








key: cord-320980-srpgcy4b
authors: Aldila, Dipo; Khoshnaw, Sarbaz H.A.; Safitri, Egi; Anwar, Yusril Rais; Bakry, Aanisah R.Q.; Samiadji, Brenda M.; Anugerah, Demas A.; Alfarizi GH, M. Farhan; Ayulani, Indri D.; Salim, Sheryl N.
title: A mathematical study on the spread of COVID-19 considering social distancing and rapid assessment : The case of Jakarta, Indonesia
date: 2020-06-28
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2020.110042
sha: 
doc_id: 320980
cord_uid: srpgcy4b

The aim of this study is to investigate the effects of rapid testing and social distancing in controlling the spread of COVID-19, particularly in the city of Jakarta, Indonesia. We formulate a modified susceptible exposed infectious recovered compartmental model considering asymptomatic individuals. Rapid testing is intended to trace the existence of asymptomatic infected individuals among the population. This asymptomatic class is categorized into two subclasses: detected and undetected asymptomatic individuals. Furthermore, the model considers the limitations of medical resources to treat an infected individual in a hospital. The model shows two types of equilibrium point: COVID-19 free and COVID-19 endemic. The COVID-19-free equilibrium point is locally and asymptotically stable if the basic reproduction number [Formula: see text] is less than unity. In contrast, COVID-19-endemic equilibrium always exists when [Formula: see text]. The model can also show a backward bifurcation at [Formula: see text] whenever the treatment saturation parameter, which describes the hospital capacity, is larger than a specific threshold. To justify the model parameters, we use the incidence data from the city of Jakarta, Indonesia. The data pertain to infected individuals who self-isolate in their homes and visit the hospital for further treatment. Our numerical experiments indicate that strict social distancing has the potential to succeed in reducing and delaying the time of an outbreak. However, if the strict social distancing policy is relaxed, a massive rapid-test intervention should be conducted to avoid a large-scale outbreak in the future.

Coronavirus disease 2019, or COVID-19, is an infectious disease caused by a new type of coronavirus named severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), which is known to have originated in the city of Wuhan, China in December 2019 [1] . This virus is transmitted from human to human and has spread widely across China and 214 other countries and territories. It is spread through droplets 5 that exit the nose or mouth when a person infected with COVID-19 coughs or exhales. These droplets then land and settle on surrounding objects and surfaces. If a person who touches any of those objects or surfaces then touches their eyes, nose, or mouth, they may become infected. Transmission can also occur if a person inhales droplets from a cough or breath of a person infected with COVID-19 [2]. On March 12, 2020 , the World Health Organization (WHO) declared COVID-19 as a pandemic. Till May 16, According to the WHO [3] , COVID-19 generally has an incubation period of 5-6 days, with a range of 1-14 days. The symptoms of COVID-19 are nonspecific and vary widely, ranging from no symptoms to severe pneumonia and eventual death. Based on specific cases, some of the symptoms of COVID-19 include fever, dry cough, fatigue, nasal congestion, diarrhea, and headache. People who are at high risk for severe illness include those aged 60 years and over and those with hypertension, diabetes, cardiovascular problems, cancer, or a chronic respiratory disease. The mortality rate increases with age, with the highest death rate among people aged over 80 years. In children, the disease is relatively rare.

Each country is at a different stage of the epidemic. In most countries where the spread of the 20 virus has caused outbreaks with exponential growth, governments have called for physical distancing and movement restrictions, commonly known as lockdown, to slow the spread of the COVID-19 outbreak [4] . Some of these countries include China, Italy, and the United Kingdom. Moreover, there are some countries that have managed to handle the COVID-19 outbreak without a lockdown; one such country is South Korea. Based on a report in [5] , South Korea conducted massive numbers of polymerase chain 25 reaction tests, which reached 726,747 by May 15, 2020 [6] .

In addition to death, this pandemic has also had negative psychological, economic, and social impacts globally. The COVID-19 pandemic has resulted in changes in the work environment, indirectly affecting gender inequality, and an increased risk of suicide owing to lockdown, social distancing, and economic crisis [7] . Panic in communities may be reduced by educating the public regarding the predictions of the 30 COVID-19 epidemic and the interventions that must be conducted. Communities must also unite and work together to help governments overcome the epidemic by following the guidelines provided by the relevant authorities [7] .

