A classical SEIR model of transmission dynamics and clinical dynamics in controlling of coronavirus disease 2019 (COVID-19) with 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.

Download Full-text

Can medication mitigate the need for a strict lock down?: A mathematical study of control strategies for COVID-19 infection

10.1101/2020.05.29.20116749 ◽

2020 ◽

Author(s):

Mohsin Ali ◽

Mudassar Imran ◽

Adnan Khan

Keyword(s):

Reproduction Number ◽

Virus Disease ◽

Control Strategies ◽

Transmission Dynamics ◽

Effective Control ◽

Mathematical Study ◽

Seir Model ◽

Epidemic Curve ◽

Asymptomatic Individuals ◽

Over Time

AbstractWe formulate a deterministic epidemic model to study the effects of medication on the transmission dynamics of Corona Virus Disease (COVID-19). We are especially interested in how the availability of medication could change the necessary quarantine measures for effective control of the disease. We model the transmission by extending the SEIR model to include asymptomatic, quarantined, isolated and medicated population compartments. We calculate the basic reproduction number R0 and show that for R0 < 1 the disease dies out and for R0 > 1 the disease is endemic. Using sensitivity analysis we establish that R0 is most sensitive to the rates of quarantine and medication. We also study how the effectiveness and the rate of medication along with the quarantine rate affect R0. We devise optimal quarantine, medication and isolation strategies, noting that availability of medication reduces the duration and severity of the lock-down needed for effective disease control. Our study also reinforces the idea that with the availability of medication, while the severity of the lock downs can be eased over time some social distancing protocols need to be observed, at least till a vaccine is found. We also analyze the COVID-109 outbreak data for four different countries, in two of these, India and Pakistan the curve is still rising, and in he other two, Italy and Spain, the epidemic curve is now falling due to effective quarantine measures. We provide estimates of R0 and the proportion of asymptomatic individuals in the population for these countries.

Download Full-text

Development of the reproduction number from coronavirus SARS-CoV-2 case data in Germany and implications for political measures

BMC Medicine ◽

10.1186/s12916-020-01884-4 ◽

2021 ◽

Vol 19 (1) ◽

Author(s):

Sahamoddin Khailaie ◽

Tanmay Mitra ◽

Arnab Bandyopadhyay ◽

Marta Schips ◽

Pietro Mascheroni ◽

...

Keyword(s):

Structural Changes ◽

Reproduction Number ◽

National Level ◽

Transmission Dynamics ◽

Federal State ◽

Parameter Estimates ◽

Time Varying ◽

Short Term ◽

Robert Koch Institute ◽

Federal States

Abstract Background SARS-CoV-2 has induced a worldwide pandemic and subsequent non-pharmaceutical interventions (NPIs) to control the spread of the virus. As in many countries, the SARS-CoV-2 pandemic in Germany has led to a consecutive roll-out of different NPIs. As these NPIs have (largely unknown) adverse effects, targeting them precisely and monitoring their effectiveness are essential. We developed a compartmental infection dynamics model with specific features of SARS-CoV-2 that allows daily estimation of a time-varying reproduction number and published this information openly since the beginning of April 2020. Here, we present the transmission dynamics in Germany over time to understand the effect of NPIs and allow adaptive forecasts of the epidemic progression. Methods We used a data-driven estimation of the evolution of the reproduction number for viral spreading in Germany as well as in all its federal states using our model. Using parameter estimates from literature and, alternatively, with parameters derived from a fit to the initial phase of COVID-19 spread in different regions of Italy, the model was optimized to fit data from the Robert Koch Institute. Results The time-varying reproduction number (Rt) in Germany decreased to <1 in early April 2020, 2–3 weeks after the implementation of NPIs. Partial release of NPIs both nationally and on federal state level correlated with moderate increases in Rt until August 2020. Implications of state-specific Rt on other states and on national level are characterized. Retrospective evaluation of the model shows excellent agreement with the data and usage of inpatient facilities well within the healthcare limit. While short-term predictions may work for a few weeks, long-term projections are complicated by unpredictable structural changes. Conclusions The estimated fraction of immunized population by August 2020 warns of a renewed outbreak upon release of measures. A low detection rate prolongs the delay reaching a low case incidence number upon release, showing the importance of an effective testing-quarantine strategy. We show that real-time monitoring of transmission dynamics is important to evaluate the extent of the outbreak, short-term projections for the burden on the healthcare system, and their response to policy changes.

