Mathematical Biosciences and Engineering

In this paper, we estimate the reproductive number R0 of COVID-19 based on Wallinga and Lipsitch framework [11] and a novel statistical time delay dynamic system. We use the observed data reported in CCDCs paper to estimate distribution of the generation interval of the infection and apply the simulation results from the time delay dynamic system as well as released data from CCDC to fit the growth rate. The conclusion is: Based our Fudan-CCDC model, the growth rate r of COVID-19 is almost in [0.30, 0.32] which is larger than the growth rate 0.1 estimated by CCDC [9], and the reproductive number R0 of COVID-19 is estimated by 3.25 [≤] R0 [≤] 3.4 if we simply use R = 1 + r * Tc with Tc = 7.5, which is bigger than that of SARS. Some evolutions and predictions are listed.

1This note models the effect of the lockdown during the first wave of COVID-19. We use SEIR type of model with a certain time lag between infection and becoming infectious. Firstly we compare the timing of the change of the coefficient of infection, growth rate of confirmed cases corresponds to the change of mobility index, and secondly we assume the change of the coefficient of infection, activity index {beta} (analogous to R0) and fit the parameter to reproduce the actual number of confirmed cases. Finally, we assume that the activity index {beta} is proportional to the square of the mobility and fit the parameters. The curves in various cuontries fits reasonably well in any cases, but estimating {beta} from various parameters (including temperature) remains as an important task.

COVID-19 is a respiratory tract infection that can range from being mild to fatal. In India, the countrywide lockdown has been imposed since 24th march, 2020, and has got multiple extensions with different guidelines for each phase. Among various models of epidemiology, we use the SIR(D) model to analyze the extent to which this multi-phased lockdown has been active in flattening the curve and lower the threat. Analyzing the effect of lockdown on the infection may give us a better insight into the evolution of epidemic while implementing the quarantine procedures as well as improving the healthcare facilities. For accurate modelling, incorporating various parameters along with sophisticated computational facilities, are required. Parallel to SIRD modelling, we tend to compare it with the Ising model and derive a quantum circuit that incorporates the rate of infection and rate of recovery, etc as its parameters. The probabilistic plots obtained from the circuit qualitatively resemble the shape of the curve for the spread of Coronavirus. We also demonstrate how the curve flattens when the lockdown is imposed. This kind of quantum computational approach can be useful in reducing space and time complexities of a huge amount of information related to the epidemic.

In this project, we study a class of fractional order generalized SEIR epidemic models. Based on the public data from Jan. 22th to May 15th, 2020, we reliably estimate key epidemic parameters and make predictions on the peak point and possible ending time for the target region. We analyze the current management strategy and predict the future implementation of different management strategies. Numerical simulations which support our analysis are also given.

We study a novel multi-strain SIR epidemic model with selective immunity by vaccination. A newer strain is made to emerge in the population when a preexisting strain has reached equilbrium. We assume that this newer strain does not exhibit cross-immunity with the original strain, hence those who are vaccinated and recovered from the original strain become susceptible to the newer strain. Recent events involving the COVID-19 virus demonstrates that it is possible for a viral strain to emerge from a population at a time when the influenza virus, a well-known virus with a vaccine readily available for some of its strains, is active in a population. We solved for four different equilibrium points and investigated the conditions for existence and local stability. The reproduction number was also determined for the epidemiological model and found to be consistent with the local stability condition for the disease-free equilibrium.

In this article, we formulate and analyse a mathematical model for the co-infection of Hepatitis B virus and COVID-19. We incorporate into our framework Hepatitis B virus prevention, COVID-19 prevention, COVID-19 vaccination, and environmental factors so as to investigate their effect on transmission dynamics. First, we derive the basic reproduction number for HBV only, COVID-19 only, and co-infection stochastic models using the next generation matrix method. Next, we establish the conditions for stability in the stochastic sense for HBV only, COVID-19 only sub-models, and the co-infection model. Furthermore, we devote our attention to finding sufficient conditions for extinction and persistence. Finally, by using the Euler-Murayama scheme, we illustrate the dynamics of the co-infection, COVID-19, HBV and the effect of some parameters on disease transmission dynamics by means of numerical simulations.

