Tracing DAY-ZERO and Forecasting the Fade out of the COVID-19 Outbreak in Lombardy, Italy: A Compartmental Modelling and Numerical Optimization Approach

AbstractItaly currently constitutes the epicenter of the novel coronavirus disease (COVID-19) pandemic, having surpassed China’s death toll. The disease is sweeping through Lombardy, which remains in lockdown since the 8th of March. As of the same day, the isolation measures taken in Lombardy have been extended to the entire country. Here, we provide estimates for: (a) the DAY-ZERO of the outbreak in Lombardy, Italy; (b) the actual number of exposed/infected cases in the total population; (c) the basic reproduction number (R0); (d) the “effective” per-day disease transmission; and, importantly, (e) a forecast for the fade out of the outbreak, on the basis of the COVID-19 Community Mobility Reports released by Google on March 29.MethodsTo deal with the uncertainty in the number of actual exposed/ infected cases in the total population, we address a compartmental Susceptible/ Exposed/ Infectious/ Recovered/ Dead (SEIRD) model with two compartments of infectious persons: one modelling the total cases in the population and another modelling the confirmed cases. The parameters of the model corresponding to the recovery period, the time from the onset of symptoms to death, the case fatality ratio, and the time from exposure to the time that an individual starts to be infectious, have been set as reported from clinical studies on COVID-For the estimation of the DAY-ZERO of the outbreak in Lombardy, as well as of the “effective” per-day transmission rate for which no clinical data are available, we have used the SEIRD simulator to fit the numbers of new daily cases from February 21 to the 8th of March, the lockdown day of Lombardy and of all Italy. This was accomplished by solving a mixed-integer optimization problem with the aid of genetic algorithms. Based on the computed values, we also provide an estimation of the basic reproduction number R0. Furthermore, based on an estimation for the reduction in the “effective” transmission rate of the disease as of March 8 that reflects the suspension of almost all activities in Italy, we ran the simulator to forecast the fade out of the epidemic. For this purpose, we considered the reduction in mobility in Lombardy as released on March 29 by Google COVID-19 Community Mobility Reports, the effect of social distancing, and the draconian measures taken by the government on March 20 and March 21, 2020.ResultsBased on the proposed methodological procedure, we estimated that the DAY-ZERO was most likely between January 5 and January 23 with the most probable date the 15th of January 2020. The actual cumulative number of exposed cases in the total population in Lombardy on March 8 was of the order of 15 times the confirmed cumulative number of infected cases. The “effective” per-day disease transmission rate for the period until March 8 was found to be 0.686 (95% CI:0.660, 0.713), while the basic reproduction number R0 was found to be 4.51 (95% CI: 4.14, 4.90).Importantly, simulations show that the COVID-19 pandemic in Lombardy is expected to fade out by the end of May -early June, 2020, if the draconian, as of March 20 and March 21, measures are maintained.

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The Effect of Seasonal Weather Variation on the Dynamics of the Plague Disease

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2017/5058085 ◽

2017 ◽

Vol 2017 ◽

pp. 1-25 ◽

Cited By ~ 3

Author(s):

Rigobert C. Ngeleja ◽

Livingstone S. Luboobi ◽

Yaw Nkansah-Gyekye

Keyword(s):

Basic Reproduction Number ◽

Disease Transmission ◽

Reproduction Number ◽

Transmission Rate ◽

Weather Conditions ◽

Effective Control ◽

Human Beings ◽

Weather Variation ◽

The Basic Reproduction Number ◽

Seasonal Weather

Plague is a historic disease which is also known to be the most devastating disease that ever occurred in human history, caused by gram-negative bacteria known as Yersinia pestis. The disease is mostly affected by variations of weather conditions as it disturbs the normal behavior of main plague disease transmission agents, namely, human beings, rodents, fleas, and pathogens, in the environment. This in turn changes the way they interact with each other and ultimately leads to a periodic transmission of plague disease. In this paper, we formulate a periodic epidemic model system by incorporating seasonal transmission rate in order to study the effect of seasonal weather variation on the dynamics of plague disease. We compute the basic reproduction number of a proposed model. We then use numerical simulation to illustrate the effect of different weather dependent parameters on the basic reproduction number. We are able to deduce that infection rate, progression rates from primary forms of plague disease to more severe forms of plague disease, and the infectious flea abundance affect, to a large extent, the number of bubonic, septicemic, and pneumonic plague infective agents. We recommend that it is more reasonable to consider these factors that have been shown to have a significant effect on RT for effective control strategies.

