key: cord-0836955-5rwcyg4z
authors: Maji, C.
title: Modes of transmission of COVID-19 outbreak- a mathematical study
date: 2020-05-20
journal: nan
DOI: 10.1101/2020.05.16.20104315
sha: 383f7249820b8c4350de506ad4edf94977b326ad
doc_id: 836955
cord_uid: 5rwcyg4z

The world has now paid a lot of attention to the outbreak of novel coronavirus (COVID-19). This virus mainly transmitted between humans through directly respiratory droplets and close contacts. However, there is currently some evidence where it has been claimed that it may be indirectly transmitted. In this work, we study the mode of transmission of COVID-19 epidemic system based on the susceptible-infected-recovered (SIR) model. We have calculated the basic reproduction number R0 by next-generation matrix method. We observed that if R0<1, then disease-free equilibrium point is locally as well as globally asymptotically stable but when $R0>1, the endemic equilibrium point exists and is globally stable. Finally, some numerical simulation is presented to validate our results.

Today an invisible emperor of the world has put human civilization in front of a big question mark which is popularly known as COVID-19 (World Health Organization (WHO)). Mankind, proud of being the most advanced intelligent creature in the world is now endangered by its tyranny. COVID-19 is an abbreviated form of Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2), which has not been previously identified in humans [1]. On Dec, 2019, this disease was first identified in Wuhan city Hubei province of china [25] , after that rapidly it has spread out all over the world. SARS [3] and MERS [4] , these two previously known corona virus disease outbreaks have already occurred in the last two decades. Till now it is reported that COVID-19 has been spread out in more than 200 countries and now it has taken the shape of an epidemic in worldwide with more than 3.8 million people infected and 2.5 lakhs death [5] . It is presumed that COVID-19 is directly transmitted through human to human via respiratory droplets among closed contact and infected people generally develop signs and symptoms, including mild respiratory discomfort and fever, on an average of 5-6 days after infection [6] . In [7] , Jiang et al. estimated that fatality rate of this virus is near about 4.5% but for the age group 70-79 it has gone up to 8%. This disease is more critical for elder people who has other diseases like diabetes, asthma, cardiovascular disease [6] . However, these transmission modes do not explain all cases. A lot of evidences have shown that COVID-19 has highly similar biological properties with severe acute respiratory syndrome coronavirus (SARS-CoV). In their study [29] , Cai et al. investigated a cluster of COVID-19 cases associated with a shopping mall in Wenzhou, China and showed that indirect transmission of the causative virus occurred from virus contamination of common objects, virus aerosolization in a confined space, or spread from asymptomatic infected persons. Zhang et al. [19] , suggested that transmission may also occur through fomites in the immediate environment around the infected person. Guo et al. [30] , tested surface and air samples from an intensive care unit (ICU) and a general COVID-19 ward at Huoshenshan Hospital in Wuhan, China and found that contamination was greater in ICU than general wards, they also have seen that virus was widely distributed on floors, computer mice, trash cans, and sickbed handrails. Therefore, it can be concluded that the transmission of the COVID-19 virus can occur by direct contact with infected people and indirect contact with surfaces in the immediate environment or with objects used on the infected person. Considering direct transmission mode, already a lot of works have been done mathematically and numerically to give an efficient prediction on COVID 19 outbreaks [9, 10, 11, 12, 13] . Researchers are trying their best to discovering vaccines and treatments for the virus but it may be far away from our imagination. In this work, we have formulated a mathematical model of direct and indirect transmission to study the dynamics of COVID-19 outbreaks.

The paper is organized as follows: In Section 2. we have formulated a mathematical model for the epidemic COVID-19. Boundedness of the solutions of system (1) is performed in Section 3. Stability analysis of disease-free equilibrium point is discussed in Section 4. Local and global stability of endemic equilibrium point is presented in section 5. Sensitivity analysis and some numerical simulations are discussed in Section 6, to validate our results. A brief discussion is given in Section 7.