Several mathematical models have been proposed by several authors to understand the spreading mechanism of COVID-19. In [8, 9, 10, 11, 12, 13, 14] , the authors proposed a modified susceptible-exposed- 35 infected-recovered (SEIR) model to understand the effects of undetected infection, hospitalization, and quarantine. The model was analyzed to determine the equilibria and the basic reproduction number. Parameter estimation was conducted by the authors in [11, 12, 15, 16] using a statistical approach involving a Bayesian or Markov Chain Monte Carlo method. In addition to compartmental deterministic modeling, the spread of COVID-19 can be predicted statistically using a time-series approach [17, 18, 19] .

A time-series model is effective owing to its ability to accommodate the factors influencing the spread of COVID-19 that cannot be calculated using other statistical approaches [17] . A frequently used time-series model is the autoregressive integrated moving average (p, d, q).

In this study, we propose a modified SEIR model that considers asymptomatic cases. These asymptomatic cases describe a hidden case in the field. As an intervention, rapid testing has been implemented 45 by many countries to detect infected individuals in this asymptomatic group. Therefore, we categorize our asymptomatic individuals into two groups: detected and undetected. Furthermore, we also accommodate the limitation of resources for medical treatment. This is an extremely important factor that will have an essential role in a successful eradication of COVID-19. Many countries are not ready for an exponential growth of COVID-19 cases because many hospitals will become overwhelmed by the high 50 number of patients. To validate the model, COVID-19 incidence data from the city of Jakarta are used for parameter estimation. The basic reproduction number is calculated, and a sensitivity analysis of the model is conducted. The model is then used to predict the effects of social and physical distancing to stop the spreading of COVID-19, and to monitor the effects of the plan to gradually ease social distancing guidelines of the government in Jakarta.

The remainder of this paper is organized as follows. In Section 2, the construction of the SEA u A d IR compartmental model is described. Next, the mathematical properties of the model, such as the equilibrium points, basic reproduction number, and existence of backward bifurcation, are detailed in Section 3. In Section 4, we explain the real-world problem using the incidence data of Jakarta, Indonesia. A discussion on the basic reproduction number and the results of the sensitivity analysis are provided in Section 6.

Finally, some conclusions are presented in Section 7.

The objective of our study is to analyze the effect of rapid testing and self-monitored isolation, and to predict the long-term dynamics of the incidence data of Jakarta, Indonesia. To achieve these purposes, our model should consider asymptomatic cases, as well as a parameter describing a rapid test intervention.

To achieve this, let us divide the human population into six categories based on their health status:

(i) Susceptible population (S(t)) := Group of susceptible individual.

(ii) Exposed population (E(t)) := Group of individual, who already infected by COVID-19, but not yet infective. (iv) Asymptomatic undetected population (A u (t)) := Infected population, have a capability to transmit COVID-19, do not show any symptoms, and undetected by the government. This individual has a larger probability of spreading the disease compared with I and A d .

(v) Asymptomatic detected population (A d (t)) = Individuals in this category are similar to those in A u 75 in terms of their health status, but have already been detected by the government through a swabtest, rapid test, or other tests. Although this group of individuals has the capability of spreading the disease, they do not isolate themselves in a hospital owing to a limited hospital capacity. Therefore, a monitored self-isolation is applicable to them.

(vi) Recovered population (R(t)) := Recovered individuals, with a short-term immunity to COVID-19.

Therefore :

We make the following assumptions for the formulation of the model. Using the transmission diagram given in Figure 1 and mentioned assumptions, the model which describes the transmission of COVID-19 considering rapid testing and asymptomatic cases is given by the following systems of equation. Here, Λ, β, µ, and φ are the natural recruitment rate, infection rate, natural death rate, and death rate from COVID-19, respectively. ξ i and ξ a represent the reduction of β for I and A d , respectively, owing to isolation at home or in a hospital. Furthermore, α, γ 0 , and δ denote the progression rate of COVID-19 based on its incubation period, natural recovery rate, and disappearance of temporal immunity, respectively. Note that p describes the proportion of exposed individuals who have progressed into asymptomatic individuals, γ 1 is the enhancement of natural recovery rate owing to treatment in a hospital, η is the rate of hospitalization from A d to I, and ν is the effort required for early detection of COVID-19 infection. Note that system (1) is supplemented with an initial non-negative condition:

