scholarly journals A Stochastic Model for Malaria Transmission Dynamics

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Rachel Waema Mbogo ◽  
Livingstone S. Luboobi ◽  
John W. Odhiambo
Keyword(s):  
Infectious Diseases ◽  
Stochastic Model ◽  
Malaria Transmission ◽  
Stochastic Models ◽  
Continuous Time ◽  
Reproduction Number ◽  
Transmission Dynamics ◽  
Infected Mosquito ◽  
Deterministic Models ◽  
Discrete State

Malaria is one of the three most dangerous infectious diseases worldwide (along with HIV/AIDS and tuberculosis). In this paper we compare the disease dynamics of the deterministic and stochastic models in order to determine the effect of randomness in malaria transmission dynamics. Relationships between the basic reproduction number for malaria transmission dynamics between humans and mosquitoes and the extinction thresholds of corresponding continuous-time Markov chain models are derived under certain assumptions. The stochastic model is formulated using the continuous-time discrete state Galton-Watson branching process (CTDSGWbp). The reproduction number of deterministic models is an essential quantity to predict whether an epidemic will spread or die out. Thresholds for disease extinction from stochastic models contribute crucial knowledge on disease control and elimination and mitigation of infectious diseases. Analytical and numerical results show some significant differences in model predictions between the stochastic and deterministic models. In particular, we find that malaria outbreak is more likely if the disease is introduced by infected mosquitoes as opposed to infected humans. These insights demonstrate the importance of a policy or intervention focusing on controlling the infected mosquito population if the control of malaria is to be realized.

scholarly journals Estimating the distribution of time to extinction of infectious diseases in mean-field approaches

2020 ◽  
Author(s):  
Maryam Aliee ◽  
Kat S. Rock ◽  
Matt J. Keeling
Keyword(s):  
Infectious Diseases ◽  
Stochastic Models ◽  
Mean Field ◽  
Practical Method ◽  
Deterministic Models ◽  
Recovery Processes ◽  
Time To Extinction ◽  
And Control ◽  
Field Approaches ◽  
Birth Death

AbstractA key challenge for many infectious diseases is to predict the time to extinction under specific interventions. In general this question requires the use of stochastic models which recognise the inherent individual-based, chance-driven nature of the dynamics; yet stochastic models are inherently computationally expensive, especially when parameter uncertainty also needs to be incorporated. Deterministic models are often used for prediction as they are more tractable, however their inability to precisely reach zero infections makes forecasting extinction times problematic. Here, we study the extinction problem in deterministic models with the help of an effective “birth-death” description of infection and recovery processes. We present a practical method to estimate the distribution, and therefore robust means and prediction intervals, of extinction times by calculating their different moments within the birth-death framework. We show these predictions agree very well with the results of stochastic models by analysing the simplified SIS dynamics as well as studying an example of more complex and realistic dynamics accounting for the infection and control of African sleeping sickness (Trypanosoma brucei gambiense).

scholarly journals Analysis of the Model on the Effect of Seasonal Factors on Malaria Transmission Dynamics

2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Victor Yiga ◽  
Hasifa Nampala ◽  
Julius Tumwiine
Keyword(s):  
Malaria Transmission ◽  
Basic Reproduction Number ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Current Control ◽  
Transmission Dynamics ◽  
Mosquito Vector ◽  
The Impact ◽  
The Basic Reproduction Number

Malaria is one of the world’s most prevalent epidemics. Current control and eradication efforts are being frustrated by rapid changes in climatic factors such as temperature and rainfall. This study is aimed at assessing the impact of temperature and rainfall abundance on the intensity of malaria transmission. A human host-mosquito vector deterministic model which incorporates temperature and rainfall dependent parameters is formulated. The model is analysed for steady states and their stability. The basic reproduction number is obtained using the next-generation method. It was established that the mosquito population depends on a threshold value θ , defined as the number of mosquitoes produced by a female Anopheles mosquito throughout its lifetime, which is governed by temperature and rainfall. The conditions for the stability of the equilibrium points are investigated, and it is shown that there exists a unique endemic equilibrium which is locally and globally asymptotically stable whenever the basic reproduction number exceeds unity. Numerical simulations show that both temperature and rainfall affect the transmission dynamics of malaria; however, temperature has more influence.