Download Full-text

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.

Download Full-text

Mathematical Analysis of Influenza A Dynamics in the Emergence of Drug Resistance

Computational and Mathematical Methods in Medicine ◽

10.1155/2018/2434560 ◽

2018 ◽

Vol 2018 ◽

pp. 1-14 ◽

Cited By ~ 1

Author(s):

Caroline W. Kanyiri ◽

Kimathi Mark ◽

Livingstone Luboobi

Keyword(s):

Mathematical Model ◽

Sensitivity Analysis ◽

Drug Resistance ◽

Stability Analysis ◽

Influenza A ◽

Reproduction Number ◽

Critical Value ◽

Backward Bifurcation ◽

Transmission Dynamics ◽

High Morbidity

Every year, influenza causes high morbidity and mortality especially among the immunocompromised persons worldwide. The emergence of drug resistance has been a major challenge in curbing the spread of influenza. In this paper, a mathematical model is formulated and used to analyze the transmission dynamics of influenza A virus having incorporated the aspect of drug resistance. The qualitative analysis of the model is given in terms of the control reproduction number,Rc. The model equilibria are computed and stability analysis carried out. The model is found to exhibit backward bifurcation prompting the need to lowerRcto a critical valueRc∗for effective disease control. Sensitivity analysis results reveal that vaccine efficacy is the parameter with the most control over the spread of influenza. Numerical simulations reveal that despite vaccination reducing the reproduction number below unity, influenza still persists in the population. Hence, it is essential, in addition to vaccination, to apply other strategies to curb the spread of influenza.

Download Full-text

Estimasi Reproduction Number Model Matematika Penyebaran Malaria di Sumba Tengah, Indonesia

Jambura Journal of Biomathematics (JJBM) ◽

10.34312/jjbm.v2i1.9971 ◽

2021 ◽

Vol 2 (1) ◽

pp. 13-19

Author(s):

Ervin Mawo Banni ◽

Maria A Kleden ◽

Maria Lobo ◽

Meksianis Zadrak Ndii

Keyword(s):

Mathematical Modeling ◽

Mathematical Model ◽

Health Center ◽

Disease Transmission ◽

Reproduction Number ◽

Transmission Dynamics ◽

Awareness Program ◽

Awareness Programs ◽

Malaria Eradication

Malaria is transmitted via a bite of mosquitoes and it is dangerous if it is not properly treated. Mathematical modeling can be formulated to understand the disease transmission dynamics. In this paper, a mathematical model with an awareness program has been formulated and the reproduction number has been estimated against the data from Weeluri Health Center, Mamboro District, Central Sumba. The calculation showed that the reproduction number is R0 = 1.2562. Results showed that if the efficacy of the awareness program is lower than 20%, the reproduction number is still above unity. If the efficacy of the awareness program is higher than 20%, the reproduction number is lower than unity. This implies that the efficacy of awareness programs is the key to the success of Malaria eradication.

Download Full-text

Transmission Dynamics of Lymphatic Filariasis: A Mathematical Approach

ISRN Biomathematics ◽

10.5402/2012/930130 ◽

2012 ◽

Vol 2012 ◽

pp. 1-9 ◽

Cited By ~ 4

Author(s):

C. P. Bhunu ◽

S. Mushayabasa

Keyword(s):

Numerical Simulations ◽

Lymphatic Filariasis ◽

Reproduction Number ◽

Epidemiological Model ◽

Transmission Dynamics ◽

Effective Control ◽

Mathematical Approach

An epidemiological model for the spread of lymphatic filariasis, a mosquito-borne infection, is developed and analysed. The epidemic thresholds known as the reproduction number and equilibria for the model are determined and stabilities analysed. Results from the analysis of the reproduction number suggest that treatment will somehow contribute to a reduction in lymphatic filariasis cases, but what it does not show is the magnitude of the reduction, a part answered by the numerical simulations. Numerical simulations show that even when all lymphatic filariasis cases displaying elephantiasis symptoms are put on treatment it will not be able to eradicate the disease. This result suggests that effective control of lymphatic filariasis may lie in treatment for those displaying symptoms as well as chemoprophylaxis for the exposed.