It is easy to prove that all variables in system (1) remain nonnegative for all time t ≥ 0 as long as the initial conditions are nonnegative. Next, we show that our model is well-posed in biological meaning. Let us consider the possible region

Summing all rate of change for each variable in system (1) yield

Clearly, if N > Λ µ , then we have that dN dt < 0. Since dN dt is bounded by Λ − µN , we have that N (t) ≤ N (0)e −µt + Λ µ (1 − e −µt ). Furthermore, we have that N (0) ≤ Λ µ → N (t) ≤ Λ µ . Also, it can be seen that 100 every solution of our COVID-19 model in (1) with initial condition in D will remains in D for all t > 0. Therefore, we have that D is positively invariant and attracting. Hence, the COVID-19 model in (1) is well-posed. Taking the right-hand side of system (1) equal to zero, we have two types of equilibrium points for system (1) . The first equilibrium is the COVID-19 free equilibrium point which given by

Using the next-generation matrix approach [20] , the basic reproduction number of system (1) is given by (See Appendix A for the derivation of the R 0 )

From results in [21] , we have the local stability criteria of E 0 depend on R 0 in the following theorem.

Theorem 1. The COVID-19 free equilibrium E * 0 is locally asymptotically stable if R * 0 < 1, and unstable otherwise.

Threshold quantity R 0 presents the expected number of new COVID-19 infections generated from one primary infection into an absolute susceptible population during a single infection period. From the results of Theorem 1, it can be shown that COVID-19 can be eliminated from the population if R 0 < 1. Please note that R 0 in (4) is a basic reproduction number when a rapid test is implemented. When a rapid test is not implemented into the model, then R 0 in (4) is reduced to the following:

This shows the multiplication among the total human population when no COVID-19 exists, ratio between the infection rate and exposed/incubation period of category E, and infection period of categories A u 110 and I. It is easy to see that reducing R * 0 is highly related to reducing β, which can be implemented by reducing the contact probability through lockdown or social distancing, and reducing ξ i by conducting a proper quarantine procedure in a hospital, such that ξ i → 0. Another way to reduce R * 0 is by increasing the recovery rate owing to hospitalization (γ 1 ). Further discussion on the complete R 0 is provided in Section 5. 

The endemic equilibrium point of system (1) is given by

where S * , E * , A * u , A * d , R * I * as a function of I * can be seen in Appendix B, while I * is taken from the positive roots of the following third order polynomial :

Here, a 2 and a 1 have significantly long expressions and are therefore omitted in this study. Because a 3 is always positive, and a 0 < 0 if R 0 > 1, we have the following theorem.

Theorem 2. System (1) has always a COVID-19 endemic equilibrium pint whenever R 0 > 1.

Proof. Let P(I) = P 1 (I) + a 0 , where P 1 (I) = a 3 I 3 + a 2 I 2 + a 1 I. Let us assume that a 0 = 0 ⇐⇒ R 0 = 1. Thus, P 1 (I) has I = 0 as one of the roots, whereas the other two roots can be positive, negative, or even imaginary. Let us consider the most extreme case in which we have no positive roots. Because we have a 3 > 0, we then have lim I→−∞ = −∞ and lim I→∞ = ∞. Therefore, when a 0 < 0 ⇐⇒ R 0 > 1, we have P I as being translated downward and providing one positive root. This completes the proof.

Theorem 1 and 2 indicate that R 0 become the endemic threshold, since when R 0 < 1, we have that the COVID-19 free equilibrium stable, but when R 0 > 1, then we have at least one positive COVID-19 endemic equilibrium. Furthermore, since P I is a third-order polynomial, we have at most three COVID-19 endemic equilibrium. Next, we analyze the possibility of having a COVID-19 endemic equilibrium when R 0 < 1. The results stated in the following theorem.

and

Proof. For the proof of the theorem, please see Appendix C.

As a direct consequence of Theorem 3 and 2, we have the following corollary.

We close this section with the following theorem, which explained the condition of backward bifurca-135 tion of system (1).