The world has now paid a lot of attention on the outbreak of novel coronavirus (COVID- 19) which has been declared a pandemic by WHO. Till date, no vaccine or medicine is available to cure the disease properly so people are getting in a panic and they are afraid of disease transmission. According to current evidence, COVID-19 virus is primarily transmitted between people through respiratory droplets and close contacts [20, 21, 22, 23, 24, 25] . But there are also some evidences who conclude that transmission may also occur through fomites in the immediate environment around the infected person [19, 26] . Inspired by above all these work, we have formulated a epidemic model for COVID-19 of direct and indirect transmission. The model is based on traditional susceptible-infected-recovered (SIR) models of disease transmission in humans. Our proposed model is described by a system of four ordinary differential 2 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted May 20, 2020. . https://doi.org/10.1101/2020.05.16.20104315 doi: medRxiv preprint equation which is as follows:

Here, S(t), I(t) denotes the density of susceptible population, infected population. R(t) is the total number of recovered population and E(t) is the mass of infectious material present in the environment. All other model parameters and their description are given in Table 1 . 

For biological validity of system (1), it is necessary to prove that all solutions of system (1) with positive initial values will remain positive for all time t > 0. Thus in this section we want to prove the positivity and boundedness of solutions of our considered system. Lemma 1. All solutions (S(t), I(t), R(t), E(t)) of system (1) with positive initial values in R 4 + will remain positive for all t > 0.

Proof. From first equation of system (1), we get

Thus,

3 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 20, 2020. . https://doi.org/10.1101/2020.05.16.20104315 doi: medRxiv preprint As, S(0) > 0 then S(t) > 0 for all t > 0.

Similarly , it can be shown that I(t) > 0, R(t) > 0, E(t) > 0 for all t > 0. Hence the interior of R 3 + is an invariant set of system (1).

Lemma 2. All solution of system (1), which initiate in R 4 + are bounded and lie in the region Ω = (S, I, R, E) ∈ R 4 + :

Proof. Define a function,

The time derivative along the solution of system (1), we get

By using differential inequality argument [27] , we get

where c is arbitrary positive constant. Hence we get,

All solution of system (1) enter into the region Ω = (S, I, R, E) ∈ R 4 + :

Basic reproduction number is one of the most important threshold parameter which can determine whether the infectious disease will die out or spread through population with time increases. Here we calculate the basic reproduction number for COVID-19 model through next generation matrix method [28] .

System (1) has a unique disease free equilibrium point E 1 (K, 0, 0, 0), where K = r µ . Then using the notation in [28] , the reproduction number R 0 of system (1) is given by

Theorem 1. Disease free equilibrium E 1 is locally as well as globally asymptotically stable if R 0 < 1 and unstable if R 0 > 1.

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Proof. The jacobian matrix at E 1 is given by

Therefore, all eigenvalues of the characteristic equation of J(E 1 ) are −µ, −µ and other two eigenvalues are the roots of the equation

where,

Clearly, p 1 , p 2 > 0 when R 0 < 1. Therefore all roots of equation (2) has negative real part, hence E 1 is locally asymptotically stable and unstable when R 0 > 1. Now, we consider the following Lyapunov function to prove global stability of disease free equilibrium point E 1 .

Since, all parameters of the model are non negative. Hence, it follows that dV dt < 0 when R 0 < 1, hence by Lasalle invariance principle [15] , the disease free equilibrium point is globally asymptotically stable.

The endemic equilibrium point E * 2 of the model (1) is given by

Clearly, we have seen that the endemic equilibrium point E * is feasible if R 0 > 1.

Theorem 2. Endemic equilibrium point E * 2 is locally symptotically stable if R 0 > 1.

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Proof. The Jacobian matrix of system (1), at endemic equilibrium point E * 2 is given by where a 11 = −µR 0 , a 12 = − τ (µ+d+γ) τ β i +αβe β i , a 13 = 0, a 14 = − τ (µ+d+γ) τ β i +αβe β e , a 21 = µ(R 0 − 1), a 22 = − αβe(µ+d+γ) τ β i +αβe , a 23 = 0, a 24 = τ (µ+d+γ) τ β i +αβe β e a 31 = a 34 = a 41 = a 43 = 0, a 32 = γ, a 33 = −µ, a 42 = α, a 44 = −τ .

Characteristic equation of J(E * 2 ) is given by

where, A 1 = −(a 11 + a 22 + a 33 + a 44 ), It follows from Routh-Hurwitz criteria [16] that all roots of equation (3) will have neg-

This will be possible when R 0 > 1. Hence, endemic equilibrium point E * 2 is locally stable.