Download Full-text

Revealing the Transmission Dynamics of COVID-19: A Bayesian Framework for Rt Estimation

10.21203/rs.3.rs-137557/v1 ◽

2021 ◽

Author(s):

Xian Yang ◽

Shuo Wang ◽

Yuting Xing ◽

Ling Li ◽

Richard Yi Da Xu ◽

...

Keyword(s):

Bayesian Inference ◽

Model Selection ◽

Reproduction Number ◽

State Of The Art ◽

Transmission Dynamics ◽

Selection Mechanism ◽

Problem Of Time ◽

Joint Inference ◽

Abrupt Changes ◽

Bayesian Smoothing

Abstract In epidemiological modelling, the instantaneous reproduction number, Rt, is important to understand the transmission dynamics of infectious diseases. Current Rt estimates often suffer from problems such as lagging, averaging and uncertainties demoting the usefulness of Rt. To address these problems, we propose a new method in the framework of sequential Bayesian inference where a Data Assimilation approach is taken for Rt estimation, resulting in the state-of-the-art ‘DARt’ system for Rt estimation. With DARt, the problem of time misalignment caused by lagging observations is tackled by incorporating observation delays into the joint inference of infections and Rt; the drawback of averaging is improved by instantaneous updating upon new observations and a model selection mechanism capturing abrupt changes caused by interventions; the uncertainty is quantified and reduced by employing Bayesian smoothing. We validate the performance of DARt through simulations and demonstrate its power in revealing the transmission dynamics of COVID-19.

Download Full-text

A Mathematical Model of a Tuberculosis Transmission Dynamics Incorporating First and Second Line Treatment

Journal of Applied Sciences and Environmental Management ◽

10.4314/jasem.v24i5.29 ◽

2020 ◽

Vol 24 (5) ◽

pp. 917-922

Author(s):

J. Andrawus ◽

F.Y. Eguda ◽

I.G. Usman ◽

S.I. Maiwa ◽

I.M. Dibal ◽

...

Keyword(s):

Mathematical Model ◽

Equilibrium Point ◽

Reproduction Number ◽

Endemic Equilibrium ◽

Transmission Dynamics ◽

Asymptotically Stable ◽

Line Treatment ◽

Second Line ◽

Tuberculosis Transmission ◽

Disease Free

This paper presents a new mathematical model of a tuberculosis transmission dynamics incorporating first and second line treatment. We calculated a control reproduction number which plays a vital role in biomathematics. The model consists of two equilibrium points namely disease free equilibrium and endemic equilibrium point, it has been shown that the disease free equilibrium point was locally asymptotically stable if thecontrol reproduction number is less than one and also the endemic equilibrium point was locally asymptotically stable if the control reproduction number is greater than one. Numerical simulation was carried out which supported the analytical results. Keywords: Mathematical Model, Biomathematics, Reproduction Number, Disease Free Equilibrium, Endemic Equilibrium Point

Download Full-text

Mathematical Study of a Class of Epidemiological Models with Multiple Infectious Stages

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2017-0244 ◽

2020 ◽

Vol 21 (3-4) ◽

pp. 259-274

Author(s):

S. Bowong ◽

A. Temgoua ◽

Y. Malong ◽

J. Mbang

Keyword(s):

Theoretical Study ◽

Reproduction Number ◽

Communicable Disease ◽

Transmission Dynamics ◽

Asymptotically Stable ◽

Epidemiological Models ◽

Mathematical Study ◽

Globally Asymptotically Stable ◽

Disease Free

AbstractThis paper deals with the mathematical analysis of a general class of epidemiological models with multiple infectious stages for the transmission dynamics of a communicable disease. We provide a theoretical study of the model. We derive the basic reproduction number $\mathcal R_0$ that determines the extinction and the persistence of the infection. We show that the disease-free equilibrium is globally asymptotically stable whenever $\mathcal R_0 \leq 1$, while when $\mathcal R_0 \gt 1$, the disease-free equilibrium is unstable and there exists a unique endemic equilibrium point which is globally asymptotically stable. A case study for tuberculosis (TB) is considered to numerically support the analytical results.

Download Full-text