key: cord-0334674-vd92639g
authors: Anggriani, Nursanti; Ndii, Meksianis Z.; Amelia, Rika; Suryaningrat, Wahyu; Aji Pratama, Mochammad Andhika
title: A mathematical COVID-19 model considering asymptomatic and symptomatic classes with waning immunity
date: 2021-05-14
journal: nan
DOI: 10.1016/j.aej.2021.04.104
sha: 60dd36fa5ddfe2786338d73273f1f1c070b619c9
doc_id: 334674
cord_uid: vd92639g

The spread of COVID-19 to more than 200 countries has shocked the public. Therefore, understanding the dynamics of transmission is very important. In this paper, the COVID-19 mathematical model has been formulated, analyzed, and validated using incident data from West Java Province, Indonesia. The model made considers the asymptomatic and symptomatic compartments and decreased immunity. The model is formulated in the form of a system of differential equations, where the population is divided into seven compartments, namely Susceptible Population ( S 0 ) , Exposed Population ( E ) , Asymptomatic Infection Population ( I A ) , Symptomatic Infection Population ( Y S ) , Recovered Population ( Z ) , Susceptible Populations previously infected ( Z 0 ) , and Quarantine population ( Q ) . The results show that there has been an outbreak of COVID-19 in West Java Province, Indonesia. This can be seen from the basic reproduction number of this model, which is 3.180126127 ( R 0 > 1 ) . Also, the numerical simulation results show that waning immunity can increase the occurrence of outbreaks; and a period of isolation can slow down the process of spreading COVID-19. So if a strict social distancing policy is enforced like a quarantine, the outbreak will lessen.

The spread of COVID-19 has shocked society and currently has transmitted to more than 200 countries [1] . As of 04 April 2021, there are 130,998,190 confirmed cases, 2,853,280 death, and 105,447,782 recovered individuals [2] . It has caused severe economic and social loss. The disease has been transmitted from human to human via droplets [3] . Infected individuals may show symptomps such as fever, cough, sore throat, rhinorrhea, myalgia or fatigue, phlegm, and headache [3] [4] [5] with the body temperature of 39 • C or above [5] . Individuals who are infected by COVID-19 can show symptoms (symptomatic) or cannot show symptoms (asymptomatic) but both types of individuals can transmit disease [3] . The incubation period has been estimated between two two fourteen days [6] .

Research showed that there is possibility for infected individuals to be reinfected by COVID-19. Currently it has been found that several recovered individuals have been re-infected by COVID-19 and this can cause death from fatal heart failure [7] . Of the 111 recovered patients, 5% of China and 10% of South Korea tested positive for COVID-19 [8] . This situation contradicts the fact that after a person catches the virus and then recovers, the individual will forman antibody that prevents the same virus from attacking twice. Research showed that reinfected individuals have experienced viral replication but did not neutralize antibodies, which implies that it is unlikely that long-term protective immunity will occur in people with COVID-19 after the first infection [9] . The virus's immune response can be reduced within four months to one year after infection [10] . The genetic basis of the innate immune response affects the severity of COVID-19, it can also lead to more severe reinfection depending on antibodies generated against the bound virus but cannot 2 neutralize the same strain [10] . The reinfection COVID-19 case has a more severe impact [11] . The reinfection occurs due to the decrease in the individual's immunity [12] . Understanding the effects of waning immunity is important.

Mathematical models can be used to understand the complex phenomena such as population dynamics problem [13] [14] [15] [16] and disease transmission dynamics [17] [18] [19] [20] [21] . A compartment-based epidemic model in the form of system of (fractional or integer) differential equations has been formulated to understand disease transmission dynamics, where the human population is divided into different stages according to their status to the diseases [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] . A mathematical SEIR model is mostly used as a basis for the model's development for transmission [37, 38] . The SEIR model has been extended to include quarantine compartment [39] , to include symptomatic and asymptomatic classes [40] . The models are mostly used to investigate the disease transmission dynamics in several countries or provinces such as Indonesia [41] , Hubei has been researched [42] , Pakistan [43] . In this paper, a modified SEIR model considering symptomatic and asymptomatic cases from [44] has been formulated. The work focuses on studying the effects of waning immunity. The model is validated against data of COVID-19 incidence from West Java Province. The basic reproduction number is calculated, and a global sensitivity analysis is performed. The model is then used to determine he effects of waning immunity or reduced immunity to an increase in the number of infected individuals.

