key: cord-0982479-3s2g3o81
authors: Mamo, Dejen Ketema
title: Model the transmission dynamics of COVID-19 propagation with public health intervention
date: 2020-04-25
journal: nan
DOI: 10.1101/2020.04.22.20075184
sha: eb000163a7f68edd9487060d4f50a4c49ed3df92
doc_id: 982479
cord_uid: 3s2g3o81

In this work, a researcher develop $SHEIQRD$ (Susceptible-Stay at home-Exposed-Infected-Quarantine-Recovery-Death) coronavirus pandemic spread model. The disease-free and endemic equilibrium points are calculated and analyzed. The basic reproductive number $R_0$ is derived and its sensitivity analysis is done. COVID-19 pandemic spread is die out when $R_0leq 1$ and its persist in the community whenever $R_0>1$. Efficient stay at home rate, high coverage of precise identification and isolation of expose and infected individuals, and redaction of transmission and stay at home return rate can be mitigate the pandemics. Finally, theoretical analysis and numerical results are consistent.

Coronaviruses are a large family of viruses which may cause illness in animals or humans. In humans, several coronaviruses are known to cause respiratory infections ranging from the common cold to more severe diseases such as Middle East Respiratory Syndrome (MERS) and Severe Acute Respiratory Syndrome (SARS). The most recently discovered coronavirus causes Coronavirus disease

In this work a researcher consider that the total population is N (t) at time t. The whole population dividing in to seven compartments. The susceptible population S(t), they stand for people who are capable of becoming infected.

The quarantine population H(t), they represent people who are stay at home. In the process of COVID-19 spreading, the spreading among these seven states is governed by the following assumptions. It is assumed that β is the contact rate of susceptible individuals with spreaders and the disease transmission follows the mass action principle. A researcher assume that susceptible individuals home quarantine or stay at home at the rate θ. And at a rate θ 0 staying at home is not fully protected from the virus due to ineffectiveness of home quarantine. The one who completed incubation period becomes to infected at a rate of σ, that means 1 σ is the average duration of incubation. According to clinical examination, the exposed and infected individuals becomes isolated at a rate of η and α respectively. It is assumed that the infectious infected individuals, leading to disease prevalence. The average duration of infectiousness is 1 γ , when γ is the transmission rate from infected to recovery or death. In my assumption recovery from isolated infected is better than and infected class due to clinical treatment. Infected and isolated infected are recover with a probability of κ 1 and κ 2 , and also they will becomes to death with a probability of (1 − κ 1 ) and

(1 − κ 2 ) respectively. The parameter Λ is the recruitment, while µ natural birth and death rate of each state individuals. The parameters are all non-negative.

Based on the above considerations, COVID-19 spreading leads to dynamic transitions among these states, shown in figure 1. The model can be described by the following system of nonlinear ordinary differential equations:

We have the non-negative initial conditions (S(0), H(0), E(0), I(0), Q(0), R(0), D(0)) ∈

To make the mathematical analysis more easier, the variables of the model

, and Λ = µN (t). After substitute it in (1) we can get the simplified form of the model

From the normalized form of the model we have to get

Now, the first equation of the system (2) can be reduced and we hold six system

where φ = (σ + η + µ) and ξ = (α + γ + µ).

So, the feasible domain of the system (3) is

For the well-posedness of the model, we have the following lemma.

Lemma 1. The set Γ is positively invariant to system (3).

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Proof. Denote x(t) = (h(t), v(t), w(t), q(t), r(t), d(t)) T and then system (3) can be rewritten as

Note that Ω is obviously a compact set. We only need to prove that if x(0) ∈ Γ, then x(t) ∈ Γ for all t ≥ 0. Note that ∂Γ consists of five plane segments:

P 5 = (h, 0, w, q, r, d)|h, w, q, r, d ∈ [0, 1], h + w + q + r + d ≤ 1, P 6 = (0, v, w, q, r, d)|v, w, q, r, d ∈ [0, 1], v + w + q + r + d ≤ 1, as their outer normal vectors, respectively. If the dot product of f (x) and nor-

of the boundary lines are less than or equal 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. (which was not certified by peer review)

The copyright holder for this preprint this version posted April 25, 2020. . to zero, then x(t) ∈ Γ for all t ≥ 0. So,

The proof is complete.

Hence, system (1) is considered mathematically and biologically well posed in Γ [18] .