Proof. For the proof of Theorem, please see Appendix D

The case study on the city of Jakarta, Indonesia 

Jakarta is the capital of Indonesia, with a population of 10,374,235 people in 2019, with a population growth rate of 0.94% per year. Jakarta consists of 6 sub-governments, namely South, East, West, North and Central Jakarta, and the Thousand Islands. The largest population is in East Jakarta at 2,916,020 people.

The first COVID-19 case in Jakarta occurred on March 3, 2020, from a patient who had contact with Japanese citizens (whom later on was confirmed to be positive for COVID -19) . The social distancing rule is the most preferred action by the Jakarta city government to reduce the spread of COVID-19. This rule was applied from April 10, 2020, which is based on Jakarta Governor Regulation No 33/2020. The regulation requires the closure of schools, public facilities such as malls, and many other places that 150 might potentially gather people in the same place.

The data used in this article is COVID-19 incident data in Jakarta, from March 3 to April 10, 2020. This data can be divided into two types of active cases, namely active cases that must be treated in the hospital (I) and independent isolation at home (A d ). Incident data used are given in Table 1 . 

To conduct the parameter estimation for data-driven in Indonesia, due to the short-term of the data, we consider model (1) but neglect the natural newborn, natural death rate, and the drop out rate from recovered compartment due to temporal immunities running out. Based on this assumption, Model (1) now read as :

To conduct the parameter estimation, we divided our data based on the date of strict social distancing in Jakarta first time implemented. The first interval is from March 3 until April 10, 2020. In this interval, we find the best fit parameters are : With these parameters, we have that R 0 in Jakarta during this interval is 1.75. This indicates that 160 COVID-19 has a big potency to be endemic in Jakarta if accurate, precise, and fast policies are not carried out soon. In the second interval, after the strict social distancing implemented, we set other parameters constant, while β and η should be re-estimated. In this second interval, we have β = 0.94 × 10 −7 and η = 0.06. This data gives R 0 = 1.22 in the second interval. It can be seen that β was reduced by 34%, which indicates the effect of social distancing. On the other hand, η reduced by 68.4%, since the number 165 of infected individual keep increasing in the hospital. Therefore, the transition from A d to I need to be reduced because of the limitation on the hospital capacity. The result of parameter estimation for system (1) respect to incidence data in Jakarta given in Figure 2 . 

We conduct numerical experiments in this section in three scenarios. The first experiment is to analyze 170 the elasticity of R 0 . From Theorem 1, 2, 3 and 4, it is clear to see that our model is depending on the size of R 0 . Understand how R 0 may be changed when parameters changed will help a more effective intervention to control the spread of COVID-19. The second experiment is to see how the estimated parameter from incidence data in Jakarta might exhibit a backward bifurcation when the quality and size of handled patients in the hospital getting worse. The last experiment is the sensitivity analysis to 175 determine the most significant parameters in determining the dynamics of each variable.

To perform the elasticity analysis on R 0 , we calculate the normalized forward sensitivity index of R 0 using the following recipe :

where ω is the set of parameters in system (1). For example, the elasticity index of R 0 respect to α is given by