At the time of the worldwide COVID-19 disaster, the author learned about the pooled (RT-) PCR test from the news. From the common sense of individual tests, the idea of mixing multiple samples seems taboo, however in fact many samples can be tested with a smaller number of tests by the method. As a retired researcher of mathematical engineering, the author was deeply interested in the idea and absorbed in the mathematical formulation and intensive analysis of the method. Later, he found that the original basic equation was already proposed in the old (1943) treatise [1] and so many related research works have been done and available as materials on the web [2], although many of those seem to be based on qualitative or intuitive analysis. In that sense, some of the analysis here seems to be already known in the field, but some results might be novel, such as boundary conditions, derivation of limit values, estimation of infection rate and adaptive optimization scheme of pool test, strict extension to multi-stage pool test, and explicit derivation of asymptotic approximate solutions of optimal pooling number and achieved efficiency measure, etc. In any case, he decided to put it together here as a material rather than a formal treatise, hoping that the results here would be useful for deeper mathematical insights into and better understanding of the pool inspection, and also in its actual practice.

An explicit solution of an initial value problem for the Susceptible-Vaccinated-Infectious-Recovered (SVIR) epidemic model is obtained, and various properties of the explicit solution are investigated. It is shown that the parametric form of the explicit solution satisfies some linear differential system including a positive solution of an integral equation. In this paper integral equations play an important role in establishing the explicit solution of the SVIR epidemic model, in particular, the number of infected individuals can be represented in a simple form by using a positive solution of an integral equation. Uniqueness of positive solutions of the SVIR epidemic model is also investigated, and it is shown that the explicit solution is a unique solution in the class of positive solutions.

In this paper, we want to simulate the COVID-19 epidemic according to the Retarded SEIRS model. One of the main questions in the human mind is whether the COVID-19 epidemic will happen again. Therefore, a criterion must be set for the occurrence or non-occurrence of the disease. With the Retarded SEIRS model and this criterion, we can predict whether the Covid-19 will re-emerge. So far, a large number of researches that have been presented in scientific groups or communities have been based on the SIR or SEIR model. But we assume that each recovered individual is immune to the disease for a limited time, and then will be susceptible again. As we know, this assumption was also true for SARS.

In December 2019, a new virus belonging to the coronavirus strain has been discovered in Wuhan, China, this virus has attracted world-wide attention and it spread rapidly in the world, reaching nearly 216 countries in the world in November 2020. In this chapter, we study the fractional incommensurate SIQR (susceptible, infections,quarantined and removed) COVID-19 model with nonlinear saturated incidence rate using Atangana-Baleanu fractional derivatives. The existence and uniqueness of the solutions for the fractional model is proved using fixed point theorem, the model are shown to have two equilibrium point (disease-free and an endemic equilibrium). Some numerical simulations using Euler method are also carried out to support our theoretical results. We estimated the value of the fractional orders and the parameters of the proposed model using the least squares method.. Further, the sensitivity analysis of the parameter is performed as a result, our incommensurate model gives a good approximation to real data of COVID-19.

In order to quantitatively characterize the epidemic of COVID-19, useful relations among parameters describing an epidemic in general are derived based on the Kermack-McKendrick model. The first relation is 1/{tau}grow =1/{tau}trans-1/{tau}inf, where{tau} grow is the time constant of the exponential growth of an epidemic,{tau} trans is the time for a pathogen to be transmitted from one patient to uninfected person, and the infectious time{tau} inf is the time during which the pathogen keeps its power of transmission. The second relation p({infty}) {approx} 1-exp(-(R0-1)/0.60) is the relation between p({infty}), the final size of the disaster defined by the ratio of the total infected people to the population of the society, and the basic reproduction number, R0, which is the number of persons infected by the transmission of the pathogen from one infected person during the infectious time. The third relation 1/{tau}end = 1/{tau}inf-(1-p({infty}))/{tau}trans gives the decay time constant{tau} end at the ending stage of the epidemic. Derived relations are applied to influenza in Japan in 2019 for characterizing the epidemic.

