From pandemic to a new normal: Strategies to optimise governmental interventions in Indonesia based on an SVEIQHR-type mathematical model.

There are five different forms of intervention presently realised by the Indonesian government in an effort to end the COVID-19 pandemic: vaccinations, social restrictions, tracings, testings, and treatments. In this paper, we construct a mathematical model of type SVEIQHR (susceptible-vaccinated-exposed-infected-quarantined-hospitalised-recovered) for the disease's spread in the country, which incorporates as parameters the rates of the above interventions, as well as the vaccine's efficacy. We determine the model's equilibria and basic reproduction number. Using the model, we formulate strategies by which the interventions should be realised in order to optimise their impact. The results show that, in a disease-free state, when the number of new cases rises, the best strategy is to implement social restrictions, whereas in an endemic state, if a near-lockdown policy is undesirable, carrying out vaccinations is the best strategy; however, efforts should be aimed not primarily towards increasing the vaccination rate, but towards the use of high-efficacy vaccines.

Introduction

Declared to be a pandemic by WHO on March 11, 2020 (Saxena, 2020, page 2), the coronavirus disease 2019 (COVID-19), reportedly originating from a seafood market in China (Saxena, 2020, page 13–14), has continued to be a global concern, with over 392 million cases recorded worldwide as of February 6, 2022 (WHO, 2022). In most countries, the initial evolution of the daily number of new cases is characterised by several successive waves (Muñoz-Fernández et al., 2021), the end of each wave seemingly indicating successfulness of certain eradicative interventions. Such successfulness was however only temporal in many countries where there occurred subsequent —often larger— waves, of which the emergence of new variants proved to be a principal cause (Page, 2021).

In Indonesia, since mid 2020, amid various disruptions caused by the disease, the government has popularised the term “a new normal” to refer to a desired form of post-pandemic life (Adjie, 2020). Its realisability, however, remained unclear. Indeed, the largest pre-omicron wave unfolded in mid 2021, the number of new cases reaching a new maximum of 56,767 on July 15, 2021 (Arief, Adji, & Suharto, 2021). The omicron variant then entered the country in late November 2021 (Ranggasari & Bhwana, 2021), before a subsequent wave emerged in January 2022 and the aforementioned maximum was surpassed as early as February 15, 2022, with 57,049 new cases (Rayda, 2022). Keeping the aim towards a new normal, the government of the country has been realising concrete eradicative interventions in the following five different forms.

Vaccinations. Indonesia's national vaccination programme commenced on January 13, 2021 (Triwadani, 2021), the first vaccinated citizen being the president, Joko Widodo, who received on the day a shot of Coronavac, a vaccine developed by China's Sinovac Biotech, approved for emergency use by the country's Food and Drug Monitoring Agency (BPOM) only two days earlier (Syakriah, 2021). The programme's progress has been tangible: by early 2022, around 45% and 21% of the population have been fully and partially vaccinated, respectively (WHO Indonesia, 2022), and 11 different vaccines with varying levels of efficacy were granted approval (COVID19 Vaccine Tracker). The reception of booster shots has also been urged (Afifa, 2022). Vaccinated citizens are provided with waivers from a number of health-related requirements1 (Junida, Ihsan, & Suharto, 2022).

Social restrictions. Besides quarantine regulations for citizens returning from abroad (Indonesia Expat, 2022), the government, in an effort to control the disease's transmission level, has set out four different levels of large-scale social restrictions (PPKM) (Departemen Dalam Negeri, 2021a, Departemen Dalam Negeri, 2021b; Saptoyo & Kurniawan, 2021; Double). Each level of social restrictions (1–4) defines a specific degree to which schools, shopping centres, public transport, etc., may operate. At any given time, every region is to implement one of these four restriction levels, carefully determined by the authority based on the region's present situation, using several indicators such as transmission and vaccination levels (Menteri Kesehatan Republik Indonesia, 2021). In particular, responding to the omicron wave, the government has re-raised the levels of social restrictions in various regions including Greater Jakarta, from 2 to 3 (Sucipto, 2022).

Tracings. Contact tracing, a procedure of interviewing a newly-diagnosed patient with the aim of identifying people who have been in contact with the patient within the last few days (WHO, 2021), has also been in operation, albeit initially at a suboptimal level. (In fact, one of the reasons for the re-raising of the social restrictions level at the start of the omicron wave was the lack of tracings (Sucipto, 2022).) Recently, however, there has been some increase in awareness and government's effort towards contact tracings, as means to break transmission chains (Hutasoit, 2022).

Testings. Since many COVID-19 patients are asymptomatic, determining whether a person —including travellers and those who have been in recent contact with a patient— is infected is best done via medical testings. At least three types of tests for COVID-19 are available in the country: the polymerase chain reaction test (PCR), the rapid antibody test, and the rapid antigen test (News Desk, 2020).