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Fractional SIR-Model for Estimating Transmission Dynamics of COVID-19 in India

J ◽

10.3390/j4020008 ◽

2021 ◽

Vol 4 (2) ◽

pp. 86-100

Author(s):

Nita H. Shah ◽

Ankush H. Suthar ◽

Ekta N. Jayswal ◽

Ankit Sikarwar

Keyword(s):

Basic Reproduction Number ◽

Reproduction Number ◽

Transmission Rate ◽

Sir Model ◽

Time Dependent ◽

Contact Rate ◽

Distribution Maps ◽

Fundamental Parameters ◽

Different Order ◽

The Basic Reproduction Number

In this article, a time-dependent susceptible-infected-recovered (SIR) model is constructed to investigate the transmission rate of COVID-19 in various regions of India. The model included the fundamental parameters on which the transmission rate of the infection is dependent, like the population density, contact rate, recovery rate, and intensity of the infection in the respective region. Looking at the great diversity in different geographic locations in India, we determined to calculate the basic reproduction number for all Indian districts based on the COVID-19 data till 7 July 2020. By preparing district-wise spatial distribution maps with the help of ArcGIS 10.2, the model was employed to show the effect of complete lockdown on the transmission rate of the COVID-19 infection in Indian districts. Moreover, with the model's transformation to the fractional ordered dynamical system, we found that the nature of the proposed SIR model is different for the different order of the systems. The sensitivity analysis of the basic reproduction number is done graphically which forecasts the change in the transmission rate of COVID-19 infection with change in different parameters. In the numerical simulation section, oscillations and variations in the model compartments are shown for two different situations, with and without lockdown.

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Mathematical analysis of a measles transmission dynamics model in Bangladesh with double dose vaccination

Scientific Reports ◽

10.1038/s41598-021-95913-8 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Md Abdul Kuddus ◽

M. Mohiuddin ◽

Azizur Rahman

Keyword(s):

Basic Reproduction Number ◽

Reproduction Number ◽

Transmission Rate ◽

Equilibrium Points ◽

Rank Correlation ◽

Dynamical Analysis ◽

Model Parameters ◽

Progression Rate ◽

Double Dose ◽

The Basic Reproduction Number

AbstractAlthough the availability of the measles vaccine, it is still epidemic in many countries globally, including Bangladesh. Eradication of measles needs to keep the basic reproduction number less than one $$(\mathrm{i}.\mathrm{e}. \, \, {\mathrm{R}}_{0}<1)$$ ( i . e . R 0 < 1 ) . This paper investigates a modified (SVEIR) measles compartmental model with double dose vaccination in Bangladesh to simulate the measles prevalence. We perform a dynamical analysis of the resulting system and find that the model contains two equilibrium points: a disease-free equilibrium and an endemic equilibrium. The disease will be died out if the basic reproduction number is less than one $$(\mathrm{i}.\mathrm{e}. \, \, {\mathrm{ R}}_{0}<1)$$ ( i . e . R 0 < 1 ) , and if greater than one $$(\mathrm{i}.\mathrm{e}. \, \, {\mathrm{R}}_{0}>1)$$ ( i . e . R 0 > 1 ) epidemic occurs. While using the Routh-Hurwitz criteria, the equilibria are found to be locally asymptotically stable under the former condition on $${\mathrm{R}}_{0}$$ R 0 . The partial rank correlation coefficients (PRCCs), a global sensitivity analysis method is used to compute $${\mathrm{R}}_{0}$$ R 0 and measles prevalence $$\left({\mathrm{I}}^{*}\right)$$ I ∗ with respect to the estimated and fitted model parameters. We found that the transmission rate $$(\upbeta )$$ ( β ) had the most significant influence on measles prevalence. Numerical simulations were carried out to commissions our analytical outcomes. These findings show that how progression rate, transmission rate and double dose vaccination rate affect the dynamics of measles prevalence. The information that we generate from this study may help government and public health professionals in making strategies to deal with the omissions of a measles outbreak and thus control and prevent an epidemic in Bangladesh.

