model ◽

Differential Equation Model ◽

The Stability ◽

Crowded Environments ◽

Disease Free

We propose a stochastic compartmental model for the population dynamics of tuberculosis. The model is applicable to crowded environments such as for people in high density camps or in prisons. We start off with a known ordinary differential equation model, and we impose stochastic perturbation. We prove the existence and uniqueness of positive solutions of a stochastic model. We introduce an invariant generalizing the basic reproduction number and prove the stability of the disease-free equilibrium when it is below unity or slightly higher than unity and the perturbation is small. Our main theorem implies that the stochastic perturbation enhances stability of the disease-free equilibrium of the underlying deterministic model. Finally, we perform some simulations to illustrate the analytical findings and the utility of the model.

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MODELING THE IMPACT OF VOLUNTARY TESTING AND TREATMENT ON TUBERCULOSIS TRANSMISSION DYNAMICS

International Journal of Biomathematics ◽

10.1142/s1793524511001726 ◽

2012 ◽

Vol 05 (04) ◽

pp. 1250029 ◽

Cited By ~ 3

Author(s):

S. MUSHAYABASA ◽

C. P. BHUNU

Keyword(s):

Deterministic Model ◽

Reproduction Number ◽

Positive Impact ◽

Transmission Dynamics ◽

Effective Control ◽

Reproductive Number ◽

Voluntary Testing ◽

Manifold Theory ◽

The Impact ◽

Disease Free

A deterministic model for evaluating the impact of voluntary testing and treatment on the transmission dynamics of tuberculosis is formulated and analyzed. The epidemiological threshold, known as the reproduction number is derived and qualitatively used to investigate the existence and stability of the associated equilibrium of the model system. The disease-free equilibrium is shown to be locally-asymptotically stable when the reproductive number is less than unity, and unstable if this threshold parameter exceeds unity. It is shown, using the Centre Manifold theory, that the model undergoes the phenomenon of backward bifurcation where the stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction number is less than unity. The analysis of the reproduction number suggests that voluntary tuberculosis testing and treatment may lead to effective control of tuberculosis. Furthermore, numerical simulations support the fact that an increase voluntary tuberculosis testing and treatment have a positive impact in controlling the spread of tuberculosis in the community.

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Threshold Dynamics of a Huanglongbing Model with Logistic Growth in Periodic Environments

Abstract and Applied Analysis ◽

10.1155/2014/841367 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10 ◽

Cited By ~ 3

Author(s):

Jianping Wang ◽

Shujing Gao ◽

Yueli Luo ◽

Dehui Xie

Keyword(s):

Basic Reproduction Number ◽

Reproduction Number ◽

Logistic Growth ◽

Threshold Dynamics ◽

Global Dynamics ◽

Globally Asymptotically Stable ◽

Linear Integral ◽

The Impact ◽

Disease Free ◽

The Basic Reproduction Number

We analyze the impact of seasonal activity of psyllid on the dynamics of Huanglongbing (HLB) infection. A new model about HLB transmission with Logistic growth in psyllid insect vectors and periodic coefficients has been investigated. It is shown that the global dynamics are determined by the basic reproduction numberR0which is defined through the spectral radius of a linear integral operator. IfR0< 1, then the disease-free periodic solution is globally asymptotically stable and ifR0> 1, then the disease persists. Numerical values of parameters of the model are evaluated taken from the literatures. Furthermore, numerical simulations support our analytical conclusions and the sensitive analysis on the basic reproduction number to the changes of average and amplitude values of the recruitment function of citrus are shown. Finally, some useful comments on controlling the transmission of HLB are given.

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Mathematical Modeling and Analysis of Covid-19 Infection spreads in India with Restricted Optimal Treatment on Disease Incidence

BIOMATH ◽

10.11145/j.biomath.2021.06.147 ◽

2021 ◽

Vol 10 (1) ◽

pp. 2106147

Author(s):

Debkumar Pal ◽

D Ghosh ◽

P K Santra ◽

G S Mahapatra

Keyword(s):

Mathematical Modeling ◽

Objective Function ◽

Reproduction Number ◽

Disease Incidence ◽

Treatment Control ◽

Modeling And Analysis ◽

Proposed Model ◽

The Cost ◽

Disease Free ◽

The Basic Reproduction Number

This paper presents the current situation and how to minimize its effect in India through a mathematical model of infectious Coronavirus disease (COVID-19). This model consists of six compartments to population classes consisting of susceptible, exposed, home quarantined, government quarantined, infected individuals in treatment, and recovered class. The basic reproduction number is calculated, and the stabilities of the proposed model at the disease-free equilibrium and endemic equilibrium are observed. The next crucial treatment control of the Covid-19 epidemic model is presented in India's situation. An objective function is considered by incorporating the optimal infected individuals and the cost of necessary treatment. Finally, optimal control is achieved that minimizes our anticipated objective function. Numerical observations are presented utilizing MATLAB software to demonstrate the consistency of present-day representation from a realistic standpoint.

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The dynamics of a delayed deterministic and stochastic epidemic SIR models with a saturated incidence rate

Random Operators and Stochastic Equations ◽

10.1515/rose-2018-0021 ◽

2018 ◽

Vol 26 (4) ◽

pp. 235-245 ◽

Cited By ~ 1

Author(s):

Modeste N’zi ◽

Ilimidi Yattara

Keyword(s):

Stochastic Model ◽

White Noise ◽

Incidence Rate ◽

Deterministic Model ◽

Contact Rate ◽

Almost Sure Stability ◽

Saturated Incidence Rate ◽

Saturated Incidence ◽

Global Positive Solution ◽

Disease Free

AbstractWe treat a delayed SIR (susceptible, infected, recovered) epidemic model with a saturated incidence rate and its perturbation through the contact rate using a white noise. We start with a deterministic model and then add a perturbation on the contact rate using a white noise to obtain a stochastic model. We prove the existence and uniqueness of the global positive solution for both deterministic and stochastic delayed differential equations. Under suitable conditions on the parameters, we study the global asymptotic stability of the disease-free equilibrium of the deterministic model and the almost sure stability of the disease-free equilibrium of the stochastic model.

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