In this sub section, we show the feasibility of all equilibria by setting the rate of change with respect to time t of all dynamical variables to zero. The model (2) has two feasible equilibria, which are listed as follows:

(i) Disease-free equilibrium (DFE)E 0 µ+θ0 µ+θ+θ0 , θ µ+θ+θ0 , 0, 0, 0, 0, 0 (ii) Endemic equilibrium (EE) E * (u * , h * , v * , w * , q * , r * , d * ).

The existence of endemic equilibrium is computed after we have the basic reproductive number R 0 .

Here, we will find the basic reproduction number (R 0 ) of the model (2) using next generation matrix approach [19] . We have the matrix of new infection 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.

The copyright holder for this preprint this version posted April 25, 2020. . (2) can be rewritten as:

The Jacobian matrices of F(X) and V(X) at the disease free equilibrium

The inverse of V is computed as

The next generation matrix K L = F V −1 is given by

Therefore, basic reproduction number is

is spectral radius of matrix K L and basic reproduction number (R 0 ) is obtained 8 . 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 April 25, 2020. . https://doi.org/10.1101/2020.04.22.20075184 doi: medRxiv preprint as follows,

In this subsection, we summarize the results of linear stability of the model (2) by finding the sign of eigenvalues of the Jacobian matrix around the equi-

Proof. In the absence of the disease, the model has a unique disease free equilibrium E 0 . Now the Jacobian matrix at equilibrium E 0 is given by:

Here, we need find the eigenvalue of the system from the Jacobian matrix (4) . We obtain the characteristic polynomial

From the characteristic polynomial in equation (5), it is easy to get five real

We get the other real negative eigenvalues from the expression

From the quadratics equation (6), we conclude that λ 6,7 are positive if R 0 > 1 and negative if R 0 < 1. Thus, the equilibrium E 0 is locally asymptotically

The proof is complete.

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The copyright holder for this preprint this version posted April 25, 2020. Physically speaking, theorem 2 implies that disease can be eliminated if the initial sizes are in the basin of attraction of the DFE E 0 . Thus the infected population can be effectively controlled if R 0 < 1. To ensure that the effective control of the infected population is independent of the initial size of the human population, a global asymptotic stability result must be established for the DFE.

Proof. Let X = (u, h, v, w, q, r, d) T and consider a Lyapunov function,

Differentiating V in the solutions of system (2) we geṫ

Thus the largest invariant set in X ∈ Γ|V(v, w) = 0 is the singleton, E 0 = µ+θ0 ψ , θ ψ , 0, 0, 0, 0, 0 . By LaSalle's Invariance Principle the disease-free equilibrium is globally asymptotically stable in Γ, completing the proof. the DFE E 0 , and the disease will die out from the community irrespective of the initial conditions. If R 0 > 1, E 0 is unstable and the system is uniformly persistent, and a disease spread will always exist.

The feasibility of the equilibrium E 0 is trivial. Here, we show the feasibility of endemic equilibrium E * . The values of u * , h * , v * , w * , q * , r * and d * are obtained by solving following set of algebraic equations:

After some algebraic calculations we get the value of E * as:

Therefore, there exists a unique positive solution only when R 0 > 1. This implies that, it has a unique endemic equilibrium, E * .

Theorem 4. If R 0 > 1, then the endemic equilibrium point E * of system (2) is locally asymptotically stable.

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The copyright holder for this preprint this version posted April 25, 2020. Proof. The Jacobian matrix of the model at E * is

where

From the Jacobian matrix (8) easily to get λ 1,2 = −µ, λ 3 = −µ − γ and the other eigenvalues of the system needs further finding. The characteristic polynomial of (8) is

Where

The polynomial (9) has negative roots (eigenvalues) if all its coefficients terms are positive, or it satisfies Routh-Hurwitz criteria of stability [20] . From (10) we can verify that c 1 > 0, c 4 > 0, c 1 c 2 − c 3 > 0 and c 3 (c 1 c 2 − c 3 ) − c 2 1 c 4 > 0, when R 0 > 1. Therefore, according to the Routh-Hurwitz criterion, we can get that all the roots of the above characteristic equation have negative real parts.

Thus, the endemic equilibrium asymptotically stable. The proof is complete.

The local stability analysis of the endemic equilibrium tells that if the initial values of any trajectory are near the equilibrium E * , the solution trajectories 12 . 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 April 25, 2020. 

We explore R 0 sensitivity analysis of system (2) to determine the model robustness to parameter values. This is a strategy to identify the most significance parameters of the model dynamics. The normalized sensitivity index Υ λ is given by

Thus normalized sensitivity indices for parameters are obtained as

From the sensitivity indices calculation results, we can identify some parameters that strongly influence the dynamics of disease spread. Parameters β, θ 0 and σ have a positive influence on the basic reproduction number R 0 , that is, an increase in these parameters implies an increase in R 0 . While parameters µ, η, α, θ and γ have a negative influence on the basic reproduction number R 0 , that is, an increase in these parameters implies a decrease in R 0 .