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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 THEORETICAL ASSESSMENT OF THE EFFECTS OF DISTRIBUTED DELAY ON THE TRANSMISSION DYNAMICS OF HEPATITIS B

Journal of Biological System ◽

10.1142/s0218339015500229 ◽

2015 ◽

Vol 23 (03) ◽

pp. 423-455

Author(s):

P. MOUOFO TCHINDA ◽

JEAN JULES TEWA ◽

BOULECHARD MEWOLI ◽

SAMUEL BOWONG

Keyword(s):

Drug Therapy ◽

Hepatitis B ◽

Basic Reproduction Number ◽

Disease Transmission ◽

Time Delays ◽

Reproduction Number ◽

Distributed Delay ◽

Global Dynamics ◽

The Impact ◽

The Basic Reproduction Number

In this paper, we investigate the global dynamics of a system of delay differential equations which describes the interaction of hepatitis B virus (HBV) with both liver and blood cells. The model has two distributed time delays describing the time needed for infection of cell and virus replication. We also include the efficiency of drug therapy in inhibiting viral production and the efficiency of drug therapy in blocking new infection. We compute the basic reproduction number and find that increasing delays will decrease the value of the basic reproduction number. We study the sensitivity analysis on the key parameters that drive the disease dynamics in order to determine their relative importance to disease transmission and prevalence. Our analysis reveals that the model exhibits the phenomenon of backward bifurcation (where a stable disease-free equilibrium (DFE) co-exists with a stable endemic equilibrium when the basic reproduction number is less than unity). Numerical simulations are presented to evaluate the impact of time-delays on the prevalence of the disease.

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The impact of media coverage with a piecewise transmission rate in the spread of diseases

International Journal of Biomathematics ◽

10.1142/s1793524517500036 ◽

2016 ◽

Vol 10 (01) ◽

pp. 1750003

Author(s):

Maoxing Liu ◽

Lixia Zuo

Keyword(s):

Basic Reproduction Number ◽

Media Coverage ◽

Reproduction Number ◽

Transmission Rate ◽

Three Dimensional ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Contact Transmission ◽

The Impact ◽

The Basic Reproduction Number

A three-dimensional compartmental model with media coverage is proposed to describe the real characteristics of its impact in the spread of infectious diseases in a given region. A piecewise continuous transmission rate is introduced to describe that media coverage exhibits its effect only when the number of the infected exceeds a certain critical level. Further, it is assumed that the impact of media coverage on the contact transmission is described by an exponential decreasing factor. Stability analysis of the model shows that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than unity. On the other hand, when the basic reproduction number is greater than unity and media coverage impact is sufficiently small, a unique endemic equilibrium exists, which is globally asymptotically stable.

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COVID-19 disease transmission model considering direct and indirect transmission

E3S Web of Conferences ◽

10.1051/e3sconf/202020212008 ◽

2020 ◽

Vol 202 ◽

pp. 12008

Author(s):

Dipo Aldila

Keyword(s):

Mathematical Model ◽

Basic Reproduction Number ◽

Disease Transmission ◽

Reproduction Number ◽

Transmission Model ◽

Indirect Transmission ◽

Variation Of Parameters ◽

The Media ◽

Disease Transmission Model ◽

The Basic Reproduction Number

A mathematical model for understanding the COVID-19 transmission mechanism proposed in this article considering two important factors: the path of transmission (direct-indirect) and human awareness. Mathematical model constructed using a four-dimensional ordinary differential equation. We find that the Covid-19 free state is locally asymptotically stable if the basic reproduction number is less than one, and unstable otherwise. Unique endemic states occur when the basic reproduction number is larger than one. From sensitivity analysis on the basic reproduction number, we find that the media campaign succeeds in suppressing the endemicity of COVID-19. Some numerical experiments conducted to show the dynamic of our model respect to the variation of parameters value.

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A first study on the impact of containment measure on COVID-19 spread in Morocco

10.1101/2020.04.26.20080770 ◽

2020 ◽

Author(s):

Aayah Hammoumi ◽

Redouane Qesmi

Keyword(s):