scholarly journals On discrete time epidemic models in Kermack-McKendrick form

2021 ◽  
Author(s):  
Odo Diekmann ◽  
Hans G. Othmer ◽  
Robert Planque ◽  
Martin CJ Bootsma
Keyword(s):  
Infectious Diseases ◽  
Discrete Time ◽  
Epidemic Model ◽  
Continuous Time ◽  
Reproduction Number ◽  
Compartmental Models ◽  
Initial Speed ◽  
Fixed Duration ◽  
Epidemic Spread ◽  
Special Cases

Surprisingly, the discrete-time version of the general 1927 Kermack-McKendrick epidemic model has, to our knowledge, not been formulated in the literature, and we rectify this omission here. The discrete time version is as general and flexible as its continuous-time counterpart, and contains numerous compartmental models as special cases. In contrast to the continuous time version, the discrete time version of the model is very easy to implement computationally, and thus promises to become a powerful tool for exploring control scenarios for specific infectious diseases. To demonstrate the potential, we investigate numerically how the incidence-peak size depends on model ingredients. We find that, with the same reproduction number and initial speed of epidemic spread, compartmental models systematically predict lower peak sizes than models that use a fixed duration for the latent and infectious periods.

scholarly journals Impact of Hygiene on Malaria Transmission Dynamics: A Mathematical Model

2021 ◽  
Author(s):  
Temidayo Oluwafemi ◽  
Emmanuel Azuaba
Keyword(s):  
Mathematical Model ◽  
Malaria Transmission ◽  
Basic Reproduction Number ◽  
Human Population ◽  
Reproduction Number ◽  
Model System ◽  
Transmission Dynamics ◽  
The Impact ◽  
Disease Free ◽  
The Basic Reproduction Number

Malaria continues to pose a major public health challenge, especially in developing countries, 219 million cases of malaria were estimated in 89 countries. In this paper, a mathematical model using non-linear differential equations is formulated to describe the impact of hygiene on Malaria transmission dynamics, the model is analyzed. The model is divided into seven compartments which includes five human compartments namely; Unhygienic susceptible human population, Hygienic Susceptible Human population, Unhygienic infected human population , hygienic infected human population and the Recovered Human population  and the mosquito population is subdivided into susceptible mosquitoes  and infected mosquitoes . The positivity of the solution shows that there exists a domain where the model is biologically meaningful and mathematically well-posed. The Disease-Free Equilibrium (DFE) point of the model is obtained, we compute the Basic Reproduction Number using the next generation method and established the condition for Local stability of the disease-free equilibrium, and we thereafter obtained the global stability of the disease-free equilibrium by constructing the Lyapunov function of the model system. Also, sensitivity analysis of the model system was carried out to identify the influence of the parameters on the Basic Reproduction Number, the result shows that the natural death rate of the mosquitoes is most sensitive to the basic reproduction number.

scholarly journals Investigation of the Stochastic Modeling of COVID-19 with Environmental Noise from the Analytical and Numerical Point of View

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3122
Author(s):  
Shah Hussain ◽  
Elissa Nadia Madi ◽  
Hasib Khan ◽  
Sina Etemad ◽  
Shahram Rezapour ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Stochastic Model ◽  
Numerical Scheme ◽  
Computational Analysis ◽  
Reproduction Number ◽  
Numerical Data ◽  
Point Of View ◽  
Deterministic Models ◽  
The Future ◽  
Future Prediction

In this article, we propose a novel mathematical model for the spread of COVID-19 involving environmental white noise. The new stochastic model was studied for the existence and persistence of the disease, as well as the extinction of the disease. We noticed that the existence and extinction of the disease are dependent on R0 (the reproduction number). Then, a numerical scheme was developed for the computational analysis of the model; with the existing values of the parameters in the literature, we obtained the related simulations, which gave us more realistic numerical data for the future prediction. The mentioned stochastic model was analyzed for different values of σ1,σ2 and β1,β2, and both the stochastic and the deterministic models were compared for the future prediction of the spread of COVID-19.

scholarly journals Estimating the distribution of time to extinction of infectious diseases in mean-field approaches

2020 ◽  
Vol 17 (173) ◽  
pp. 20200540
Author(s):  
Maryam Aliee ◽  
Kat S. Rock ◽  
Matt J. Keeling
Keyword(s):  
Infectious Diseases ◽  
Stochastic Models ◽  
Mean Field ◽  
Practical Method ◽  
Deterministic Models ◽  
Recovery Processes ◽  
Time To Extinction ◽  
And Control ◽  
Field Approaches ◽  
Birth Death