Here, we illustrate graphically the relationship between the basic reproductive number and the parameters in model (2) .

A researcher can find some interesting results, which have been showed in figure 2, and figure 3 , it can be seen that big β or σ can lead to large R 0 . That is to say, the larger contact or short incubation period can increase the opportunity of disease spreading. If we reduce the transmission rate by quarantine or any appropriate control measure, then the disease outbreak will end.

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The copyright holder for this preprint this version posted April 25, 2020. That is to say, effective quarantine of incubated and infectious individuals can reduce the opportunity of disease spreading.

From figure 8, and figure 9 , we find that, short average time from the symptom onset to recovery or death γ and large value of µ can reduce the COVID-19

spread.

14 . 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 April 25, 2020. 

In this section, we conduct numerical simulation of the model (2) by using Matlab standard ordinary differential equations (ODEs) solver function ode45.

We numerically illustrate the asymptotic behavior of the model (2) . We 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 April 25, 2020. . https://doi.org/10.1101/2020.04.22.20075184 doi: medRxiv preprint whenever R 0 > 1.

In order to investigate the impact of the transmission rate on the spread of COVID-19, we carry out a numerical simulation to show the contribution of transmission rate β in fractional infection population density. to reduce the transmission rate in the current pandemic are stay at home and quarantine or isolation of exposed and infected individuals by clinical tests.

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The copyright holder for this preprint this version posted April 25, 2020. 

Study the recommended containment strategies of the pandemic, we conduct some numerical simulations to show the contribution of public health interventions.

One of the recommended control measure to reduce the pandemic is quarantine or isolation. Here, we observe the isolation of exposed and infected individuals within different rate: Now, we set the exposed population isolation rates η as 0.6, 0.2, 0.1, 0.05 and 0.0. In figure 13 , we can see that the infectiousness increase as η decrease. This implies that effective isolation of exposed individuals by clinical identification before the symptom onset can mitigates the COVID-19 pandemic. Similarly, infected isolation rates α set as 0.25, 0.1, 0.05, 0.03 and 0.0. In figure 14 , we observe that the infectiousness density approaches to highest peak level as α value decreases. This implies that ineffective quarantine of symptomatic individuals can lead the prevalence of the pandemic.

In the current critical time the public health experts and government officials announced that every individuals must stay at home. Due to food security and ineffectiveness of stay at home peoples my lose this recommendation. To observe the impact of stay at home efficiency and its lose in the following numerical 18 . 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 April 25, 2020. θ=0.0 (R 0 =3.5) Figure 15 : Impacts of θ on w(t). 

In this paper, a researcher investigated the dynamics of the COVID-19

spreading with control measure. An SHEIQRD Corona pandemic model with public health intervention has been presented, and analyzed theoretically as will as numerically. The theoretical analysis of the model are done. An essential epidemiological parameter value R 0 is derived by using the next generation matrix approach. Furthermore, we have shown that the disease free equilibrium globally asymptotically stable if R 0 ≤ 1 and unstable otherwise. For the case where R 0 > 1, the exists a unique endemic equilibrium E * , which is locally 19 . 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 April 25, 2020. . https://doi.org/10.1101/2020.04.22.20075184 doi: medRxiv preprint asymptotically stable. The sensitivity analysis of basic reproductive number was conducted. Its result suggested that transmission rate β, isolation rates η and α, stay at home rate θ and stay at home return rate θ 0 are the basic control parameter of the model.

Numerical simulations are conducted aim to support theoretical analysis and shows the significance of public health intervention to containment these pandemics. The general dynamics of the model with time is illustrated that the disease is die out when R 0 ≤ 1 (see figure 10 ), but its persists in the community whenever R 0 > 1 (see figure 11 ). Moreover, socioeconomically crisis caused by these pandemic can be minimized and eliminated when we implemented appropriate control measure.

Also of importance in mitigation of the pandemics are reduced a transmission rate β (see figure 12 ) and the stay at home return rate θ 0 , an efficient identification and isolation of exposed and infected individual with rate of η and α (see figures 13, 14) respectively, and enhance the ability of stay at home rate θ (see figure 15 ). Finally, robust public health intervention end the current pandemic and minimizing crisis caused by these outbreak.

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The copyright holder for this preprint this version posted April 25, 2020. . https://doi.org/10.1101/2020.04.22.20075184 doi: medRxiv preprint

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