A MODEL OF THE TRANSMISSION DYNAMICS OF LEISHMANIASIS

2011 ◽  
Vol 19 (02) ◽  
pp. 237-250 ◽  
Author(s):  
EPHRAIM O. AGYINGI ◽  
DAVID S. ROSS ◽  
KARTHIK BATHENA
Keyword(s):  
Global Attractor ◽  
Basic Reproduction Number ◽  
Human Population ◽  
Reproduction Number ◽  
Transmission Dynamics ◽  
Sis Model ◽  
Vector Population ◽  
Single Animal ◽  
Reservoir Hosts ◽  
The Basic Reproduction Number

In this paper we present a susceptible–infectious–susceptible (SIS) model that describes the transmission dynamics of cutaneous Leishmaniasis. The model treats a vector population and several populations of different mammals. Members of the human population serve as the incidental hosts, and members of the various animals populations serve as reservoir hosts. We establish the basic reproduction number and the equilibrium conditions of the system. We use a generalization of the Lyapunov function approach to show that when the basic reproduction number is less than or equal to one, the diseases-free equilibrium is a global attractor, and that when it is greater than one the endemic equilibrium is a global attractor. We present numerical simulations that demonstrate the dynamics of the model for a system containing a human population and a single animal population.

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

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.

scholarly journals Seasonality Impact on the Transmission Dynamics of Tuberculosis

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Yali Yang ◽  
Chenping Guo ◽  
Luju Liu ◽  
Tianhua Zhang ◽  
Weiping Liu
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Autonomous Systems ◽  
Positive Periodic Solution ◽  
Transmission Dynamics ◽  
Uniform Persistence ◽  
Globally Asymptotically Stable ◽  
The Impact ◽  
Disease Free ◽  
The Basic Reproduction Number

The statistical data of monthly pulmonary tuberculosis (TB) incidence cases from January 2004 to December 2012 show the seasonality fluctuations in Shaanxi of China. A seasonality TB epidemic model with periodic varying contact rate, reactivation rate, and disease-induced death rate is proposed to explore the impact of seasonality on the transmission dynamics of TB. Simulations show that the basic reproduction number of time-averaged autonomous systems may underestimate or overestimate infection risks in some cases, which may be up to the value of period. The basic reproduction number of the seasonality model is appropriately given, which determines the extinction and uniform persistence of TB disease. If it is less than one, then the disease-free equilibrium is globally asymptotically stable; if it is greater than one, the system at least has a positive periodic solution and the disease will persist. Moreover, numerical simulations demonstrate these theorem results.

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 Global Dynamics of an SIS Model on Metapopulation Networks with Demographics

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Maoxing Liu ◽  
Xinjie Fu ◽  
Jie Zhang ◽  
Donghua Zhao
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Sufficient Conditions ◽  
Active Role ◽  
Endemic Equilibrium ◽  
Global Dynamics ◽  
Sis Model ◽  
Globally Asymptotically Stable ◽  
Sis Epidemic Model ◽  
The Basic Reproduction Number

In this paper, we propose a susceptible-infected-susceptible (SIS) epidemic model with demographics on heterogeneous metapopulation networks. We analytically derive the basic reproduction number, which determines not only the existence of endemic equilibrium but also the global dynamics of the model. The model always has the disease-free equilibrium, which is globally asymptotically stable when the basic reproduction number is less than unity and otherwise unstable. We also provide sufficient conditions on the global stability of the unique endemic equilibrium. Numerical simulations are performed to illustrate the theoretical results and the effects of the connectivity and diffusion. Furthermore, we find that diffusion rates play an active role in controlling the spread of infectious diseases.

scholarly journals SEIITR Model for Diabetes Mellitus Distribution in Case of Insulin and Care Factors

2020 ◽  
Vol 5 (2) ◽  
pp. 100-106
Author(s):  
Nur Fajri ◽  
Sanusi ◽  
Asmaidi
Keyword(s):  
Diabetes Mellitus ◽  
Mathematical Model ◽  
Fixed Point ◽  
Fixed Points ◽  
Basic Reproduction Number ◽  
Human Population ◽  
Reproduction Number ◽  
Model Type ◽  
Reproduction Numbers ◽  
The Basic Reproduction Number

This research is done to learn diabetes mellitus type SEIITR with insulin and care factors. Mathematical model type SEIITR is a mathematical model of diabetes in which the human population is divided into five groups: susceptible humans (Susceptible) S, exposed (Exposed) E, infected I without treatment, infected (Infected) IT  with treatment dan recovered (Recovery) R. The SEIITR model has two fixed points, namely, a fixed point without disease and an endemic fixed point. By using basic reproduction numbers (R0), it is found that the fixed point without disease is stable if R0 < 1 and when R0 > 1. Then the fixed point without disease is unstable. The simulation shows the effect of giving insulin to changes in the value of the basic reproduction number. If the effectiveness of β decreases, the basic reproduction number decreases too. Thus, a decrease in the value of this parameter will be able to help reduce the rate of diabetes mellitus in the population.