Treatments. Being a populous country, Indonesia has faced some considerable difficulty in optimising medical treatments for COVID-19 patients as hospitals became increasingly crowded during the country's largest pre-omicron wave in mid 2021 (WHO Indonesia, 2021). A number of makeshift hospitals were set up, so as to keep the overall bed-occupancy rate within a safe level (WHO Indonesia, 2020). Fortunately, such an effort appeared to be successful in maintaining an acceptable quality of treatment and preventing hospitals from being overwhelmed (Shofa, 2022a). At the start of the omicron wave, the government has, in addition, looked into providing citizens with antiviral medicines, securing 400,000 tablets of molnupiravir (Shofa, 2022b).

In the literature, one finds a large number of studies on the spread of COVID-19, with or without mathematical models, which take into consideration some of the above forms of governmental intervention. In the first place, with regards to vaccinations, studies have confirmed the importance of using high-efficacy vaccines (Dashtbali & Mirzaie, 2021; Moghadas et al., 2021; Sadarangani et al., 2021), starting vaccinations as early as possible (Albani, Loria, Massad, & Zubelli, 2021a, 2021b; Amaku et al., 2021a), and devising vaccination strategies such as age-based prioritisations (Bubar et al., 2021; LimaCampos, Cysne, Madureira, & Mendes, 2021). Some studies have also looked into the necessity of considering the possibility that the number of infection cases is underreported, in the determination of appropriate vaccination strategies (Albani et al., 2021a, 2021b; Gibbons et al., 2014; Lau et al., 2021; Saberi et al., 2020). On the other hand, the impact of social restrictions (Aldila et al., 2020a; Choi & Ki, 2020; Mwalili, Kimathi, Ojiambo, Gathungu, & Mbogo, 2020), as well as that of tracings, testings, and treatments (Alharbi, Alqahtani, Albalawi, & Bakouri, 2020; Amaku et al., 2021b; Aronnaa, Guglielmi, & Moschen, 2021; Chen, Fang, & Huang, 2021 ; Cui, Ni, & Shen, 2021; Gatto & Schellhorn, 2021; Grimm, Mengel, & Schimdt, 2021), have also been a subject of numerous studies. In some of these studies, the employed mathematical models were made more realistic by assuming certain key parameters to be time-dependent, thereby allowing the implementation of parametric fittings which use actual daily data and accommodate fluctuations caused by factors which are not explicitly incorporated to the model (Albani et al., 2021c; Amaku et al., 2021a, 2021b; Aronnaa et al., 2021; Calvetti, Hoover, Rose, & Somersalo, 2020; Cui et al., 2021; LimaCampos et al., 2021).

In (Yong et al., 2022a), we have constructed a SIR-type mathematical model for the spread of COVID-19 in Indonesia, which incorporates —among others— a parameter measuring the aforementioned hospitals’ bed-occupancy rate. We have also constructed a discrete version of this model, in which some parameters were assumed to be time-dependent, and use it to design a quantitative method for determining the appropriate level(s) of social restrictions to be enforced in Jakarta on any given day (Yong et al., 2022b). In this paper, taking into consideration the above five forms of intervention, as well as the idea proposed in (Mu ñ oz-Fern á ndez et al., 2021, section 4) of incorporating more compartments and the possibility of reinfection, we aim to construct a new model which is more comprehensive and realistic, with the hope of formulating strategies by which the above forms of intervention should be realised in order to optimise their impact, so that a new normal can be embraced as soon as possible.

The incorporation of additional compartments implies that the present model is no longer SIR-type. Indeed, we shall take into account, at any given time t ⩾ 0, the numbers S = S(t) of (unvaccinated) susceptible individuals, V = V(t) of (susceptible) vaccinated individuals, E = E(t) of (non-transmitting) exposed individuals, I = I(t) of infected individuals, Q = Q(t) of quarantined individuals, H = H(t) of hospitalised individuals, and R = R(t) of recovered individuals, thereby building a seven-compartment SVEIQHR-type model. We assume that all quarantines are centralised, so that the quarantined —as well as the hospitalised— individuals are never in contact with the susceptible and vaccinated individuals, meaning that only the infected individuals transmit the disease. We also assume that social restrictions are waived for vaccinated citizens2; see, e.g., (Junida, Ihsan, & Suharto, 2022). The above five forms of intervention shall be incorporated to the model as parameters u1, u2, u3, u4, and u5, all belonging to [0, 1], which represent, respectively, the rates of vaccine, mobility, contact-tracing, rapid-testing, and treatment interventions.3 We shall also incorporate a parameter δ ∈ [0, 1] representing the vaccine efficacy. These and all other parameters, together with their values used in our numerical analysis, are described in Table 1.

Table 1

Parameters used in the model (1) and their values chosen for our numerical simulations (section 3).