A key challenge for many infectious diseases is to predict the time to extinction under specific interventions. In general, this question requires the use of stochastic models which recognize the inherent individual-based, chance-driven nature of the dynamics; yet stochastic models are inherently computationally expensive, especially when parameter uncertainty also needs to be incorporated. Deterministic models are often used for prediction as they are more tractable; however, their inability to precisely reach zero infections makes forecasting extinction times problematic. Here, we study the extinction problem in deterministic models with the help of an effective ‘birth–death’ description of infection and recovery processes. We present a practical method to estimate the distribution, and therefore robust means and prediction intervals, of extinction times by calculating their different moments within the birth–death framework. We show that these predictions agree very well with the results of stochastic models by analysing the simplified susceptible–infected–susceptible (SIS) dynamics as well as studying an example of more complex and realistic dynamics accounting for the infection and control of African sleeping sickness ( Trypanosoma brucei gambiense ).

scholarly journals Dynamics of infectious diseases: A review of the main biological aspects and their mathematical translation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Deccy Y. Trejos ◽  
Jose C. Valverde ◽  
Ezio Venturino
Keyword(s):  
Mathematical Model ◽  
Infectious Diseases ◽  
Discrete Time ◽  
Continuous Time ◽  
Control Strategies ◽  
Transmission Dynamics ◽  
Model Parameters ◽  
Simulation Techniques ◽  
Biological Aspects ◽  
Tools And Methods

Abstract In this paper, the main biological aspects of infectious diseases and their mathematical translation for modeling their transmission dynamics are revised. In particular, some heterogeneity factors which could influence the fitting of the model to reality are pointed out. Mathematical tools and methods needed to qualitatively analyze deterministic continuous-time models, formulated by ordinary differential equations, are also introduced, while its discrete-time counterparts are properly referenced. In addition, some simulation techniques to validate a mathematical model and to estimate the model parameters are shown. Finally, we present some control strategies usually considered to prevent epidemic outbreaks and their implementation in the model.

scholarly journals Deterministic and Stochastic Models of the Transmission Dynamics of Chicken Pox.

2019 ◽  
Vol 1 ◽  
pp. 184-192
Author(s):  
Bright O Osu ◽  
O Andrew ◽  
A I Victory
Keyword(s):  
Stochastic Model ◽  
Numerical Simulations ◽  
Stochastic Models ◽  
Deterministic Model ◽  
Transmission Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Chicken Pox ◽  
Number Of Individuals ◽  
Disease Free

In this work a deterministic and stochastic model is developed and used to investigate the transmission dynamics of chicken pox. The models involve the Susceptible, Vaccinated, Exposed, Infectious and Recovered individuals. In the deterministic model the Disease free Equilibrium is computed and proved to be globally asymptotically stable when R0 < 1. The deterministic model is transformed into a stochastic model which was solved using the Euler Maruyama method. Numerical simulations of the stochastic Model show that as the vaccine rate wanes, the number of individuals susceptible to the chicken pox epidemic increases.

scholarly journals Continuous-time stochastic processes for the spread of COVID-19 disease simulated via a Monte Carlo approach and comparison with deterministic models

2021 ◽  
Vol 83 (4) ◽  
Author(s):  
Fabiana Calleri ◽  
Giovanni Nastasi ◽  
Vittorio Romano
Keyword(s):  
Monte Carlo ◽  
Stochastic Models ◽  
Continuous Time ◽  
Cross Validation ◽  
Numerical Solutions ◽  
Herd Immunity ◽  
Deterministic Models ◽  
Infected People ◽  
Asymptomatic Individuals ◽  
Second Wave

AbstractTwo stochastic models are proposed to describe the evolution of the COVID-19 pandemic. In the first model the population is partitioned into four compartments: susceptible S, infected I, removed R and dead people D. In order to have a cross validation, a deterministic version of such a model is also devised which is represented by a system of ordinary differential equations with delays. In the second stochastic model two further compartments are added: the class A of asymptomatic individuals and the class L of isolated infected people. Effects such as social distancing measures are easily included and the consequences are analyzed. Numerical solutions are obtained with Monte Carlo simulations. Quantitative predictions are provided which can be useful for the evaluation of political measures, e.g. the obtained results suggest that strategies based on herd immunity are too risky. Finally, the models are calibrated on data referring to the second wave of infection in Italy.

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