In a similar way, we can find all elasticity indices of R 0 for the rest of parameters. Substituting parameters value found from section 4 and assuming Λ = 10467629/(65 × 365), µ = 1/(65 × 365) and δ = 1/90, the elasticity value of all parameters in system (1) respect to R 0 is given in table 2. Positive and negative sign in 2 indicates increasing and decreasing of R 0 respect to the parameters, respectively. Therefore, we can say that R 0 increase when Λ, β, ξ i , ξ a , α or p increase. In the other hand, whenever µ, η, ν, γ 0 , γ 1 , or φ increase, then R 0 will decrease. Furthermore, for an example, since Γ R0 ν = −0.2174486541, we have that increasing ν for 10% will reduce R 0 2.174486541%. A similar interpretation is given for the rest parameters in Table 2 . It is interesting to see that the saturation 185 parameter b, which describes the capacity of the hospital or the number of a medical officer do not affect the size of R 0 since Γ R0 b = 0. However, b plays an important role in determining whether the system (1) undergoes a forward or a backward bifurcation when R 0 = 1. From Table 2 , it can be seen that β is the most positive significant parameter that can be used to control R 0 . Therefore, social/physical distancing is a very reasonable intervention to control the spread of COVID-19. Furthermore, it can be seen that 190 reducing ξ i and ξ a will reduce R 0 , which indicates that the better the reduction of contact from infected humans being isolated in the hospital or at home is able to reduce R 0 . The most negative indices of R − 0 is given by µ. However, this parameter can not be changed in the field. It can be seen that the most negative indices parameters that controllable in the field is ν, which describe the rate of the rapid test. Therefore, we can conclude that more massive the government to find the asymptomatic individual, 195 and then ask them to do independent isolation at home will reduce R 0 . Furthermore, we can see that increasing the additional recovery rate caused by hospitalization (γ 1 ) will reduce R 0 . Figure 3 presents an area for a combination between ν and β that will determine the size of R 0 . It can be seen that the increased value of β will increase R 0 , while an increased value of ν will reduce R 0 . The area of β can differ into three intervals. The first interval is when β ∈ (1.010006455 × 10 −7 , ∞). In this interval, R 0 is always larger than unity for all value of ν between 0 and 1. This means that the intervention of rapid tests will not make the COVID-19 free equilibrium stable. The second interval is when β ∈ (5.421388711 × 10 −8 , 1.010006455 × 10 −7 ). In this area, a combination of ν and β should be considered carefully to reach the condition of a stable COVID-19 free equilibrium point. For more precisely, for a random β = β 0 in the second interval, it needs ν > ν * where

.

For an example, if β 0 = 0.9 × 10 −7 , then it requires ν > 0.5630546276 to achieve a stable COVID-19 free equilibrium point. The last interval is when β ∈ [0, 5.421388711 × 10 −8 ). In the third interval, the intervention of rapid test is not needed to reach a stable COVID-19 free equilibrium point. However, 200 giving a positive value of ν will accelerate the time needed to achieve a stable COVID-19 free equilibrium point.

In the other hand, when b > b * , then system (1) undergoes forward bifurcation in R 0 = 1. Using estimated parameters value 205 from previous section, we have that R 0 = 1 when β = 8.599684570 × 10 −8 , and b * = 1/573. Therefore, we have that our system (1) undergoes forward bifurcation when R 0 = 1 since b = 1/10000, which illustrated in Figure 4 (a) . On the other hand, backward bifurcation appears when we choose b = 1/100, which describes a low capacity of the hospital to take care of the infected individual. This is illustrated in Figure 4 

We compute the local sensitivity for our suggested model equations of the COVID-19 in system (1). Computational results here are obtained using three different techniques: non-normalizations, half normalizations and full normalizations using SimBiology Toolbox for MATLAB ; see results show that the exposed infected, asymptomatic undetected individuals, asymptomatic detected individuals, symptomatic infected and recovered individuals are more sensitive to the set of parameters {β, p, α, ν, γ 0 } while they are less sensitive to the other model parameters, see Figures 7a and 7b . This gives us how public health partners pay more attention priority on interventions for such groups.

As a result, identifying critical model parameters in this study based on computational simulations 225 is an effective way to further study the model practically and theoretically and give some suggestions for future improvements of the COVID-19 transmissions, interventions and controlling the spread of disease. It can be concluded that the contact between person-to-person, transmission rate between exposed and asymptomatic, progression rate of incubation period, the effort for early detection test and natural recovery rate may have a great role in controlling this disease. The other factors have also role to infect 230 people in different levels, this is clearly occurred in our computational simulations. 6. Analyzing the plan for gradual relaxing of strict social distancing in the city of Jakarta, Indonesia

In Figure 8 , we present estimation and actual incidence data of COVID-19 in Jakarta, from March 3 until May 10, and then the simulation continues for a longer period of time. Note that policy from the government to conduct physical and social distancing conducted on April 10, 2020. Since April 10, the reduction of incidence occurs significantly. The physical distancing in Jakarta called PSBB (in Indonesian: Pembatasan Sosial Berskala Besar (large-scale social restriction)). It can be seen that if social distancing intervention maintained for a longer period of time, then the outbreak of COVID-19 in Jakarta will be reduced significantly, and delayed. With this intervention, the hospital can treat infected 240 individuals maximally. Recently, the Jakarta government planning to relax the strict social distancing policy. This policy will be conducted in five-phase: Based on this description, it is assumed that at the transmission rate will increasing step by step from β = 1.05 × 10 −7 (when social distancing implemented in April 10) to β = 1.44 × 10 −7 (early infection period of data). To handle this, we assume β as a step function as follows : 

Using above β(t), the dynamic of I and A d (t) is given in Figure 9 . It appears that the policy of relaxing the strict social distancing can result in an increasing number of new infections. However, it will not be as it was before social distancing was implemented. The possible explanation is because of the policy to relaxing the social distancing is too early to take place. 

