key: cord-0481763-y2punilz
authors: Tamilalagan, P.; Krithika, B.; Manivannan, P.
title: A SEIRUC mathematical model for transmission dynamics of COVID-19
date: 2021-06-15
journal: nan
DOI: nan
sha: fb99727029e6e6a0c94a832a80ee898a4296de67
doc_id: 481763
cord_uid: y2punilz

The world is still fighting against COVID-19, which has been lasting for more than a year. Till date, it has been a greatest challenge to human beings in fighting against COVID-19 since, the pathogen SARS-COV-2 that causes COVID-19 has significant biological and transmission characteristics when compared to SARS-COV and MERS-COV pathogens. In spite of many control strategies that are implemented to reduce the disease spread, there is a rise in the number of infected cases around the world. Hence, a mathematical model which can describe the real nature and impact of COVID-19 is necessary for the better understanding of disease transmission dynamics of COVID-19. This article proposes a new compartmental SEIRUC mathematical model, which includes the new state called convalesce (C). The basic reproduction number $mathcal{R}_0$ is identified for the proposed model. The stability analysis are performed for the disease free equilibrium ($mathcal{E}_0$) as well for the endemic equilibrium ($mathcal{E}_*$) by using the Routh-Hurwitz criterion. The graphical illustrations of the proposed mathematical results are provided to validate the theoretical results.

Mathematical modelling of infectious disease using nonlinear dynamical systems can give significant insight into the transmission dynamics or dynamical behaviour of disease spread. Epidemiological modelling of diseases has improved drastically over the past decades and continues to rise up in several fields [1] - [3] . In particular, the differential equation models have been utilized to develop biological and physical problems [4] - [9] . COVID-19 is a disease caused by a new virus, which is generating a pandemic worldwide and needs a model taking into account its known specification characteristics.

Due to the significance and powerful nature of differential equation models in epidemiology, some recent studies in literature have considered the mathematical modelling of the COVID-19 pandemic using nonlinear differential equations [10] - [15] . Since pandemics are large-scale outbreaks of infectious disease, it can produce an important risk to human life over a wide geographic area and can cause economic and social disruption. Mathematical models comprising derivatives aid in estimating the effect of precautionary measures adopted against novel coronavirus. Recently, few mathematical models have been investigated by many researchers to understand the transmission dynamics of COVID-19 pandemic and some of these are listed in our references. There are some mathematical models in the literature that try to describe the dynamics of the evolution of COVID-19 [10] - [15] . Other works [11] , [15] propose SIR and SEIR type models with little variations. Piu Samui et al. [10] proposed a SAIU model for the spread of COVID-19 using data from case study of India, taking into account the asymptomatic, reported symptomatic infectious and unreported symptomatic infectious class. Abdullah et al. [12] introduced a mathematical model by including resistive class together with quarantine class and use it to investigate the transmission dynamics of novel corona virus disease. Liu et al. [14] introduced a COVID-19 epidemic model taking into account the latency period. As identified by the World Health Organisation (WHO), the mathematical models, mainly those formulated on a timely basis plays a vital role in allowing public health decision and decision makers with evidence-based statistics [16] - [18] .

According to Worldometer data, 159,068,471 total confirmed cases, 3,308,750 confirmed deaths and 136,633,409 recovered cases has been recorded throughout the world as of May 10, 2021 [19] . Also, as per the report of World Health Organization (WHO) as of 23rd December, 2020, the individuals infected to SARS-Cov-2 virus, which causes COVID-19, develop antibodies after infection [16] . It has been reported that, the infected individuals who have even severe and mild disease also develop these antibodies. Hence, Serological studies and research are underway to recognize the stability of this immune response and also to investigate, how far these antibodies last. [18] This present article try to incorporate the antibody response of immune system to the COVID-19 disease as a separate compartment namely, Convalesce. Specifically, the immune system of the individuals belonging to the Convalesce class is strong enough to be safe from pandemic, so that they are not infected again. It also accounts the population recovered by taking the home herbal medicine or region-specific traditional medicines in various forms in different countries. Traditional Chinese medicine substances used in clinical trials includes Polygonum cuspidatum(also known as Asian knotweed), Honey suckle, Ligustrum lucidum(an evergreen tree) etc, [20] . The World Health Organisation has also recommended inclusion of traditional medicine in its COVID-19 strategic preparedness and response [21] . Indian traditional system, Ayurveda has a clear concept of the cause and treatment of pandemics, [22] provides information on the potential antiviral traditional medicines along with their immunomodulatory pathways and also described seven most important Indian traditional plants with antiviral properties. Recently, an antiviral drug, Clevira has been approved by Government of India, as a supporting measure for mild to moderate condition of Covid-19. The trial outcomes revealed that Clevira has shown 86 per cent recovery rate on fifty days of treatment in mild to moderate Covid-19 cases [23] . Based on these discussions, the individuals belonging to the Convalesce class are assumed to be immune to reinfection.