scholarly journals The basic reproduction number of COVID-19 across Africa

2021 ◽  
Author(s):  
Sarafa Adewale Iyaniwura ◽  
Muhammad Rabiu Musa ◽  
Jummy F. David ◽  
Jude Dzevela Kong
Keyword(s):  
Bayesian Inference ◽  
Severe Acute Respiratory Syndrome ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Transmission Dynamics ◽  
Clear Understanding ◽  
Seir Model ◽  
The World ◽  
Bayesian Inference Framework ◽  
The Basic Reproduction Number

The pandemic of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) took the world by surprise. Following the first outbreak of COVID-19 in December 2019, several models have been developed to study and understand its transmission dynamics. Although the spread of COVID-19 is being slowed down by vaccination and other interventions, there is still a need to have a clear understanding of the evolution of the pandemic across countries, states and communities. To this end, there is a need to have a clearer picture of the initial spread of the disease in different regions. In this project, we used a simple SEIR model and a Bayesian inference framework to estimate the basic reproduction number of COVID-19 across Africa. Our estimates vary between 1.98 (Sudan) and 9.66 (Mauritius), with a median of 3.67 (90% CrI: 3.31 - 4.12). The estimates provided in this paper will help to inform COVID-19 modeling in the respective countries/regions.

scholarly journals Mathematical Model of the Transmission Dynamics of Corona Virus Disease (COVID-19) and Its Control

2021 ◽  
pp. 69-88
Author(s):  
Atokolo William ◽  
Omale David ◽  
Bashir Sezuo Tenuche ◽  
Olayemi Kehinde Samuel ◽  
Daniel Musa Alih ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Equilibrium State ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Virus Disease ◽  
Control Measure ◽  
Transmission Dynamics ◽  
Asymptotically Stable ◽  
Corona Virus ◽  
The Basic Reproduction Number

This work is aimed at formulating a mathematical model for the transmission dynamics and control of corona virus disease in a population. The Disease Free Equilibrium state of the model was determined and shown to be locally asymptotically stable. The Endemic Equilibrium state of the model was also established and proved to be locally asymptotically stable using the trace and determinant method, after which we determined the basic reproduction number ( ) of the model using the next generation method. When ( ), the disease is wiped out of a population, but if ( ), the disease invades such population. Local sensitivity analysis result shows that the rate at which the exposed are quarantined ( ), the rate at which the infected are isolated ( ), the rate at which the quarantined are isolated ( ), and the treatment rate ( ) should be targeted by the control intervention strategies as an increase in the values of these parameters (  and ) will reduce the basic reproduction number  ( ) of the COVID-19 and as such will eliminate the disease from the population with time. Numerical simulation of the model shows that the disease will be eradicated with time when enlightenment control measure for the susceptible individuals to observe social distance, frequent use of hand sanitizers, covering of mouth when coughing or sneezing are properly observed. Moreso, increasing the rates at which the suspected and confirmed cases of COVID-19 are quarantined and isolated respectively reduce the spread of the global pandemic.

scholarly journals A Data Driven Analysis and Forecast of COVID-19 Dynamics during the Third Wave Using SIRD Model in Bangladesh

COVID ◽  
2021 ◽  
Vol 1 (2) ◽  
pp. 503-517
Author(s):  
Omar Faruk ◽  
Suman Kar
Keyword(s):  
Numerical Simulations ◽  
Basic Reproduction Number ◽  
Goodness Of Fit ◽  
Reproduction Number ◽  
Transmission Rate ◽  
Transmission Dynamics ◽  
Third Wave ◽  
The Third ◽  
The Government ◽  
The Basic Reproduction Number

In this study, we developed a compartmental SIRD model to analyze and forecast the transmission dynamics of the COVID-19 pandemic in Bangladesh during the third wave caused by the Indian delta variant. With the help of the nonlinear system of differential equations, this model can analyze the trends and provide reliable predictions regarding how the epidemic would evolve. The basic reproduction number regarding the pandemic has been determined analytically. The parameters used in this model have been estimated by fitting our model to the reported data for the months of May, June, and July 2021 and the goodness of fit of the parameter’s value has been found by the respective regression coefficients. Further, we conducted a sensitivity analysis of the basic reproduction number and observed that decreasing the transmission rate is the most significant factor in disease prevention. Our proposed model’s appropriateness for the available COVID-19 data in Bangladesh has been demonstrated through numerical simulations. According to the numerical simulation, it is evident that a rise in the transmission rate leads to a significant increase in the infected number of the population. Numerical simulations have also been performed by using our proposed model to forecast the future transmission dynamics for COVID-19 over a longer period of time. Knowledge of these forecasts may help the government in adopting appropriate measures to prepare for unforeseen situations that may arise in Bangladesh as well as to minimize detrimental impacts during the outbreak.

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