In this study, we investigate the stochastic dynamics of an extended SIS epidemic model in densely populated environments within a Markov jump process framework. We solve the master equation in closed form and obtain exact solutions of the time-dependent distribution of the number of infected individuals, the quasi-stationary distribution, the extinction time distribution of the epidemic, and the distribution of the first-passage time at which the number of infections reaches a certain threshold. The approximated quasi-stationary distribution and mean extinction time are also derived using the large deviation theory. Interestingly, we find that the first nonzero eigenvalue of the generator matrix of the Markovian model characterizes the extinction rate of the epidemic, while the second nonzero eigenvalue characterizes its outbreak rate. We also examine the stochastic bifurcation for our model based on the time evolution of the probability distribution and the bifurcation threshold of the basic reproduction number for the stochastic SIS model is shown to be large than that for its deterministic counterpart. Finally, we demonstrate that analyzing the first-passage time distribution can offer early warning for interventions and optimize the allocation of emergency beds.

Sequel to [10], who studied the dynamics of COVID-19 using an SEIRUS model. We consider an SEIRS model capturing saturated incidence with treatment response. In this theoretical model, we assumed that the treatment response is proportional to the number of infected as long as the incidence cases are within the capacity of the healthcare system, after which the value becomes constant, when the number of confirmed cases exceed the carrying capacity of the available medical facilities. Thus, we obtain the reproduction number stating that when R0 < 1, the disease free equilibrium is globally asymptotically stable. Also, we studied the existence of the local and global stability of the disease free and endemic equilibria and found that the kind of treatment response and inhibitory measures deployed in tackling the COVID-19 pandemic determines whether the disease will die out or become endemic.

Rate parameters are critical in estimating the covid burden using mathematical models. In the Covid-19 mathematical models, these parameters are assumed to be constant. However, uncertainties in these rate parameters are almost inevitable. In this paper, we study a stochastic epidemic model of the SARS-CoV-2 virus infection in the presence of vaccination in which some parameters fluctuate around its average value. Our analysis shows that if the stochastic basic reproduction number (SBRN) of the system is greater than unity, then there is a stationary distribution, implying the long-time disease persistence. A sufficient condition for disease eradication is also prescribed for which the exposed class goes extinct, followed by the infected class. The disease eradication criterion may not hold if the rate of vaccine-induced immunity loss increases or/and the force of infection increases. Using the Indian Covid-19 data, we estimated the model parameters and showed the future disease progression in the presence of vaccination. The disease extinction time is estimated under various conditions. It is revealed that the mean extinction time is an increasing function of both the force of infection and immunity loss rate and shows the lognormal distribution. We point out that disease eradication might not be possible even at a higher vaccination rate if the vaccine-induced immunity loss rate is high. Our observation thus indicates the endemicity of the disease for the existing vaccine efficacy. The disease eradication is possible only with a higher vaccine efficacy or a reduced infection rate.

We investigate time scale separation in the vector borne disease model SIRUV, as previously described in the literature [1], and recently reanalyzed with the singular perturbation technique [2]. We focus on the analysis with a single small parameter, the birth and death rate {micro}, whereas all other model parameters are much larger and describe fast transitions. The scaling of the endemic stationary state, the Jacobian matrix around it and its eigenvalues with this small parameter {micro} is calculated and the center manifold analysis performed with the method described in [3] which goes back to earlier work [4, 5], namely a transformation of the Jacobian matrix to block structure in zeroth order in the parameter {micro} is used and then a family of center manifolds with {micro} larger than zero is obtained.

The goal of this study is to establish a general framework for predicting the so-called critical "Turning Period" in an infectious disease epidemic such as the COVID-19 outbreak in China early this year. This framework enabled a timely prediction of the turning period when applied to Wuhan COVID-19 epidemic and informed the relevant authority for taking appropriate and timely actions to control the epidemic. It is expected to provide insightful information on turning period for the worlds current battle against the COVID-19 pandemic. The underlying mathematical model in our framework is the individual Susceptible-Exposed-Infective-Removed (iSEIR) model, which is a set of differential equations extending the classic SEIR model. We used the observed daily cases of COVID-19 in Wuhan from February 6 to 10, 2020 as the input to the iSEIR model and were able to generate the trajectory of COVID-19 cases dynamics for the following days at midnight of February 10 based on the updated model, from which we predicted that the turning period of CIVID-19 outbreak in Wuhan would arrive within one week after February 14. This prediction turned to be timely and accurate, providing adequate time for the government, hospitals, essential industry sectors and services to meet peak demands and to prepare aftermath planning. Our study also supports the observed effectiveness on flatting the epidemic curve by decisively imposing the "Lockdown and Isolation Control Program" in Wuhan since January 23, 2020. The Wuhan experience provides an exemplary lesson for the whole world to learn in combating COVID-19.

When we consider a probability distribution about how many COVID-19 infected people will transmit the disease, two points become important. First, there should be super-spreaders in these distributions/networks and secondly, the Pareto principle should be valid in these distributions/networks. When we accept that these two points are valid, the distribution of transmission becomes a discrete Pareto distribution, which is a kind of power law. Having such a transmission distribution, then we can simulate COVID-19 networks and find super-spreaders using the centricity measurements in these networks. In this research, in the first we transformed a transmission distribution of statistics and epidemiology into a transmission network of network science and secondly we try to determine who the super-spreaders are by using this network and eigenvalue centrality measure. We underline that determination of transmission probability distribution is a very important point in the analysis of the epidemic and determining the precautions to be taken.

A logistic formula in biology is applied to analyze deaths by COVID-19 for both of the first and the second waves in Japan. We then discuss ends of both waves and their mortality ratios. The meaning of population N in an epidemic is discussed.

The COVID-19 pandemic has already consumed few months of indolence all over the world. Almost every part of the world from which the victim of COVID 19 are, have not yet been able to find out a strong way to combat corona virus. Therefore, the main aim is to minimize the spreading of the COVID-19 by detecting most of the affected people during lockdown. Hence, it is necessary to understand what the nature of growth is of spreading of this corona virus with time after almost one month (30 days) of lockdown. In this paper we have developed a very simple mathematical model to describe the growth of spreading of corona virus in human being. This model is based on realistic fact and the statistics we have so far. For controlling the spread of the COVID-19, minimization of the growth with minimum number of days of lockdown is necessary. We have established a relation between the long-term recovery coefficient and the long-term infected coefficient. The growth can be minimized if such condition satisfies. We have also discussed how the different age of the people can be cured by applying different types of medicine. We have presented the data of new cases, recovery and deaths per day to visualize the different coefficient for India and establish our theory. We have also explained how the medicine could be effective to sustain and improve such condition for country having large population like India.

A kinetic approach is developed, in a "tutorial style" to describe the evolution of an epidemic with spread taking place through contact. The "infection - rate" is calculated from the rate at which an infected person approaches an uninfected susceptible individual, i.e. a potential recipient of the disease, up to a distance p, where the value of p may lie between pmin[&le;] p [&le;] pmax. We consider a situation with a total population of N individuals, living in an area A, x(t) amongst them being infected while xd(t) = {beta}'x(t) is the number that have died in the course of transmission and evolution of the epidemic. The evolution is developed under the conditions (1) a faction (t) of the [N-x(t) - xd(t)] uninfected individuals and (2) a {beta}(t) fraction of the x(t) infected population are quarantined, while the "source events" that spread the infection are considered to occur with frequency{upsilon} 0. The processes of contact and transmission are considered to be Markovian. Transmission is assumed to be inhibited by several processes like the use of "masks", "hand washing or use of sanitizers" while "physical distancing" is described by p. The evolution equation for x(t) is a Riccati - type differential equation whose coefficients are time-dependent quantities, being determined by an interplay between the above parameters. A formal solution for x(t) is presented, for a "graded lockdown" with the parameters, 0[&le;] (t), {beta}(t)[&le;]1 reaching their respective saturation values in time scales,{tau} 1,{tau} 2 respectively, from their initial values (0)={beta}(0)=0. The growth is predicted for several BBMP wards in Bengaluru and in urban centers in Chikkaballapur district, as an illustrative case. Above selections serve as model cases for high, moderate and thin population densities. It is seen that the evolution of [x(t)/N] with time depends upon (a) the initial time scale of evolution, (b) the time scale of cure and (c) on the time dependence of the Lockdown function Q(t) = {[1-(t)]{middle dot}[1-{beta}(t)]}. The formulae are amenable to simple computations and show that in order to curb the spread one must ensure that Q({infty}) must be below a critical value and the vigilance has to be continued for a long time (at least 100 to 150 days) after the decay starts, to avoid all chances of the infection reappearing.