Parameter	Description	Unit	Value for simulation	Source
λ	recruitment rate of newborns	individual/day	27352362165×365	estimated as μ N(0) World Bank (2020)
λ′	recruitment rate of foreigners	individual/day	3000	Indonesia Expat (2021)
μ	natural death rate	1/day	165×365	Aldila, Samiadji, Simorangkir, Khosnaw, and Shahzad (2021)
μ′	death rate increment due to COVID-19	1/day	0.0291	Gugus Tugas Percepatan Penanganan COVID
β	transmission coefficient	1/(individual × day)	4.74396 × 10−8	Aldila et al. (2021)
δ	vaccine efficacy	dimensionless	see subsection 3.1
α	temporary immunity rate	1/day	0.011	Shakhany and Salimifard (2021)
θ	incubation rate	1/day	0.4	Prem et al. (2020)
γ	recovery rate of infected individuals	1/day	0.1	Ferguson et al. (2020)
φ	recovery rate of hospitalised individuals	1/day	0.8198	Gugus Tugas Percepatan Penanganan COVID
κ	recovery rate of quarantined individuals	1/day	0.1	Ferguson et al. (2020)
τ	hospitalisation rate of quarantined individuals	1/day	0.01	Aldila et al. (2020b)
u1	vaccination rate	1/day	0.4	Diagne, Rwezaura, Tchoumi, and Tchuenche (2021)
u2	mobility intervention rate	dimensionless	see subsection 3.1
u3	contact-tracing intervention rate	1/day	0.5	assumed
u4	rapid-testing intervention rate	1/day	0.3	assumed
u5	treatment intervention rate	1/day	0.0833	Babaei, Jafari, Banihashemi, and Ahmadi (2021)

Let us now construct the model itself, by detailing the changes assumed to be experienced at any given time by each of the above seven time-dependent variables, which are summarised in the compartment diagram in Fig. 1.

Fig. 1

The compartment diagram of our SVEIQHR-type model.

The number S of susceptible individuals increases due to the entry of newborns at the rate λ > 0 and of recovered individuals at the rate αR, where α > 0, and decreases due to the exit of those who become exposed at the rate , where u2 ∈ [0, 1] and β > 0, vaccinated at the rate u1S, where u1 ∈ [0, 1], and dead at the rate μS, where μ > 0.

The number V of vaccinated individuals increases due to the entry of susceptible individuals at the rate u1S, and decreases due to the exit of those who become exposed at the rate (1 − δ)βVI, where δ ∈ [0, 1], and dead at the rate μV. Notice that the former does not contain the mobility restriction factor 1 − u2 as in the rate at which susceptible individuals become exposed; this manifests our assumption that social restrictions are waived for vaccinated citizens.

The number E of exposed individuals increase due to the entry of susceptible individuals at the rate and of vaccinated individuals at the rate (1 − δ)βVI, and decreases due to the exit of those who become infected at the rate θE, where θ > 0, quarantined at the rate u3E, where u3 ∈ [0, 1], and dead at the rate μE.

The number I of infected individuals increase due to the entry of exposed individuals at the rate θE, and decreases due to the exit of those who become recovered at the rate γI, where γ > 0, quarantined at the rate u4I, where u4 ∈ [0, 1], hospitalised at the rate u5I, where u5 ∈ [0, 1], and dead at the rate , where μ′ > 0.

The number Q of quarantined individuals increase due to the entry of foreigners at the rate λ′ > 0, of exposed individuals at the rate u3E, and of infected individuals at the rate u4I, and decreases due to the exit of those who become recovered at the rate κQ, where κ > 0, hospitalised at the rate τQ, where τ > 0, and dead at the rate μQ.

The number H of hospitalised individuals increase due to the entry of quarantined individuals at the rate κQ and of infected individuals at the rate u5I, and decreases due to the exit of those who become recovered at the rate φH, where φ > 0, and dead at the rate .

The number R of recovered individuals increases due to the entry of infected individuals at the rate γI, quarantined individuals at the rate κQ, and hospitalised individuals at the rate φH, and decreases due to the exit of those who become susceptible at the rate αR and dead at the rate μR.

We therefore obtain the model

The rest of the paper is organised as follows. In the upcoming section 2, we analyse the model (1) dynamically. We first establish the non-negativity and boundedness of its solutions, and determine a subdomain which is positively invariant under the model (subsection 2.1). Next, we show that, for every set of parameter values, the model possesses a unique disease-free equilibrium, and determine an explicit expression of this equilibrium (subsection 2.2). We also derive the model's basic reproduction number and show that, if , the disease-free equilibrium is stable, whereas if , the disease-free equilibrium is unstable and a unique positive endemic equilibrium exists (subsections 2.2 and 2.3).