Based on the elasticity analysis of R 0 in the previous section, the rapid test is one of a promising alternative for the eradication of COVID-19. Therefore, we simulate how if the policy to relaxing the strict social distancing combined with more rapid test intervention. To simulate this scenario, we use the same β as in (11) , but increasing the rapid test and hospitalization rate twice larger. The results are shown in Figure 10 . It can be seen that increasing rapid tests and hospitalization as a tolerance policy of relaxing 260 the social distancing success to reduce the number of the infected population. Unfortunately, when the social distancing completely stops (after July 27), then the effect of rapid test and hospitalization no longer able to compensate for the impact of relaxing the strict social distancing in a purpose to reduce the spread of COVID-19. Therefore, the number of the infected population start to re-increase and produce a new outbreak. 265 Figure 10 : long-time simulation for prediction of incidence of COVID-19 in Jakarta with easing the social distancing policy combined with more massive rapid test and hospitalization.

A new deterministic compartmental model was constructed in this study to evaluate the spreading of COVID-19 among the human population. The model considers many important factors, such as hidden cases, rapid testing to trace hidden cases, limitation of medical resources, social distancing, quarantine/isolation, and parameter estimation for the incidence date from the city of Jakarta, Indonesia. 3. The model undergoes a backward bifurcation phenomenon when the associated R 0 is less than unity, and the saturation parameter for hospitalization (b) is larger than a specific threshold (b * ). This means that whenever the medical resources are insufficient (larger b), the risk of the appearance of a 280 backward bifurcation increases; this is related to the existence of the COVID-19-endemic equilibrium despite R 0 < 1. Therefore, the success of the intervention also depends on the initial condition when the response is implemented. The study shows that increasing the capacity of a hospital or providing a considerably better quality of treatment in the hospital increases the probability of avoiding a backward bifurcation.

Numerical experiments of the model based on the incidence data of the city of Jakarta suggest the following.

1. The basic reproduction number in Jakarta during the early spread of COVID-19 is 1.75, which is larger than unity. This means that COVID-19 will persist in the population if no intervention is implemented by the government or the community. 2. From an analysis of the elasticity of R 0 , we observe that the infection rate (β) is the most significant controllable parameter to reduce R 0 , followed by the effectiveness of self-isolation and quarantine. Smaller values reduce R 0 , thereby increasing the chance of eradicating COVID-19 from the community.

3. The government must be careful when relaxing the policy of strict social distancing, particularly in 295 terms of when it should be initiated. Mistakes in the prediction of when to start relaxing the social distancing policy can affect the emergence of a second outbreak.

A rapid test-based intervention has been proven to have potential in reducing R 0 as an alternative approach, instead of relying solely on a lockdown or strict social distancing. Significantly better results might be obtained if these interventions can be implemented simultaneously.

During this pandemic, it is important to avoid overconfidence in the capabilities of the model for the long term prediction of the data. Many assumptions were made in the study to simplify the model without compromising the main objective. Although many important qualitative features were found from the model applied in this study, several limitations can still be found, and an alternative way to improve the model should be developed. One of the limitations in this study is that the applied model 305 does not include the spatial spread of COVID-19 and the possibility of a relapse for recovered individuals. Further research is required in this field to address this limitation and a better modeling is needed to understand and anticipate the outcome of the COVID-19 pandemic. 