The coming sections of this article is sorted out as follows: We propose a dynamical model and the model description in section 2. The qualitative properties of the proposed SEIRUC model have been discussed in section 3. In the same section, the basic reproduction number for the SEIRUC model have been computed and the stability analysis of both disease free equilibrium and endemic equilibrium points have been performed. In section 4, we conduct sensitivity analysis for the basic reproduction number. In section 5, we present numerical simulation to verify our analytical findings and a discussion in section 6 concludes our manuscript.

The SEIRUC mathematical model for transmissions dynamics of COVID-19, which is under consideration, is given below, In the above model (1), Λ denotes net inflow of susceptible or uninfected population, the uninfected population S(t) becomes infected at a rate β by exposed with infected infectious individuals namely I, R, and U , the natural decay rate of S(t) is δ. In the population E(t), the parameter a denotes the rate at which the infected noninfectious population E(t) becomes infectious I(t). The asymptomatic infectious population I(t) is asymptomatic for the period 1 γ . The populations E(t) and I(t) are recovered respectively at a rate c 1 and c 2 due to protective immune response to the infection by human immune system and as the infection is not acute at this stage, it also depends on the age of the individual. The natural decay rates of the populations E(t) and I(t) are respectively δ 1 and δ 2 , this accounts a fraction of death's caused by COVID-19 without any symptoms. The asymptomatic infectious population I(t) becomes reported symptomatic infectious R(t) at a rate γq, where q is the fraction of asymptomatic infectious that become reported symptomatic infectious, correspondingly (1 − q)γ is the rate of asymptomatic infectious population I(t) becomes unreported symptomatic population U (t). η is the rate at which the unreported symptomatic infectious population, U (t) becomes reported symptomatic infectious population, R(t). The recovery rate of the population R(t)

is c 3 , it accounts, the recovered population due to severe treatment by hospitals. The natural decay rates of the populations R(t) and U (t) are respectively δ 3 and δ 4 . The population U (t) recovered at a rate c 4 . The mortality rate of the population C(t) is δ 5 , it accounts the individuals recovered from COVID-19 but died because of other reasons/diseases.

For the SEIRUC model to be epidemiologically realistic, it is necessary to prove that the model solutions are positive under non-negative initial conditions.

where Υ 1 (t) = β N (I(t) + U (t) + R(t)) + δ. Thereafter, we obtain the following expression.

This implies, S(t) is non-negative for all t.

Similarly, it can also be shown that E(t), I(t), R(t), U (t), C(t) > 0 for all t > 0., which implies that the disease is uniformly persistent for every positive solution.

) be the solution of the system with initial condi-

Solving the above equation, we obtain 0

Thus, Φ is positively invariant and attractive set. Therefore, all the feasible soluions in the model converge in Φ.

The proposed model has two exclusively different steady states namely endemic free or disease free equilibrium E 0 = (S 0 , 0, 0, 0, 0, 0) and the endemic equilibrium E * = (S * , E * , I * , R * , U * , C * ), where

and S * is given by the quadratic equation

The roots of (2) are S * = δ A and S 0 = Λ δ . Further, the second root S 0 = Λ δ entails the endemic free steady state E 0 .

Theorem 3.3. The dynamical system (1) is locally asymptotically stable at the endemic-free equilib- The basic reproduction number R 0 corresponding to the proposed SEIRUC model (1) is obtained by using the procedure in [23] and [24] . It is given by the spectral radius of the next generation matrix

Here, we consider the matrices representing the production of new infection and transition part of our proposed SEIRUC model as follows 

where, the matrices F and V are the Jacobians of the matrices F and V respectively. Thus, we have,

Where, P = γ(1 − q)(η + θ 3 ) + θ 4 (θ 3 + qγ), θ 1 = a + δ 1 + c 1 , θ 2 = γ + δ 2 + c 2 , θ 3 = δ 3 + c 3 θ 4 = δ 4 + c 4 + η.