System (1) is said to be uniformly persistent if there exist a constantā such that any solution of (1) satisfies

From Theorem 1, we get the disease free equilibrium E 1 is unstable for R 0 > 1. Now apply the uniform persistence result defined in [17] , and then using the approach used to prove Proposition 3.3 of [18] , it can be proved that sysytem (2) is uniformly persistent in the region Ω .

Theorem 3. Assume R 0 > 1, then endemic equilibrium point E * 2 is globally asymptotically stable if the following conditions hold:

and det(M ) > 0.

Proof. Consider a positive definite function

6 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 20, 2020. . https://doi.org/10.1101/2020.05. 16.20104315 doi: medRxiv preprint Now differentiating V with respect to t along the solutions of system (2), we get Thus, dV dt < 0 and consequently V is a Lyapunov function and hence endemic equilibrium point E * is globally asymptotically stable.

Sensitivity analysis is one of the most important part to get an overview of most influential parameters in modelling of a infectious disease. As system stability determined by its reproduction number (R 0 ), so we want to verify how the sensitive parametrs are related with R 0 and therefore we compute the sensitivity analysis of R 0 with repect to the model parameters. By the definition of the normalized sensitivity index of R 0 with respect to β is given by

The sensitivity indices of R 0 with respect to other parameters are as follows:

From this analysis we observed that, J R 0 β i = 0.87, J R 0 βe = 0.128, which implies if the direct and indirect disease transmission rate increase by 1% then it will increase the 7 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. value of R 0 by 0.87% and 0.128% repectively. Again, we see that J R 0 γ = −0.84 < 0 that means increase the value of γ by 1% then the value of R 0 will be reduced by 0.84%. Thus, it can be concluded that reproduction number is positively corelated with β i and β e and negatively corelated with γ.

To observe the dynamics of our proposed model through numerically we set the following set of parameters hypothetically : r = 0.00045308; β i = 1. 7 [5] ; β e = 0.5 [5]; α = 0.1; µ = 0.005; γ = 0.05; τ = 0.2; d = 0.0039 [25] . For this set of parameters, we obtain R 0 = 3.1794 > 1, which satisfy the conditions of Theorem 3 and consequently the endemic equilibrium point is globally stable (Figure 1 ). From this figure we have seen that, susceptible population is decreasing but infected popuation is increasing over time that means more people are being infected within a short time period and as a result there is an outbreak within a short time span.

Till date, no vaccine or medicine is available to cure the disease properly so to prevent the disease transmission it is necessary to obey social distancing or lockdown which have already been applied by all Countries . The effectiveness of lockdown of 21 days or more days is described in Figure 2 . From this figure, we observed that 21 days' lockdown is not sufficient to control the disease but if we would applied 54 days' lockdown or more then it shows that the lowest number of people would be infected and the epidemic might be under control. In Figure 3 , we observed that how infected population is related with disease transmission rate.

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The copyright holder for this preprint this version posted May 20, 2020. . https://doi.org/10.1101/2020.05.16.20104315 doi: medRxiv preprint

Mathematical modelling is one of the best way to express the dynamics of an epidemic. In this paper, we have developed a mathematical model considering the direct and indirect disease transmission rates for COVID-19 outbreaks. The basic reproduction number for our considered model is given by

The asymptotic behaviour of our model is determined by it's basic reproduction number. We observed that if R 0 < 1, then the disease-free equilibrium is globally asymptotically stable. If R 0 > 1, then disease persists in the system and endemic equilibrium point E * is globally asymptotically stable (see Figure 1. ). From equation (9), we have also seen that the basic reproduction number is proportional to the recruitment rate of the susceptible population and directly as well as indirectly disease transmission rate, that means the probability of disease transmission is higher among the close contact or indirect contact with surfaces in the immediate environment or with objects used on the infected person. From sensitivity analysis it is noticed that basic reproduction number is positive corelated with β i while negatively corelated with γ. Thus to control the disease we must keep the basic reproduction number below the unity.

The author declare that there is no conflict of interest.

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The copyright holder for this preprint this version posted May 20, 2020. . https://doi.org/10.1101/2020.05.16.20104315 doi: medRxiv preprint

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