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The basic reproduction number R 0 is calculated here with taking E(t), A u (t), A d (t), I(t) as the infected compartments and then using notation in [20] We define T as the transmission matrix and Σ as the transition matrix. The transmission and transition matrices of the corresponding linearized subsystem are four-dimensional, with

It can be seen that T has three zero rows in row 2, 3 and 4. Therefore, the auxiliary matrix E is given 320 by E = [1, 0, 0, 0] T . Hence, we have the next-generation matrix is given by

The basic reproduction number as the spectral radius of K is given by

The COVID-19 endemic equilibrium point is given by

,

,

and 325 k s = p (ξ a ν + η + µ + γ 0 ) (I * bµ + I * bφ + I * bγ 0 + µ + φ + γ 0 + γ 1 ) + I * bη µ ξ i + η µ ξ i , k r = η µ γ 0 + η ν γ 1 + η γ 0 2 + η γ 0 γ 1 + µ 2 γ 0 + µ ν γ 0 + 2 µ γ 0 2 + ν γ 0 2 + ν γ 0 γ 1 + γ 0 3 + γ 0 2 γ 1 + ν γ 0 .

where I * is taken from the positive roots of P(I).

To show the possible existence of the COVID-19 endemic equilibrium point when R 0 < 1, we will analyze the sign of ∂I ∂R0 when R 0 = 1 and I = 0. If the sign is negative, then we have at least one COVID-19 endemic equilibrium point when R 0 < 1 but close to 1. First step, let rewrite each coefficient of a i in P(I) as a function of R 0 . TO do this, let

Substitute this β * into a i , then taking the implicit derivative of I respect to R 0 from P(I), we get :

.

is always negative , then we have that ∂I ∂R0 < 0 ⇐⇒ a 1 (R 0 ) < 0, or equivalently b < b * where b * = β((1 − p) ξ i (µ + γ 0 ) (η + µ + ν + γ 0 ) − p (µ + φ + γ 0 + γ 1 ) (ν ξ a + η + µ + γ 0 ) − η ν ξ i )k b1 (µ + δ) ((1 − p) (µ + γ 0 ) (η + µ + ν + γ 0 ) − η ν) k b2 and β = β * . with k b1 = (η + µ + ν + γ 0 ) ((1 − p) α δ φ γ 0 + α δ µ γ 1 p) + (µ + φ + γ 0 + γ 1 ) (µ + ν + γ 0 ) η µ 2 + µ 2 (µ + γ 0 ) + α η µ + γ 0 (φ + γ 0 + γ 1 ) (ν + γ 0 ) + α δ µ 2 + µ ν + µ φ + µ γ 0 + ν φ η + µ (µ + φ + γ 0 ) (µ + ν + γ 0 ) + (µ + ν + φ + 2 γ 0 + γ 1 ) η + µ 2 µ + η (φ + γ 0 + γ 1 ) (ν + γ 0 ) + α 2 µ 5 γ 0 2 γ 1 + α µ 3 ν + α µ 3 φ + 3 α µ 3 γ 0 + α µ 3 γ 1 + α µ γ 0 3 + α µ 2 ν φ + 2 α µ 2 ν γ 0 + α µ 2 ν γ 1 + 2 α µ 2 φ γ 0 + 3 α µ 2 γ 0 2 + 2 α µ 2 γ 0 γ 1 + α µ ν φ γ 0 + α µ ν γ 0 2 + α µ ν γ 0 γ 1 + α µ φ γ 0 2 k b2 = 2 (1 − p) Λ α ξ i (µ + γ 0 ) (η + µ + ν + γ 0 ) − Λ α (2 µ + 2 φ + 2 γ 0 + γ 1 ) (ν ξ a + η + µ + γ 0 ) p + µ (2 µ + 2 φ + 2 γ 0 + γ 1 ) (µ + ν + γ 0 ) (µ + η + γ 0 ) (µ + α) .

we have ω 1 = ω 5 α δ (µ + η + ν + γ 0 ) (µ pγ 1 + (1 − p) φ γ 0 ) + C ((p − 1) (µ + γ 0 ) (µ + η + ν + γ 0 ) − ν η) (δ + µ) α µ , ω 2 = (γ 0 + γ 1 + µ + φ) (µ + ν + γ 0 ) (η + µ + γ 0 ) ω 5 α (ν η + (1 − p) (µ + γ 0 ) (µ + η + ν + γ 0 )) , ω 3 = (η + µ + γ 0 ) ω 5 p (γ 0 + γ 1 + µ + φ) ν η + (1 − p) (µ + γ 0 ) (µ + η + ν + γ 0 ) , ω 4 = ω 5 ν p (γ 0 + γ 1 + µ + φ) ν η + (1 − p) (µ + γ 0 ) (µ + η + ν + γ 0 )

,

− ω 5 ((µ + η) (µ + ν) γ 0 − η ν γ 1 − ((+µ + ν + η) γ 1 ) γ 0 ) ν η + (1 − p) (µ + γ 0 ) (η + µ + ν + γ 0 ) (δ + µ)

where C is long expression with positive sign, and 340 v 1 = v 6 = 0, v 2 = (1 − p) pξ i (µ + γ 0 ) (µ + η + ν + γ 0 ) + η ν ξ i + p (γ 0 + γ 1 + µ + φ) (ν ξ a + η + µ + γ 0 ) ξ i (µ + ν + γ 0 ) (η + µ + γ 0 ) (α + µ) v 3 = γ 0 2 + (ν ξ a + η + 2 µ + φ + γ 1 ) γ 0 + µ 2 + (ν ξ a + η + φ + γ 1 ) µ + (ν ξ i + φ + γ 1 ) η + ν ξ a (γ 1 + φ) v 5

Next we will calculate the values of A and B. Since v 1 = v 6 = 0, we only need to compute the partial derivatives of f 2 , f 3 , f 4 , f 5 at the DFE. For system (D.1) the associated non-zero partial derivatives of f 2 , f 3 , f 4 , f 5 are given by ∂ 2 f 2 ∂x 1 ∂x 3 = ∂ 2 f 2 ∂x 3 ∂x 1 = β, ∂ 2 f 2 ∂x 1 ∂x 4 = ∂ 2 f 2 ∂x 4 ∂x 1 = β ξ a , ∂ 2 f 2 ∂x 1 ∂x 5 = ∂ 2 f 2 ∂x 5 ∂x 1 = β ξ i , ∂ 2 f 5 ∂x 5 ∂x 5 = 2 γ 1 b.

= v 2 ω 1 2 ν p (γ 0 + γ 1 + µ + φ) β ξ a ν η + (1 − p) (µ + γ 0 ) (µ + η + ν + γ 0 ) + 2 (η + µ + γ 0 ) p (γ 0 + γ 1 + µ + φ) β ν η + (1 − p) (µ + γ 0 ) (µ + η + ν + γ 0 ) + v 2 ω 1 2 β ξ i + 2 γ 1 b.

For the sign B, the association non-vanishing partial derivatives of f 2 , f 3 , f 4 , f 5 are

It also follows that

= v 2 (γ 0 + γ 1 + µ + φ) (µ + ν + γ 0 ) (η + µ + γ 0 ) Λ α (ν η + (1 − p) (µ + γ 0 ) (µ + η + ν + γ 0 )) µ + ν p (γ 0 + γ 1 + µ + φ) Λ ξ a (ν η + (1 − p) (µ + γ 0 ) (µ + η + ν + γ 0 )) µ

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The coefficient B is clearly positive; the presence of backward bifurcation in the model (1) is determined by the sign of coefficient A. Therefore, to conduct a backward bifurcation A should be positive, which gives us b > b * where b * is given in Theorem 3.

The backward bifurcation is shown using the concept of center manifold theory on system (1) . 330 To use the center manifold theory, we make the following changes in variables. Let S = x 1 , E = x 2 , A u = x 3 , A d = x 4 , I = x 5 and R = x 6 and let β the bifurcation parameter. Denote x = (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ) T and dx dt = (f 1 , f 2 , f 3 , f 4 , f 5 , f 6 ) T as given belowThe Jacobian of system (D.1) evaluated at the disease-free equilibrium, is given byConsider the case R 0 = 1. Suppose that β is chosen as a bifurcation parameter. Setting R 0 = 1 and solving for β givesThe Jacobian matrix D x f evaluated at β = β * has a simple zero eigenvalue and the other eigenvalues 335 having negative real parts. Computing the right and the left eigenvector of D x f . The right and the left eigenvector associated with zero eigenvalue denoted respectively by ω = (ω 1 , ω 2 , ω 3 , ω 4 , ω 5 , ω 6 )