Basic Reproduction Number ◽

High Speed ◽

Social Economic ◽

Reproduction Number ◽

Cumulative Number ◽

First Case ◽

Containment Measure ◽

The Impact ◽

The Basic Reproduction Number ◽

Economic Damages

AbstractBackgroundSince the appearance of the first case of COVID-19 in Morocco, the cumulative number of reported infectious cases continues to increase and, consequently, the government imposed the containment measure within the country. Our aim is to predict the impact of the compulsory containment on COVID-19 spread. Earlier knowledge of the epidemic characteristics of COVID-19 transmission related to Morocco will be of great interest to establish an optimal plan-of-action to control the epidemic.MethodUsing a Susceptible-Asymptomatic-Infectious model and the data of reported cumulative confirmed cases in Morocco from March 2nd to April 9, 2020, we determined the basic and control reproduction numbers and we estimated the model parameter values. Furthermore, simulations of different scenarios of containment are performed.ResultsEpidemic characteristics are predicted according to different rates of containment. The basic reproduction number is estimated to be 2.9949, with CI(2.6729–3.1485). Furthermore, a threshold value of containment rate, below which the epidemic duration is postponed, is determined.ConclusionOur findings show that the basic reproduction number reflects a high speed of spread of the epidemic. Furthermore, the compulsory containment can be efficient if more than 73% of population are confined. However, even with 90% of containment, the end-time is estimated to happen on July 4th which can be harmful and lead to consequent social-economic damages. Thus, containment need to be accompanied by other measures such as mass testing to reduce the size of asymptomatic population. Indeed, our sensitivity analysis investigation shows that the COVID-19 dynamics depends strongly on the asymptomatic duration as well as the contact and containment rates. Our results can help the Moroccan government to anticipate the spread of COVID-19 and avoid human loses and consequent social-economic damages as well.

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Basic reproduction number estimation and forecasting of COVID-19: A case study of India, Brazil and Peru

Communications on Pure and Applied Analysis ◽

10.3934/cpaa.2021170 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Nitu Kumari ◽

Sumit Kumar ◽

Sandeep Sharma ◽

Fateh Singh ◽

Rana Parshad

Keyword(s):

Basic Reproduction Number ◽

Reproduction Number ◽

Transmission Rate ◽

Equilibrium Points ◽

Real Data ◽

Time Frame ◽

Complete Analysis ◽

Data Set ◽

Definition Of ◽

The Basic Reproduction Number

<p style='text-indent:20px;'>Since the start of COVID-19 pandemic, the definition of normal life has changed drastically. The number of cases of this pandemic is rising everyday across the globe. In this study, we propose a compartmental model, which considers the isolation factor of Coronavirus infected individuals. The model consists of five compartments: susceptible (S), exposed (E), Infected (I), Isolated (L) and recovered (R). We have estimated the parameters of the model system and the expression of the basic reproduction number <inline-formula><tex-math id="M1">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula> using real data set. The exact value of the basic reproduction number is computed for India, Brazil and Peru. The local and global stability analysis of disease-free equilibrium and endemic equilibrium points is carried out. The forecasting of the pandemic is done using real data. It has been observed that to understand the pandemic the time frame has to be divided into small intervals as the parameters of the pandemic are changing with time. Within a time frame of approximately four months (i.e. from July to October 2020), the transmission rate of India has been reduced by approximately 84%. Whereas the transmission rate in Brazil and Peru has increased by 79% and 45% respectively. The sensitivity of various parameters involved in the model has been analyzed. We have presented a complete analysis to check the existence of backward bifurcation.</p>

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Mathematical modelling and optimal cost-effective control of COVID-19 transmission dynamics

10.21203/rs.3.rs-84331/v1 ◽

2020 ◽

Author(s):

S. Olaniyi ◽

O.S. Obabiyi ◽

K.O. Okosun ◽

A.T. Oladipo ◽

S.O. Adewale

Keyword(s):

Basic Reproduction Number ◽

Least Squares Method ◽

Reproduction Number ◽

Cost Effective ◽

Effective Control ◽

Cumulative Number ◽

The Novel ◽

Novel Coronavirus ◽

Disease Free ◽

The Basic Reproduction Number

Abstract The novel coronavirus disease (COVID-19) caused by a new strain of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) remains the current global health challenge. In this paper, an epidemic model based on system of ordinary differential equations is formulated by taking into account the transmission routes from symptomatic, asymptomatic and hospitalized individuals. The model is fitted to the corresponding cumulative number of hospitalized individuals (active cases) reported by the Nigeria Centre for Disease Control (NCDC), and parameterized using the least squares method. The basic reproduction number which measures the potential spread of COVID-19 in the population is computed using the next generation operator method. Further, Lyapunov function is constructed to investigate the stability of the model around a disease-free equilibrium point. It is shown that the model has a globally asymptotically stable disease-free equilibrium if the basic reproduction number of the novel coronavirus transmission is less than one. Sensitivities of the model to changes in parameters are explored. It is revealed further that the basic reproduction number can be brought to a value less than one in Nigeria, if the current effective transmission rate of the disease can be reduced by 50%. Otherwise, the number of active cases may get up to 2.5% of the total estimated population. In addition, two time-dependent control variables, namely preventive and management measures, are considered to mitigate the damaging effects of the disease using Pontryagin's maximum principle. The most cost-effective control measure is determined through cost-effectiveness analysis. Numerical simulations of the overall system are implemented in MatLab® for demonstration of the theoretical results.

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