The remainder of this paper is organized as follows. In Section 2, the construction of the SEIR compartmental model. Next, the model's mathematical properties, such as the equilibrium points, Basic Reproduction Number, and the existence of backward bifurcation, are detailed in Section 3. In Section 4, we explain the real-world problem using the incidence data of West Java Province, Indonesia. A discussion on the Basic Reproduction Number and the sensitivity analysis results are provided in Section 5. Finally, some conclusions are presented in Section 6. 3 

We developed model of transmission of COVID-19 by considering asymptomatic and symptomatic compartments and decreased immunity. The total population is divided into Susceptible population (S 0 ), Exposed population (E), Asymptomatic infected population (I A ), Symptomatic infected population (Y S ), Recovered population (Z), Susceptible that previously infected (Z 0 ), Quarantine population (Q). The total number of population at time t is given by:

The assumption used in the formulation of a mathematical model for the spread of the COVID-19 disease is that individuals with symptoms will undergo hospitalization or quarantine. Deaths experienced by latent, symptomatic, asymptomatic, and quarantine individuals are caused by disease [45] . This means that the death of the three individuals is a combination of natural death and death due to disease. We assume that the µ 1 parameter contained in compartments E, I a , Y s is a death caused by COVID-19 plus a natural death factor.

People who have decreased immunity can catch COVID-19 again with a high severity [11] . Hence, the second person infected will develop symptoms and be hospitalized. The model is represented by the diagrams shown in Figure 1 , with the description of the parameters given in Table 1 . The probability of exposed people become infected p The proportion of exposed people become infected κ The rate of asymptomatic infected people become infected symptomatic q

The rate of quarantine

The natural recovery rate of infected asymptomatic people 

of system (1-7) are positif for all t > 0.

Proof. Assume that

Clearly, X (0) > 0.

Assuming that there exist a t 1 > 0 such that

From the equation of model (1), we can obtain

Thus, we have

which will be positive since exponential functions and initial solutions S 0 (0)are non-negative. Thus, S 0 (t) > 0 for all t ≥ 0.

Similarly, we can also prove that 

We get:

Assume that µ = µ 1 , to simplify the analysis process.

Then:

Thus we have

so all solutions of system (1-7) are ultimately bounded for all t ∈ [0, t 0 ].

The non-endemic equilibrium point of the COVID-19 disease model is obtained by setting I A = 0, E = 0, Y S = 0, and substituting it into (1-7) to obtain:

Theorem 3.3. The non-endemic equilibrium point of system (1-7) is locally asymptotically stable whenever it exists.

Proof. By following Diekmann (2000) [46] substituting P 0 from 8 into the Jacobian matrix for the non-endemic equilibrium point is obtained: 2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63 64

The characteristics of the polynomial is

From the polynomial (P(λ)) we get λ 1 = −µ and for λ i with i = 2, 3, . . . , 7

will be negative if a j > 0 where j = 0, 1, 2, . . . , 6, R 0 < 1, a 1 a 2 > a 0 a 3 , a 1 (a 2 a 3 + a 0 a 5 ) > a 2 1 a 4 + a 0 a 2 3 , and a 1 a 2 a 4 > a 0 (a 1 a 6 + a 2 a 5 ). Since the coefficients in the characteristic equation P 1 (λ) are complex, we proceed to analyze the coefficient values numerically with β 1 = β 2 . The results of the numerical analysis obtained (see Appendix A.), show that for λ i with i = 2, 3, . . . , 7 negative. Because λ j with j = 1, 2, 3, . . . , 7 is negative, it can be concluded that the non-endemic equilibrium point of the system (1-7) is locally stable, so Theorem 3.3 is proven.

The Basic Reproduction Ratio (R 0 ) is an important number in epidemiology, which is defined as the number of secondary infections caused by one primary infection in a population. We use the next-generation method to determine R 0 , the value of R 0 an be obtained by finding the dominant eigenvalue F V −1 .

Where F and V are Jacobian matrices of f (newly infected matrices) and v (exiting matrices) that are evaluated at the disease-free equilibrium point (P 0 ) 9 1 2 3 4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 from 8. From the models (1-7) are obtained:

The eigenvalues of (F V −1 ) are: 

Under certain conditions where the probability of transmission from infected people same as from asymptomatic infected people hold β 1 = β 2 and the natural recovery rate of infected people asymptomatic and symptomatic γ 1 = γ 2 = γ.