In this subsection, the existence and stability conditions for the endemic steady state are presented.

The endemic steady state can be rewritten as follows

The term, P = γ(1−q)(η+θ 3 )+θ 4 (θ 3 +γq) is positive since q is the fraction of asymptomatic infectious that become reported symptomatic infectious, it lies between 0 and 1. Hence, it is obvious from the above that, the positive endemic steady state of system (1) exist only when R 0 > 1. Hence, the model system (1) has an endemic equilibrium point whenever R 0 > 1 and has no endemic steady state for R 0 ≤ 1. The stability results are proved using Routh-Hurwitz criterion in the following theorem.

Theorem 3.4. The dynamical system (1) is locally asymptotically stable at the endemic equilibrium point E * = (S * , E * , I * , R * , U * , C * ), for R 0 > 1 and further if the following inequalities hold

Proof. The stability nature of the endemic steady state E * = (S * , E * , I * , R * , U * , C * ) can be determined using the eigenvalues of the Jacobian matrix J(E * ). Hence, evaluating the Jacobian matrix around the endemic equilibrium E * (S * , E * , I * , R * , U * , C * ), we obtain,

From the characteristic equation of the Jacobian matrix J(E * ), we obtain, one of the eigenvalues as −δ 5 and the remaining eigenvalues are the roots of the following polynomial

Where

Then, according to Routh-Hurwitz criterion, when R 0 > 1, the system (1) has eigenvalues with negative real parts if

Here, we obtain

Similarly, we obtain

Hence, it can be seen that t 1 > 0 and d 1 > 0 whenever A 2 and A 3 holds and the last coefficient a 5 > 0 whenever R 0 > 1. Thus, by Routh-Hurwitz criterion, when R 0 > 1, the endemic steady state, E * = (S * , E * , I * , R * , U * , C * ) is locally asymptotically stable if s 1 > 0, t 1 > 0 and d 1 > 0 which holds whenever A 1 , A 2 and A 3 holds. .

In literature, sensitivity analysis is proposed to understand the relative importance of the different factors responsible for transmission and prevalence of the disease. In order to reduce the disease transmission, it is necessary to control the fluctuations in the SEIRUC model parameters to make R 0 < 1. The sensitivity index of a variable to a parameter is the ratio of the relative change in the variable to the relative change in the parameter, which can be estimated from S[h] = h R0 × ∂R0 ∂h . The normalised sensitivity indices of the reproduction number with respect to the system parameters are given in the following table 

From Table 1 and Figure (3) , it is obvious that the most sensitive parameter to the basic reproduction R 0 for the SEIRUC model system is the disease transmission rate, β and the least sensitive parameter is the natural decay rate of the population, δ. The value of R 0 increases as β increases. Thus, R 0 increases proportionally with the increase in transmission rate of infection (β).

The aim of this study is to determine how the model parameters such as transmission rate, nat- For the endemic-free steady state, we consider the parameter values given in Table. 1 along with the disease transmission rate β = 0.07. We obtain the endemic-free equilibrium point at (1.3526 × 10 9 , 0, 0, 0, 0, 0). We For the endemic steady state, we consider the model parameter values given in Table.2 In order to better understand the transmission dynamics of our SEIRUC model, In Figure ( 2) we present a 3-dimensional plot for the basic reproduction number, R0 with respect to the rate of transmission of disease, β and the rate at which the asymptomatic infectious individuals turns into reported symptomatic individuals (q). 

We have proposed a new mathematical model by considering the new class called convalesce class along with asymptomatic infectious and symptomatic infectious class for the transmission dynamics of COVID-19.

By exploiting Routh-Hurwitz criteria for higher order polynomials, we established the sufficient conditions for the local stability of the disease-free and endemic equilibrium. Using the concept of next generation matrices, the threshold quantity, R0 has been computed. The established qualitative behaviour of the proposed model has been verified by the numerical simulations.

In future work, we aim to study the qualitative behaviour and numerical aspects of the proposed SEIRUC model to understand the transmission dynamics of COVID-19 under different fractional order derivatives.

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