As the algebraic computations required to establish further dynamical properties of the model —such as the endemic equilibrium's stability— appear to be inaccessibly complicated, we shift from analytical to numerical methods (section 3), whose flexibility allows us to achieve our ultimate goal: formulating strategies by which the aforementioned forms of governmental intervention (vaccinations, social restrictions, tracings, testings, and treatments) should be implemented for an optimal impact. The first stage of our analysis yields results which strongly point towards vaccinations, and more specifically, towards the importance of a high vaccine efficacy, in addition to the necessity of unwaiving social restrictions for vaccinated citizens (subsection 3.1). This is confirmed quantitatively in our second stage (subsection 3.2) via sensitivity analysis, from which we conclude that the optimal intervention strategy is to implement social restrictions in the case of , and, if a lockdown is undesirable, vaccinations using high-efficacy vaccines in the case of . These conclusions are reasserted in section 4, where we also describe a number of ways in which the model (1) could be modified for further research.

Dynamical analysis

Let us first analyse the model (1) from the viewpoint of dynamical systems theory; see (Martcheva, 2015; Robinson, 2012) for background. First, we establish the non-negativity and boundedness of the model's solutions associated to non-negative initial conditions, and the positive-invariance of a bounded subdomain (Theorem 1). Subsequently, we show that the model has a unique disease-free equilibrium for every set of parameter values, which is stable if and unstable if , where is the model's basic reproduction number (Theorem 2). Finally, we show that in the case of , in which the model's solutions do not approach the disease-free equilibrium, a unique positive endemic equilibrium exists (Theorem 3).

Non-negativity and boundedness of solutions

Let us first establish the non-negativity and boundedness of the solutions of the model (1) associated to non-negative initial conditions. Letbe such an initial condition, where , and let be the solution associated to this initial condition. For every t∗⩾0 satisfying , we have, from the model's first equation,which means that the function S is increasing at t∗. Since S(0)⩾0, it follows that S(t)⩾0 for every t ⩾ 0. Similar arguments show thatfor every t ⩾ 0.

Next, adding all equations in (1), one obtains that the time-dependent total population N≔S + V + E + I + Q + H + R satisfieswhich is equivalent to

Now, the functions N(t)e and have the same value at t = 0, namely, N(0), and, by (2), at every point, the slope of the former function does not exceed that of the latter function. Consequently, for every t ⩾ 0 we havei.e.,

This implies that the solution is bounded, and that the subsetis positively invariant (60, Definition 4.4) under the model.

We summarise our results in the following theorem.

Every solution of the model (1) associated to an initial condition in is bounded and remains forever in .

Every solution of the model (1) associated to an initial condition in remains forever in .

Disease-free equilibrium and basic reproduction number

Let us now study the equilibria of the model (1), i.e., the solutions of the system

We shall begin by showing that, for every set of parameter values, the model possesses a unique disease-free equilibrium, which admits an explicit description, and relating its stability to the model's basic reproduction number.

Let be a disease-free equilibrium of the model (1), i.e., a solution of (3) satisfying I0 = 0. The fourth equation in (3) gives E0 = 0. The fifth, sixth, seventh, first, and second equations then give, respectively, Q0, H0, R0, S0, and V0. In explicit form,where

The disease-free equilibrium e0 thus exists —since all its components are positive— and is unique, for every set of parameter values.

Let us now compute the model's basic reproduction number, using the so-called next-generation matrix method (van den Driessche & Watmough, 2002, page 33), taking into account as infected compartments those of exposed, infected, quarantined, and hospitalised individuals, whose numbers evolve at the rates given by the third, fourth, fifth, and sixth equations of the model (1). Letting , we first defineand

Next, we define the 4 × 4 matricesand

The basic reproduction number of the model (1) is the spectral radius of the model's next-generation matrix FV−1:

Therefore, the basic reproduction number grows only sublinearly with the vaccination rate u1. This means that, for the eradication of COVID-19, it is not advisable to strive only towards a high vaccination rate; indeed, many of the countries with high percentages of citizens vaccinated (Wolf et al., 2022) retain their pandemic status. Instead, since grows linearly with the mobility intervention rate u2, and with the vaccine efficacy δ, these parameters deserve more attention. In subsection 3.2, we shall confirm quantitatively that this is the case, i.e., that these are the parameters upon which depends most sensitively in the cases of and , respectively.

Direct computation shows that the characteristic polynomial of the Jacobian matrix of the model (1) evaluated at e0 is given bywheremeaning that − μ, − u1 − μ, − k3, − k4, and − k5 are five negative roots of P(x). Therefore, the equilibrium e0 is locally asymptotically stable if the other two roots x1 and x2, i.e., those of x2 + bx + c, have negative real parts (Robinson, 2012, Theorem 4.6(a)), and is unstable if at least one of x1 and x2 have a positive real part. Since b > 0, the former holds if c > 0 (by the Routh-Hurwitz criterion (Allen, 2007, section 4.5)), while the latter holds if c < 0 (in which case x1 and x2 are real and have opposite signs). Direct computation shows that c > 0 is equivalent to , while c < 0 is equivalent to . This proves the following theorem.

For every set of parameter values, the model (1) has a unique disease-free equilibrium, which is locally asymptotically stable if , and unstable if .

Endemic equilibria

Let us now seek all equilibria with I ≠ 0, . First, solving the fifth equation in (3) for Q gives

Substituting (6) into the sixth equation in (3) and solving the resulting equation for H gives

Next, substituting (6) and (7) into the seventh equation in (3) and solving the resulting equation for R gives

Subsequently, substituting (8) into the first equation in (3) and solving the resulting equation for S gives

In addition, substituting (9) into the second equation in (3) and solving the resulting equation for V gives

Solving the third and fourth equations in (3) for E gives, respectively,

Equating the two equations in (11) yields

Equating this and (10) yields the following expression of S as a function of I:

On the other hand, substituting the second equation in (11) and into (9) yields another expression of S as a function of I:

Equating (12) and (13), one finds that the values of I are the roots of the quadratic polynomialwhere

Direct computation shows that the condition is equivalent to f/d < 0. If this holds, then the values of I are real (since fd < 0) and have opposite signs: I1 > 0 and I2 < 0, say. Substituting I1 into (6), (7), (8), (9), (10), and (11), one obtains a unique endemic equilibrium of the model (1), with all components positive. We have therefore proved the following theorem.

If, then the model (1) has a unique positive endemic equilibrium.

In the case of, we havef/d > 0, and so no immediate conclusion can be drawn on whether the values of I are real. Obtaining an analytic expression for the polynomial’s discriminant e2 − 4df requires tedious computations, let alone examining its non-negativity. The same situation is faced as we attempt to characterise the stability of these endemic equilibria in the case of their existence, since the analytic expression of I given by the quadratic formula is already complicated. This forces us to migrate from analytical to numerical techniques. Preliminary numerical experiments show that, for the parameter values shown in Table 1, δ = 0.653, u1 = 10−8, and u2 = 0.93, in which case is less than but very close to 1 (cf. first case in subsection 3.2), the polynomial (14) has two negative real roots, suggesting that the bifurcation occurring at is a forward transcritical bifurcation (Dashtbali and Mirzaie, 2021, subsection 3.4.3). Subsequently, replacing u1 and u2 with 0.4 and 0.278, respectively, we observe that is greater than 1 (cf. second case in subsection 3.2) and that a solution of the model (1) converges towards the unique positive endemic equilibrium guaranteed to exist by Theorem 3 (cf. red graph in Fig. 7), suggesting that this equilibrium is stable in the case of its existence.

Fig. 7

Time-evolution of the number E + I + Q + H of non-healthy individuals, for the parameter values shown in Table 1, u1 = 0.4, and u2 = 0.278, with δ = 0.653 (red), δ = 0.9 (green), and δ = 0.93 (blue). Notice that, as δ is increased, both the maximum and the limit decrease drastically, showing the significance of δ. For δ = 0.93, we observe convergence to the disease-free equilibrium.

Formulation of intervention strategies

We have mentioned the five concrete forms of intervention presently realised by the Indonesian government to strive towards a new normal: vaccinations, social restrictions, tracings, testings, and treatments. Now, we are ready to exploit the model (1) to formulate strategies for realising these forms of intervention, in order to optimise their impact. This will be achieved via a two-stage analysis —which is both numerical and interpretative— of the model's basic reproduction number . The first stage is the preliminary analysis, where we identify the set of parameter values corresponding to the disease-free region —that in which — and its realisability, in various epidemic scenarios. The results, as we shall see, point towards the necessity of vaccinations, and, more importantly, the vaccine efficacy, as important keys to achieve a new normal. The second stage consists in an analysis of the sensitivity of with respect to each parameter. For each of the two cases and , we choose a set of parameter values and compute the sensitivity indices of with respect to each parameter, using the results to rank the above five intervention forms in order of significance.

Preliminary analysis

The five intervention forms are not all equal in the current degree of realisation: tracings, testings, and treatments —the so-called “3 Ts”— are reportedly suboptimal (Muthiariny & Murti, 2021), while vaccinations and social restrictions seem to be given primary attention (WHO Indonesia, 2022; COVID19 Vaccine Tracker; Afifa, 2022; Hutasoit, 2022). Accordingly, in this stage of our analysis, let us assume that the parameters u3, u4, and u5, which represent the rates of the 3Ts, have fixed values. Furthermore, let us fix the values of all parameters except δ, u1, and u2.

As noted in section 1, u2 = 0 represents normal mobility, while u2 = 1 represents a total lockdown. The values of u2 which represent social restrictions of level 1, 2, 3, and 4 can be estimated in the following way. First, we deal with level 1 social restrictions, which, as detailed in (Saptoyo & Kurniawan, 2021), consist of the following regulations:

businesses in non-essential sectors are to implement the work-from-office policy at up to p1 = 75% capacity;

businesses in essential sectors are to implement the work-from-office policy at up to p2 = 100% capacity;

daily-need shops are to operate at up to p3 = 75% capacity;

non-daily-need shops are to operate at up to p4 = 75% capacity;

malls and shopping centres are to operate at up to p5 = 75% capacity;

roadside stalls and street vendors are to operate at up to p6 = 75% capacity;

restaurants are to operate at up to p7 = 75% capacity;

educational activities are to be carried out p8 = 50% onsite and 50% online;

places of worship are to operate at up to p9 = 50% capacity.

We estimate u2 as the average percentage of restrictions in the case of level 1 social restrictions according to the above data: . In a similar way, we obtain the following values of u2 representing social restrictions of level 2, 3, and 4: 0.389, 0.694, and 0.861, respectively. The values of δ, on the other hand, will be chosen in view of the efficacies of the actual COVID-19 vaccines (Globe, 2021; Jara et al., 2021; Mascellino, Timoteo, Angelis, & Oliva, 2021).

It is apparent from (5) and the definition of the ks in (4) that, for any given δ, the graph of the equation on the u1u2-plane is a straight line, the abscissa and ordinate intercepts being, respectively,and

Notice that ℓ2 is independent of δ, since so are the ks, by (4). Moreover, a direct computation shows that the denominator of ℓ1 is equal to zero if and only if δ = ℓ2. Furthermore, we letbe the abscissa of the point of ordinate 1 on the line. Substituting the values shown in Table 1 of all parameters except δ, u1, and u2, one obtains

Recently, the vaccine most sought-after in the country is reportedly Sinovac's Coronavac (Bona, 2021), which has demonstrated a 65.3% efficacy (Globe, 2021; Jara et al., 2021). For this value of δ, we have ℓ1 = −0.0001417358 and ℓ3 = 0.0000107698, and the line on the u1u2-plane is plotted in Fig. 2 (1). The shaded region is the feasible disease-free region, i.e., the region . Therefore, according to our model, using a vaccine with only a 65.3% efficacy, the pandemic can only be resolved if u1⩽ℓ3 = 0.0000107698, i.e., the vaccination rate is made extremely low, and u2⩾ℓ2 = 0.9293807942, i.e., a near-lockdown policy is implemented. The latter is uncompromisable: even level 4 social restrictions are insufficient; see the red curve in Fig. 2 (2). Likewise, if the vaccination rate is increased even only to a moderate level, say u1 = 0.4 (Table 1), then the policy of raising the level of social restrictions becomes insignificant: such a policy suppresses the endemic-valued basic reproduction number —and thus the number of daily new cases— only insignificantly; see Fig. 2 (3). A major reason for this is that, for vaccinated citizens, social restrictions are waived,4 allowing them to travel, visit public places, etc. more unrestrictedly than unvaccinated citizens, bringing about a high risk in the case of high vaccination rate but low vaccine efficacy. We thus find it unsurprising that the omicron wave remained unavoidable despite the notable progress of the country's vaccination programme (see section 1).

Fig. 2

(1) Plot of the line on the u1u2-plane in the case of δ = 0.653, with the feasible disease-free region shaded; (2) plot of as a function of u1 in the case of δ = 0.653, for u2 = 0.861 (red) and for u2 = 0.999995 (blue); (3) plot of as a function of u2 in the case of δ = 0.653, for u1 = 0.4 (red) and for u1 = 0.000005 (blue).

Now let us suppose that the country utilises a vaccine with a higher efficacy: say, δ = 0.9. In this case, we have ℓ1 = −0.0013332879 and ℓ3 = 0.0001013102, and the line on the u1u2-plane is plotted in Fig. 3 (1). Since no qualitative change is observed here, the message remains the same: the disease's transmission can only be halted with an extremely low vaccination rate and an extremely high level of mobility restrictions. However, comparing Fig. 2 (2) and Fig. 3 (2), we see a qualitative change: the red curve, which, in both figures, correspond to level 4 social restrictions, i.e., u1 = 0.861, has changed monotonicity. Furthermore, comparing the quantitative properties of the red lines in Fig. 2 (3) and Fig. 3 (3), both corresponding to the moderate vaccination rate u1 = 0.4, we can see that the improvement of the vaccine efficacy, from 0.653 to 0.9, drastically decreases the value of the basic reproduction number, from above 4 to below 2. We infer therefore that the improvement of the quality of COVID-19 vaccines should take precedence over that of the rate at which vaccinations are carried out.

Fig. 3

(1) Plot of the line on the u1u2-plane in the case of δ = 0.9, with the feasible disease-free region shaded; (2) plot of as a function of u1 in the case of δ = 0.9, for u2 = 0.861 (red) and for u2 = 0.999995 (blue); (3) plot of as a function of u2 in the case of δ = 0.9, for u1 = 0.4 (red) and for u1 = 0.9 (blue).

Let us further increase the vaccine efficacy: δ = 0.93. In this case, we have ℓ1 = 0.0632634203 and ℓ3 = −0.0048070860, and the line on the u1u2-plane is plotted in Fig. 4 (1). Now, we see a radical qualitative change —a much desirable one— from Fig. 4 (1): the line's slope is now negative, and a disease-free state can be achieved even with a complete removal of social restrictions, i.e., u2 = 0, and a very low vaccination rate, i.e., any low value of u1 satisfying u1⩾ℓ1 = 0.0632634203, say, u1 = 0.064 (i.e., 6.4% of the current number of susceptible individuals are vaccinated each day); see Fig. 4 (2) and (3).

Fig. 4

(1) Plot of the line on the u1u2-plane in the case of δ = 0.93, with the feasible disease-free region shaded; (2) plot of as a function of u1 in the case of δ = 0.93, for u2 = 0.861 (red) and for u2 = 0 (blue); (3) plot of as a function of u2 in the case of δ = 0.93, for u1 = 0.4 (red) and for u1 = 0.064 (blue).

A follow-up question naturally arises: is it possible to achieve a disease-free state without vaccinations, i.e., with u1 = 0? In this case, the expression (5) for is independent of δ:a plot of versus u2 is shown in Fig. 5: a line with a negative slope which is rather large in absolute value. Therefore, in absence of vaccinations, raising the level of social restrictions suppresses the basic reproduction number significantly. This justifies the effectiveness of the government's social restriction policies prior to the commencement of the vaccination programme. Notice that, near u2 = 1, the values of shown Fig. 5 are lower than those shown in Fig. 2 (3). That is, in the cases where social restrictions are imposed on level 3 or 4, not administering vaccines results in lower values of than administering vaccines with low efficacy, justifying that the assumed policy of waiving restrictions for vaccinated citizens is imprudent, since these citizens are not completely immune. Nevertheless, in absence of vaccinations, even level 4 social restrictions are not sufficient to bring the country to the disease-free state. The latter requires, again, a near-lockdown policy: u2⩾ℓ2 = 0.9293807942.

Fig. 5

Plot of as a function of u2 in the case of u1 = 0.

Let us summarise the recommended strategies arising from this first-stage analysis. Firstly, if a lockdown is undesirable, it is necessary to administer vaccinations. However, one should strive not primarily towards the increase of the rate at which they are administered, but towards the use of high-efficacy vaccines,5 such as Pfizer-BioNTech or Moderna (Mascellino et al., 2021). Secondly, it is necessary to set out and implement an appropriate level of social restrictions to vaccinated citizens,6 especially those who received vaccines with limited efficacy and/or have only been vaccinated partially.

Sensitivity analysis

Let us now complement the above analysis with a quantitative assessment of the significance of each parameter. We compute the sensitivity index (Chitnis, Hyman, & Cushing, 2008) of the basic reproduction number with respect to a parameter p, i.e.,for every , wheredenotes the set of all parameters in the model (1), obtaining, e.g., for p = δ and p = u2,respectively.

We choose two sets of parameter values: one representing a disease-free case , and another representing an endemic case . In both sets, the values of λ, λ′, μ, μ′, β, α, θ, γ, φ, κ, τ, u3, u4, u5 are as shown in Table 1, and δ = 0.653. In the former case, we set , so that , while in the latter, we set , so that .

In each caese, we substitute the parameter values to all sensitivity indices. For each index , the following aspects are essential.

The sign , which is positive (negative) if and only if is monotonically increasing (decreasing) with p.

The absolute value , which measures the relative change of with respect to p: a P% change of p results in a change of . Thus, the higher the value of , the more significant the parameter p.

It was our intention to visualise the results using a bidirectional bar chart of the values of for every p in each case, but the rather unusual distribution of these values, especially in the disease-free case where an extreme outlier is present, makes such a chart ineffective.

For a more effective visualisation, let us first define the significance rank of a parameter p to be r(p), where is the unique bijection for which

Thus, the values of 18 − r(p) carry the same qualitative information as : the higher the value of 18 − r(p), the more significant the parameter p. Therefore, instead of visualising the values of , we visualise the values of , by bars which are labelled by the associated values of to retain the quantitative information (Fig. 6). We can see that the sequence of the parameters in decreasing order of significance isin the disease-free case, andin the endemic case. Let us now infer from these orderings the appropriate strategies of intervention in each case.

Fig. 6

Bidirectional bar charts, in which, for every parameter p, the value of is represented by a bar which is labelled by the value of , in the disease-free case (top) and endemic case (bottom) specified in subsection 3.2. Bars to the left (right) of the ordinate axis are associated with parameters with which the basic reproduction number is monotonically decreasing (increasing). The longer the bar, the more significant the associated parameter. Consequently, the ordering of the parameters according to significance in each case is given by (16), (17).

The disease-free case

In the disease-free case, effort must be made in order to maintain the low value of the basic reproduction number. When the number of daily new cases rises, as the ordering (16) and Fig. 6 suggest, a committed implementation of social restrictions should be sufficient. Indeed, remarkably, tightening social restrictions only by 1% suppresses the basic reproduction number by 13.2701075492%. Other forms of intervention, if at all desired, are recommended in the following order: testings, tracings, treatments, and vaccinations. The latter is rather inessential, let alone when not supported by a high level of vaccine efficacy.

The endemic case

In the endemic case, significant effort is necessary for a transition to a disease-free state. The ordering (17) strongly supports our main finding in subsection 3.1: the vaccine efficacy being the parameter upon which the basic reproduction number depends most sensitively. Accordingly, we reiterate our primary finding in the previous subsection: that

raising the efficacy of vaccines

must be given the highest priority. With regards to the five forms of intervention, we recommend, in order of importance:

expanding and accelerating testings and tracings, so that infected and exposed individuals may be quarantined more immediately;

optimising treatments for infected individuals, by ensuring that health facilities and services (medications, hospital beds, medical practitioners, etc.) are in adequate availability;

administering vaccinations;

social restrictions, being the least important form of intervention, albeit, as previously remarked, may become significant if also applied to some degree to vaccinated individuals.

There is however a substantial difference in significance between recommendations (1) and (2)–(5), i.e., the rates of all five forms of intervention are far less significant than the vaccine efficacy. Indeed, increasing the vaccine efficacy from 65.3%, firstly to 90%, and subsequently to 93%, as narrated in subsection 3.1, results in significant drops of both the peak and limiting numbers of non-healthy individuals, the final value being sufficient for a transition from endemic to disease-free; see Fig. 7. On the other hand, assuming the original value of the vaccine efficacy, 65.3%, increasing any of the five intervention rates by 30%, or even by 60%, gives rise to barely any tangible impact; see Fig. 8. Therefore, the main key for a successful eradication of the pandemic is not a high rate of implementation of any of the five intervention forms, but a high-efficacy vaccine.

Fig. 8

Time-evolution of the number E + I + Q + H of non-healthy individuals, for the parameter values shown in Table 1, δ = 0.653, u1 = 0.4, u2 = 0.278, u3 = 0.5, u4 = 0.3, and u5 = 0.0833 (red graphs on all panels), together with, on panel (i), where i ∈ {1, 2, 3, 4, 5}, the same in the cases of u being increased by 30% (green) and by 60% (blue). The magnifications near the maxima reflect the previously obtained order of importance of the intervention parameters: u4, u3, u5, u1, u2. However, comparing with Fig. 7, we see that all these parameters are far less significant than δ.

Conclusions and future research

We have constructed an SVEIQHR-type mathematical model for the spread of COVID-19, which incorporates as parameters the rates of the five forms of intervention presently realised by the government of Indonesia: vaccinations, social restrictions, tracings, testings, and treatments. We have computed the model's basic reproduction number , and show that the model possesses a unique disease-free equilibrium, which exists for all sets of parameter values and is stable (unstable) if , as well as a unique endemic equilibrium, which exists if .

We have also analysed the model numerically, with the aim of determining strategies by which the five intervention forms should be realised in order to optimise their impact. The analysis results in the following two major conclusions.

In a disease-free state, social restrictions proved to be the best form of intervention in the case of a rise in the number of new cases.

In an endemic state, a transition to disease-free state without vaccinations requires a near-lockdown policy. Since the country's government has refused to impose such a policy (Bulletin, 2021; Gorbiano & Sutrisno, 2020), vaccinations are necessary. However, efforts should be focused not primarily on increasing the vaccination rate (or even the rate of any other form of intervention), but on the use of vaccines with a high efficacy.

Finally, our model is open to further analysis. Indeed, one could continue the study of how the existence and local asymptotic stability of the model's endemic equilibria depend on its basic reproduction number, and investigate whether the equilibria's global (asymptotic) stability can be established using, e.g., Lyapunov functions (Korobeinikov, 2009; Melesse & Gumel, 2020; Safi & Gumel, 2010). Our model is also open to a number of modifications. One could incorporate a specified level of social restrictions for vaccinated individuals, and confirm whether, as a result, vaccination rate becomes more significant. Besides, the vaccinated compartment itself could be split into several compartments, in order to allow different assumptions on recipients of different vaccines and/or, in the case of multi-dose vaccines, recipients of different numbers of vaccine doses. Similarly, the quarantined compartment could be split into several compartments, in order to distinguish the isolated individuals (i.e., the separated infected individuals) from the quarantined individuals (i.e., the separated exposed individuals) —which may further be split according to their vaccination histories— so that different recovery rates and/or time-delays may be employed. Finally, one could devote more explicit attention to a new virus variant by splitting the infected compartment into that of individuals infected by the new variant and that of those infected otherwise.

Declaration of interests

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.