It obtained the reproduction number for this condition symbolized by R 0β .

Where:

Theorem 3.4. An endemic equilibrium point of system

and H < 0. 10   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 Proof. The endemic point of this disease is endemic in certain areas for a certain period, which releases the COVID-19 in the population. It is indicated by the presence of compartments exposed to virus transmission E * , I * a , Y * S at steady state. By calculating model (1, (5) (6) (7) and setting the right hand side zero we obtained:

By substituting S * 0 , I * A , Y * S , Z * 0 , Z * 0 , Q * to equation (2.2) and set the right hand side equal to zero, obtained:

which this polynomial have to roots E = 0 or E = E * which can be written by

where G and H written on Appendix B. .

Because the denomerator of H always positif, the steady state E * will exsist if R 0β > 1 and pq > 0, see attachment for the proof. The system of equations 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62 63 64

Theorem 3.5. The endemic equilibrium point of the system (P 1 ) is locally asymptotically stable whenever it exists.

Proof. Following method on the proof of Theorem 2, Based on the method of proof of Theorem 2, by substituting P 1 is obtained characteistic polynomial

From the polynomial (Q(λ)) we get λ i with i = 1, 2, 3, . . . , 7 will be negative if a j > 0 where j = 0, 1, 2, . . . , 7, a 1 a 2 > a 0 a 3 , a 1 (a 2 a 3 + a 0 a 5 ) > a 2 1 a 4 + a 0 a 2 3 , a 1 a 2 (a 3 a 4 + a 0 a 7 ) > a 0 a 3 (a 1 a 6 + a 2 a 5 ), and a 2 a 5 > a 0 a 7 . Since the coefficients in the characteristic equation Q(λ) are complex, we proceed to analyze the coefficient values numerically. The results of the numerical analysis obtained can be seen in the Appendix C. It satisfies the Routh-Hurwitz's criteria so that the endemic point is locally asymptotically stable whenever it exists. These results will remain consistent using the parameter values in Table 2. 3.6. Global Stability of the equilibria Theorem 3.6. The non-endemic equilibrium point (P 0 ) is globally asymptoti-

Proof. Refer to global proving by Tewa et al. (2009) [48] , let Define the Lyapunov function

Differentiating with respect to time yields 12   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63 64

The value of dV dt will be negative if β 1 Λ µ < µ 1 and β 2 Λ µ < µ 1 .

By following LaSalle's extension on Lyapunov's method [49] , disease-free equilibrium P 0 is globally asymptotically stable.

This concludes the proof.

This section presents a global sensitivity analysis of the model. We use the combination of Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coefficient (PRCC) to determine the most influential parameters of the model [50] . LHS is stratified sampling without replacement. The parameter distribution is divided into equation probability intervals and then is sampled. Each 13   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62 of sensitivity analysis is given in Figure 2 .

It showed that the waning immunity (ξ) is one of the influential parameters.

When the value of waning immunity increases, the number of infected individuals also increases. This means that waning immunity would contribute to the increasing number of infected individuals. Therefore, an analysis of effects of 14   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 waning immunity is of importance. The parameters k, p are also influential and has positive relationship. This means that the rate of asymptomatic become infected and the proportion of exposed individuals become infected contributes to an increasing number of infected individuals. When these parameter values increases, the number of infected individuals decreases.

In this section, we estimated the parameters β 1 ,β 2 , and γ against data Table 2 . The lsqnonlin built-in function in MATLAB is used for the parameter estimation. 15   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 We minimize the sum of squared error as

where Q t is the number of active cases of Q up to day t, respectively, while g t (x) is the number of active cases for Q up to day t from the model's solution,

respectively. The transmission rate, β 0 and β 1 , the quarantine rate q are then estimated using the "lsqnonlin" built-in function in MATLAB. The case fatality rate is estimated using the linear regression method.

The initial conditions used Table 3 

This numerical simulation is designed to support the results of the analysis discussed in the previous section. We set the parameter by curve fitting from actual case of COVID-19 in West Java Province, Indonesia. We applied Runge-Kutta-Fehlberg (RKF) method in MAPLE software, to solve the ordinary differential equations of model 1-7 using the parameters in Table 2 and Table 3 . RKF method is one of the most popular numerical approach because it is quite accurate, stable, to program [51] . 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62 63 64 2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62 63 64 where the number reaches 70 and after that decreases. At the end of the 400 th day, the number of cumulative quarantine reaches 3200.

In this section, we simulate the sensitivity analysis for the effect of parameter ξ, related to waning immunity issue, which describes the probability rate of recovered people become susceptible, and the probability rate of susceptible people that previously infected become asymptomatic infected, respectively.

Using the parameters and initial values in Table 2 and Table 3 , except for ξ, we choose ξ = 0.001, 0.01, 0.1, 1. 18   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62 63 64 However, the higher the value of the parameter ξ, the higher the population 19   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62 63 64 of Y S after passing the peak of the spread, and the lower the population Z.

When this parameter is greater, the recovered population (Z) decreases due to the loss of immunity and returns to the susceptible population that previously infected (Z 0 ). Where population Z 0 can be re-infected to become asymptomatic infected population (Y S ). 20   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62 63 64 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 the value of q. Meanwhile the population in the quarantine compartment (Q)

is increasing from population of Symptomatic Infected which did Quarantine. 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 7

We have formulated a mathematical model of COVID-19 transmission by considering infected individuals with symptoms and asymptomatic, as well as decreased immunity, validated with data from West Java Province, Indonesia.

The compartment-based model is formulated as a system of differential equa- 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 ing and rapid assessment: The case of Jakarta, Indonesia, Chaos, Solitons and Fractals, 139 (2020), 110042. 31   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 Appendix A. Proof of numerical analysis of Theorem 3.3

Coefficient polynomial (P 1 (λ)) : Appendix B. Characteristic polynomial of Exposed Compartment G = √ K + (((−µ 1 2 γ + ((−q − γ)δ − κ γ − γ 2 )µ 1 + (−γ 2 + (−κ − q)γ − κ q)δ)µ + β Λ µ 1 2 + (δ β Λ + Λ β γ + (κ + qp)Λ β)µ 1 + (Λ β γ + (2 qp + κ)Λ β)δ)ξ + (β Λ µ 1 2 + (δ β Λ + Λ β γ + (κ + qp)Λ β)µ 1 + (Λ β γ + (κ + qp)Λ β)δ)µ)α + (−µ 1 3 γ + ((−q − γ)δ − κ γ − γ 2 )µ 1 2 + (−γ 2 + (−κ − q)γ − κ q)δ µ 1 )µ ξ, 32   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 where K =((µ 1 γ + (q + γ)δ) 2 (κ + γ + µ 1 ) 2 (α + µ 1 ) 2 ξ 2 + 2 (κ + γ + µ 1 )(δ + µ 1 )β Λ α (α + µ 1 ) (−µ 1 2 γ + ((−γ + (2 p − 1)q)δ − γ (κ + γ + qp))µ 1 + δ (q + γ)(qp − κ − γ))ξ + β 2 α 2 Λ 2 (δ + µ 1 ) 2 (κ + γ + qp + µ 1 ) 2 )µ 2 + 2 ξ β Λ α ((κ + γ + µ 1 )(−µ 1 3 γ+ ((−2 γ + (2 p − 1)q)δ − γ (κ + γ + qp))µ 1 2 + ((−γ + (2 p − 1)q)δ − 2 γ 2 + ((−p − 1)q − 2 κ)γ + q(qp − κ))δ µ 1 − δ 2 (κ + γ)(q + γ))(α + µ 1 )ξ + (µ 1 2 + (κ + γ + δ + qp)µ 1 + δ (κ + γ))(δ + µ 1 )β Λ α (κ + γ + qp + µ 1 ))µ + ξ 2 (µ 1 2 + (κ + γ + δ + qp)µ 1 + δ (κ + γ)) 2 β 2 Λ 2 α 2 , and H ={2 α β δ pqξ (µ 1 + α)}.

. By substituting the parameter value from Table 2 Table 2 , except β 1 = β 2 = 0.7477942169036 × 10 −10 . 33   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62 63 64 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 

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The work was supported by Universitas Padjadjaran, with contract number

The authors declare that they have no conflict of interest.

Please check the following as appropriate:✓ All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version.

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The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript