Sontag Notes on Systems Biology Solutions

An explicit formula for minimizing the infected peak in an SIR epidemic model when using a fixed number of complete lockdowns

doi: https://doi.org/10.1101/2021.04.11.21255289

ABSTRACT

A too-early start of NPIs (non-pharmaceutical interventions) such as social distancing may lead to high "second waves" of infections of COVID-19. This paper asks what should be the timing of a set of k complete-lockdowns of prespecified lengths (such as two weeks) so as to minimize the peak of the infective compartment. Perhaps surprisingly, it is possible to give an explicit and easily computable rule for when each lockdown should commence. Simulations are used to show that the rule remains fairly accurate even if lockdowns are not perfect.

1 Introduction

The year 2020 will be remembered for the COVID-19 (coronavirus disease 2019) pandemic, which is an individual-to-individual infection by SARS-CoV-2 (severe acute respiratory syndrome coronavirus 2). An immediate way to stop transmissibility is to establish total lockdowns, as China and Northern Italy did early on in the outbreak [1]. Lockdowns and other NPIs (non-pharmaceutical interventions) such as quarantine and social distancing were soon implemented around the world. On the other hand, frustration with isolation rules and economic costs mean that, in most countries, long lockdowns are not feasible, nor is it easy to enforce even milder forms of NPIs [2, 3, 4]. Moreover, a too-early start of NPIs often led to large "second waves" of infections when the NPIs where relaxed. This motivates the search for optimally timing the start of multiple short NPIs so as to minimize the maximum peak of infective individuals. This paper deals with that problem.

To be precise, we consider an SIR model with strict (no-contact) lockdowns, and assume that policymakers want to decide when to start each one of K of lockdowns, with respective lengths T _k , k = 1,…, K. For example, the T _k 's may all be equal, say two weeks. We provide an exact and very simple rule, which says that lockdowns should commence whenever the number of infectious individuals reaches a certain level, namely , where V ₀ is a number that can be easily computed from the infectivity rate, recovery rate (τ), and the initial populations of susceptible and infected individuals. The formula for I _start simplifies in the case in which all T _k 's are equal, T _k = T, to just (1 + K − Ke ^−τT )⁻¹ V ₀. In addition, we show that there will be exactly K peaks of infected populations, all equal to this value. Observe that as the lockdown intervals become larger, T → ∞, the best possible maximum peak is V ₀ /(1 + K). Obviously, a perfect or near-perfect lockdown is not practical, except in jurisdictions where complete obedience can be strongly enforced, but this theoretical problem is nonetheless of interest. Moreover, studying this case also helps understand the problem with non-strict lockdowns, as we will discuss.

Infections and mathematical modeling

Infectious agents have critically influenced the history of mankind, with disease-causing pathogens constantly emerging or evolving. From the Plague of Athens (430-428 BC), to the fourteenth century Black Death that killed about a third of Europe's population, to the Yellow Fever epidemic in Philadelphia in 1793, in which a tenth of the population of the city perished, to the 1918 "Spanish flu" pandemic (which did not originate in Spain) that resulted in about 3-5% of the world population dying, to the COVID-19 2020-2021 pandemic, infectious diseases have had major impacts on health, psychological and social well-being, medical advances (mRNA vaccines, for example), economics, politics, military history, and religious and racial persecution. Different types of pathogens are involved in infectious diseases. Viruses cause the common cold, influenza, measles, West Nile, and COVID-19, while anthrax, salmonella, chlamydia, and cholera are caused by bacteria, and protozoa give rise to malaria and trypanosomiasis (sleeping sickness). There are many mechanisms for transmission, including respiratory droplets (influenza, colds), body secretions (chlamydia), flies (trypanosomiasis), mosquitoes (malaria), and food or water (cholera). Control strategies include behavioral and sanitation changes (NPIs), vaccines, antibiotics, antiviral drugs. Notwithstanding this variety, there is a common mathematical structure.

The modeling of infectious diseases and their spread is an important part of mathematical biology, specifically mathematical epidemiology. Modeling is an important tool for gauging the impact of NPIs such as social distancing, masking, lock-downs, or school closings, as well as predicting/attenuating magnitude of peak infections ("flattening the curve" so as not to overwhelm ICU capacities), predicting/delaying peak infections (until vaccine/treatments available), and devising strategies for vaccination, control, or eradication of diseases. The social and political use of epidemic models must take into account their degree of realism. Good models do not incorporate all possible effects, but rather focus on the basic mechanisms in their simplest possible fashion. Not only it is difficult to model every detail, but the more details the more the likelihood of making the model sensitive to parameters and assumptions, and the more difficult it is to understand and interpret the model as well as to play "what-if" scenarios to compare alternative containment policies. It turns out that even simple models help pose important questions about the underlying mechanisms of infection spread and possible means of control of an epidemic.

Most mathematical epidemiology models incorporate some version of the classical SIR model proposed by Kermack and McKendrick in 1927 [5]. We will restrict attention to this core model, which is suitable for describing initial stages of an infection in a single city, and also for modeling later stages when community spread becomes dominant. Mathematical models have long played a central role in epidemiology, and this has been especially true with the COVID-19 pandemic [6, 7, 8, 9, 10, 11, 12, 13, 14]. This includes control-theoretic aspects, especially optimal control [15, 16, 17, 18].

Mathematical models had a major impact on the political response to the COVID-19 pandemic. To quote from "Behind the virus report that jarred the U.S. and the U.K. to action" (New York Times, 17 March 2020):

The report [from Imperial College London], which warned that an uncontrolled spread of the disease could cause as many as 510,000 deaths in Britain, triggered a sudden shift in the government's comparatively relaxed response to the virus. American officials said the report, which projected up to 2.2 million deaths in the United States from such a spread, also influenced the White House to strengthen its measures to isolate members of the public.

Of course, one should always keep in mind as well the following quote from Dr. Anthony Fauci, Director, National Institute of Allergy and Infectious Diseases, United States (CNN, 05 May 2020):

I have skepticism about models [of COVID-19], and they are only as good as the assumptionsyou put into them, but they are not completely misleading. They are telling you something that is a reality, that when you have mitigation that is containing something, and unless it is down, in the right direction, and you pull back prematurely, you are going to get a rebound of cases.

2 Formulation of problem

We consider a standard SIR model for epidemics, which consists of three groups of individuals: those who are susceptible and can be passed on the pathogen by the infectious individuals, and the removed individuals, who are have either developed immunity after infection or who have died. In SIR models, one does not include a flow back from individuals into the susceptible compartment. On longer time-scales, one may also allow for the fact those individuals in the removed group who are immune may eventually return to the susceptible population, which would happen if immunity is only temporary or if a pathogen has evolved substantially. The numbers of individuals in the three classes will be denoted by S, I, and R respectively, and hence the name "SIR" model. See Figure 1, where we use the symbol "⊗" to indicate that the number of new infected will depend both on the number of susceptibles and infectious (specifically, it will be a product in the classical SIR model to be discussed next, with a proportionality constant β), and ν denotes the flow to the recovered compartment. Observe that the "feedback" term implicit in the ⊗ effect means that this is not exactly a compartmental system, because for those, the flow into a compartment does not depend on the number of individuals in that compartment.

Mathematically, assuming that new infections are due to contacts between S and I individuals, and that the rate at which this happens is proportional to the numbers of such individuals, there results a system of three coupled ordinary differential equations as follows: From now on we will ignore the last equation since it does not affect the number of infectives. The initial state satisfies S(0) = S ₀ > 0, I(0) = I ₀ > 0. The positive parameter β (infectivity/contact rate) quantifies the transmission rate between the susceptible S and infected I individuals in a well-mixed population, and ν (recovery/death rate) is the rate of flow into the removed R compartment. We will assume that ℛ₀ := βS ₀ /ν > 1; otherwise, the problem to be discussed is not interesting, as I decreases monotonically to zero if the condition does not hold. (This and other well known facts about SIR systems are reviewed in Section 5; see also e.g. [19].)

In the SIR model, NPIs are viewed as reducing the contact rate β. (More sophisticated models of social distancing have been widely studied, see for example [20, 21, 22, 23, 24, 25, 26].) The reduction of β is modeled by a time-varying β(t) where where 0 ≤ β ₀ < β are two fixed values, and J ⊆ [0, ∞). In our work, J will be the union of a number K of intervals and we will be taking β ₀ = 0, representing a strict or full lockdown. The lengths T _k of the respective intervals will be allowed to be arbitrary (but fixed) in our theory, though the most elegant formulas are obtained when they are all equal (we might have T _k = 14 or 28 days for all k, for instance). The objective will be to minimize the maximum of I(t) for t ≥ 0 ("flatten the curve") by appropriately choosing the start times t _k .

As discused in the introduction, even if perfect lockdowns are not completely realistic, studying this case helps understand the case β ₀ > 0. Indeed, we show via simulations that the conclusions for β ₀ = 0 are very relevant to the case of a small but nonzero β ₀, even one that is about 20% of the value of β.

Previous work on this and related problems includes [27], which treats an optimal schedule minimizing a combination of the total number of deaths and the peak of the infected compartment, [17] which shows that a single interval (K = 1) is optimal if the objective is to minimize the total number of susceptible individuals at the end of the epidemic, [15] with numerical studies of optimally timing fixed-duration "one-shot" strategies, and the very nice theoretical paper [16] which showed that the optimal strategy for minimizing peak infection is a combination of a strict lockdown ("full suppression") with a feedback strategy which keeps ℛ₀=1. Also closely related to this work is [28], which studies timing of lockdowns, including periodic strategies, through a combination of theoretical and numerical methods,

Let r := ν/β, and consider the "virtual peak" of I(t) if no lockdown were imposed: (cf. Section 5). Define this expression: The main result is as follows:

Theorem.

Suppose that I(0) < I _start . Then, in order to minimize the maximum of I(t), t ≥ 0, lockdowns should start whenever Moreover, under this policy, the maximum of I(t) will equal I _start.

In other words, any time that the infective population level reaches the value I _start, the next time-T _k lockdown interval I _k should start.

For example, if T _k = T for i = 1,…, K then the formula is Note that as T → ∞ the best possible maximum peak is We prove the theorem next.

For mathematical elegance, we will include the theoretical possibility of a lockdown starting at time exactly t ₀ = 0 (and later make T ₀ = 0, so that there is in effect no initial lockdown).

For any initial population (σ, ι), we introduce the following function: which gives the peak value of I(t) if we start at the initial population (σ, ι), and let: (so t ₀ = 0 and (S ₀, I ₀) is the state at the start of the epidemic). Observe that Consider these equalities: where the last equality follows from the conservation law I(t) + S(t) — r ln S(t) ≡ constant, applied to a solution that starts at ι = p _k I _k and σ = S _k . Adding (5) to (7) and subtracting (6), we obtain: from which we conclude the following recursion: where for convenience we let a _k := 1 − p _k . Applying this formula recursively, we conclude: The largest value of I(t) will occur at the maximum of the peaks occurring at the start of each kth lockdown, or the last "virtual peak" (which is then a real local maximum). In other words, the maximum of I is Under the epidemic assumption ℛ₀ > 1, İ(0) > 0, so we can drop I ₀ from the list. We replace the value for V _K from formula (8). Letting T ₀ = 0 means that a ₀ = 1 − p ₀ = 0, so we conclude that To minimize the peak, we need to pick non-negative I ₁,…, I _K in such a way that this expression is minimized. This is achieved at a unique global minimum, at which (see Lemma below), namely This completes the proof of the theorem.

A lemma on linear optimization

Let a ₁ ≥ 0,…, a _K ≥ 0, b > 0. Consider the following function, defined on : Let Note these properties:

1. When all x _k = ξ, g(x ₁,…, x _k ) = ξ. To prove this, note that so all terms in the max are the same.

2. If g(x ₁,…, x _k ) ≤ ξ then (by definition of the max)

   1. x _k ≤ ξ for all k, and

   2. .

3. If g(x ₁,…, x _k ) < ξ for some (x ₁,…, x _k ), then at least one of the inequalities (a) or (b) must be strict. Note that (a) implies that and this is a strict inequality if one of the inequalities in (a) is strict, which then contradicts (b). If instead (b) is strict, then is contradicted. In summary, for all (x ₁,…, x _k ).

We conclude from (1) and (3) that ξ is the minimum value of g, and it is achieved when all x _k = ξ.

We next prove that the minimum is achieved only at this point.

Indeed, suppose that g(x ₁,…, x _k ) = ξ, so that (a) and (b) above hold. If any x _k ≠ ξ, then x _k < ξ, because if x _k > ξ for any k, then g(x ₁,…, x _k ) > ξ (by definition of g). But we already remarked that this contradicts (b).

We next show some illustrations of the use of the formula for several lengths of lockdowns as well as simulations for various scenarios of 1, 2, 3, or 4 lockdowns. We use reasonable parameters in each case. Finally, we will show computationally that a small positive β ₀ does not change conclusions much, at least for the case of a single lockdown.

3 Optimal reduction of "virtual peak" with K perfect lockdowns

The use of a small number of lockdowns results in a drastic reduction of the peak that would occur (the "virtual peak") if there were no lockdowns. The marginal benefit of additional lockdowns is relatively minor, after a certain number of them. We plot here the fraction (1 + K(1 −e ^−νT ) ⁻¹ for ν = 0.05 and T = 7, 14, 21, 28 days.

4 Simulations

4.1 Simulations using optimal formula

We simulate various lockdown lengths using our optimal formulas. Parameters are, β = 0.00025, ν = 0.05 (so ℛ₀ = 5), initial conditions are S ₀ = 1000, I ₀ = 1, and the peak if there are no lockdowns is I = 479. The maximum from the formula coincides with the maximum in simulations, up to numerical error.

4.1.1 Lockdown length is T = 14 days, perfect lockdown

• Download figure
• Open in new tab

Figure 6:

# lockdowns: 1, peak I from formula: 318.682808, maximum of I on last (no lockdown) period: 318.601530. Lockdown start time(s): 32.42.

• Download figure
• Open in new tab

Figure 7:

# lockdowns: 2, peak I from formula: 238.740981, maximum of I on last (no lockdown) period: 238.755779. Lockdown start time(s): 29.73, 50.69.

• Download figure
• Open in new tab

Figure 8:

# lockdowns: 3, peak I from formula: 190.862880, maximum of I on last (no lockdown) period: 191.019085. Lockdown start time(s): 28.01, 47.71, 69.80.

• Download figure
• Open in new tab

Figure 9:

# lockdowns: 4, peak I from formula: 158.980313, maximum of I on last (no lockdown) period: 159.342577. Lockdown start time(s): 26.74, 45.87, 66.33, 89.44.

4.1.2 Lockdown length is T = 28 days, perfect lockdown

• Download figure
• Open in new tab

Figure 10:

# lockdowns: 1, peak I from formula: 273.247170, maximum of I on last (no lockdown) period: 273.286548. Lockdown start time(s): 30.9.

• Download figure
• Open in new tab

Figure 11:

# lockdowns: 2, peak I from formula: 191.124644, maximum of I on last (no lockdown) period: 191.048654. Lockdown start time(s): 28.02, 68.02.

• Download figure
• Open in new tab

Figure 12:

# lockdowns: 3, peak I from formula: 146.957573, maximum of I on last (no lockdown) period: 146.913773. Lockdown start time(s): 26.22, 64.39, 106.74.

• Download figure
• Open in new tab

Figure 13:

# lockdowns: 4, peak I from formula: 119.371878, maximum of I on last (no lockdown) period: 119.395364. Lockdown start time(s): 24.91, 62.23, 101.98, 146.47.

4.2 Testing β ₀ ≠ 0 with formula derived for β ₀ = 0

We simulate the use of the formula derived for β ₀ = 0, under the lockdown value β ₀ = 0.00005, which represents a 20% value of the normal contact rate. Again, β = 0.00025, ν = 0.05 (so ℛ₀ = 5), initial conditions are S ₀ = 1000, I ₀ = 1, and the peak if there are no lockdowns is I = 479.

4.2.1 Lockdown length is T = 28 days, β ₀ = 0.00005

• Download figure
• Open in new tab

Figure 14:

# lockdowns: 1, peak I from formula: 273.247170, maximum of I on last (no lockdown) period: 248.383407. Lockdown start time(s): 30.90.

• Download figure
• Open in new tab

Figure 15:

# lockdowns: 2, peak I from formula: 191.124644, maximum of I on last (no lockdown) period: 154.387915. Lockdown start time(s): 28.02, 61.2.

• Download figure
• Open in new tab

Figure 16:

# lockdowns: 3, peak I from formula: 146.957573, maximum of I on last (no lockdown) period: 103.506053. Lockdown start time(s): 26.22, 57.43, 94.30.

• Download figure
• Open in new tab

Figure 17:

# lockdowns: 4, peak I from formula: 119.371878, maximum of I on last (no lockdown) period: 71.792718. Lockdown start time(s): 24.91, 55.24, 88.63, 129.63.

4.2.2 Lockdown length is T = 14 days, β ₀ = 0.00005

• Download figure
• Open in new tab

Figure 18:

# lockdowns: 1, peak I from formula: 318.682808, maximum of I on last (no lockdown) period: 326.846639. Lockdown start time(s): 32.4.

• Download figure
• Open in new tab

Figure 19:

# lockdowns: 2, peak I from formula: 238.740981, maximum of I on last (no lockdown) period: 248.153424. Lockdown start time(s): 29.73, 46.6.

• Download figure
• Open in new tab

Figure 20:

# lockdowns: 3, peak I from formula: 190.862880, maximum of I on last (no lockdown) period: 200.218534. Lockdown start time(s): 28.01, 43.86, 61.6.

• Download figure
• Open in new tab

Figure 21:

# lockdowns: 4, peak I from formula: 158.980313, maximum of I on last (no lockdown) period: 167.977200. Lockdown start time(s): 26.74, 42.11, 58.60, 77.22.

4.3 Comparison with optimal strategy for β ₀ ≠ 0

For comparison with the use of the optimal formulas derived for β ₀ = 0, we show here numerically the optimal solution when β ₀ ≠ 0, specifically β ₀ = 0.00005 as above. As earlier, parameters are β = 0.00025, ν = 0.05 (so ℛ₀ = 5), initial conditions are S ₀ = 1000, I ₀ = 1, and the peak if there are no lockdowns is I = 479. We take only the case of a single lockdown, for simplicity, and lockdown lengths of 14 or 28 days.

We find that the formula predicts the optimal timing extremely well for 14-day lockdowns (error less than 1% in maximum infectives), and is fairly good for 28-day lockdowns as well (about 5% error).

4.3.1 Lockdown length is T = 14 days, β ₀ = 0.00005

For 14-day lockdowns, the plot in Figure 18 suggests that the formula derived for β ₀ = 0 triggered the first lockdown too early, so we explored by what fraction > 1 to increase the trigger point, from which the optimal strategy is clear.

• Download figure
• Open in new tab

Figure 22:

Magnitude of first and second peak, with the lockdown time parametrized by the percentage of the ideal (perfect lockdown) formula. Note that the second peak decreases as the first peak happens later, so the minimum of the maximum among them will occur when the curves intersect.

• Download figure
• Open in new tab

Figure 24:

Simulation with an imperfect lockdown. Using now the optimal fraction 1.018 of the optimal for perfect lockdowns, obtained by minimizing the plot in Figure 23. The optimal peak value is now approximately 324, which can be compared with the suboptimal plot (using the formula that assumed perfect lockdowns) shown in Figure 18, which had a peak of approximately 327. Observe how the two peaks are now balanced. The optimal result is not that different from the one obtained from our formulas, and the lockdown start is at time 30.39.

4.3.2 Lockdown length is T = 14 days, β ₀ = 0.00005

In contrast, for 28-day lockdowns the plot in Figure 14 suggests that the formula derived for β ₀ = 0 waited too long for the first lockdown, so we first explore by what fraction < 1 to decrease the trigger point, from which the optimal strategy is clear.

• Download figure
• Open in new tab

Figure 25:

Magnitude of first and second peak, with the lockdown time parametrized by the percentage of the ideal (perfect lockdown) formula. Note that the second peak decreases as the first peak happens later, so the minimum of the maximum among them will occur when the curves intersect.

• Download figure
• Open in new tab

Figure 27:

Simulation with an imperfect lockdown. Using now the optimal fraction 0.944 of the optimal for perfect lockdowns, obtained by minimizing the plot in Figure 26. The optimal peak value is now approximately 258, which can be compared with the suboptimal plot (using the formula that assumed perfect lockdowns) shown in Figure 14, which had a peak of approximately 273. Observe how the two peaks are now balanced. The optimal result is very close to the one obtained from our formulas.

5 Review of SIR model

To make this paper self-contained, we review here some facts about SIR models, see e.g. [19] or mathematical epidemiology texts for more details. We wish to analyze solutions, from initial conditions S(0) = S ₀, I(0) = I ₀.

Infections always die-out in SIR model

Note that if I ₀ = 0, then S(t) ≡ S ₀ and I(t) ≡ 0; in other words, every point of the form (S, 0) is an equilibrium. Similarly, if S ₀ = 0, then S(t) ≡ 0 and I(t) = e ^−νt I ₀ → 0, so the case S ₀ = 0 is not interesting either. So we study the only interesting cases, I ₀ > 0 and S ₀ > 0.

Since , S(t) is a nonincreasing function of time, and thus S(t) ↘ S _∞ for some S _∞ ≥ 0. A most important result is this one: This says that the infection will end (asymptotically), and there will remain a number of "naive" individuals at the end.

We will show that I(t) → 0 and defer the proof that S _∞ > 0 to later.

To prove this result, we will use this theorem: if x(t) is a solution of a system of ODEs , and if the solution converges, x(t) → x ^∗, then x ^∗ must be an equilibrium point, i.e. f (x ^∗) = 0. This is true because the omega-limit set of a trajectory is an invariant set, the LaSalle Invariance Principle (see e.g. [29]).

We apply this theorem as follows. First we define V (t) := S(t) + I(t), and notice that , which means that V (t) is nonincreasing, and thus there is a limit V (t) ↘ V _∞ as t → ∞. Therefore I(t) = V (t) − S(t) → V _∞ − S _∞ =: I _∞ also has a limit. So the state x(t) = (S(t), I(t)) converges to x ^∗ := (S _∞, I _∞). It follows that f (x ^∗) = 0, which means that and from there we conclude that I _∞= 0 because ν ≠ 0.

We still have to show that S _∞ > 0; we will in fact provide a formula for S _∞.

ℛ₀ and epidemics

A central role in epidemiology is played by the "intrinsic reproductive rate" The epidemiological (and non-mathematically rigorous) definition of ℛ₀ is "the average number of secondary cases produced by one infected individual introduced into a population of susceptible individuals," where by a susceptible individual one means one who can acquire the disease. This can be made precise with a stochastic model, but an intuitive argument can be found in [19]. We remark ℛ₀ has a generalization to more complex epidemics models, and it characterizes the local stability of the set of "disease-free" steady states (DFSS) (those for there are no infectives). One may compute ℛ₀ using the so-called "next generation matrix" built from the differential equations. which was introduced in [30] (see e.g [31] and also the worked examples in [19]).

We discuss ℛ₀ below in more detail, but for now note the following fact. From the ODE for I, we have that This means that an epidemic will happen, meaning that I(t) will increase when starting from any I(0) > 0, if and only if ℛ₀ > 1.

Moreover, the initial growth of I(t) will be exponential, with rate λ = ν (ℛ₀ − 1). (For small times and a large susceptible population S ₀, we may assume that S(t) remains roughly constant.) Logarithmically plotting infections, we can estimate λ, and from there we may estimate (assuming that we know ν, the recovery/death rate of infecteds), and When ℛ₀ ≤ 1 and t > 0, (because S(t) < S ₀) and so I(t) monotonically decreases to zero.

From now on, when discussing the SIR model, we assume that ℛ₀ > 1.

Peak infection time t _p and susceptibles at that time

The derivative İ = (βS − ν)I is positive for small t, because at zero it equals ν (ℛ₀ − 1)I > 0.

On the other hand, since I(t) → 0 as t → ∞, the derivative must eventually become negative, which means that there is some time t _p (p for "peak" infectivity) at which İ(t _p ) = 0. Since S(t) decreases monotonically, the derivative of I can only change sign from positive to negative at exactly one such time t _p . So t _p is the point at which I(t) attains is maximum.

From İ = 0 at t _p , we have that where we define for convenience Thus, at the peak infection time, there is a precise formula for the number of susceptibles.

A formula for the final number S _∞ > 0 of susceptibles

Let us know derive an (implicit) equation for the limit S _∞ of the susceptible population. We introduce the following function, along a given solution: Taking derivatives, which means that H is constant along trajectories (a conserved quantity): for all t > 0. It follows, in particular, that and therefore S(t) ≥ e ^pβ/ν for all t, so taking limits S _∞ ≥ e ^pβ/ν > 0 as claimed.

Even better, we can obtain an equation for S _∞ by passing to the limit in the conservation law, which gives (taking into account that I _∞ = 0): Dividing by S ₀ and using that ℛ ₀ = βS ₀ /ν, and definining for convenience , we obtain: or, letting x := S _∞ /S ₀ and : Observe that, since S(t) is decreasing, x < 1. We claim that there is exactly one solution of the equation f (x) = c with x ∈ (0, 1). By computing this solution, we can retrieve the final value of the susceptibles, S _∞ = xS ₀. To prove that there is a solution x and it is unique, note that and f (1) = 1, and is an increasing function of x, with and f ′ (1) =−∞ 1 —q > 0 (because we assumed ℛ₀ > 1). Therefore, f decreases until some x ^∗ and then increases back to 1. Since c > 1, it follows that f (x) = c has a unique solution, as we wanted to prove.

There is in fact a solution of this equation that employs a classical function. For simplicity let us write s := q ⁻¹ = ℛ₀. Multiplying by −s, we write the equation as ln x − sx = −sc. Taking exponentials and multiplying again by −s results in we ^w = y, where w := −sx and y := −se ^−sc . Note that, since s > 1 and c > 1, y ∈ (−1/e, 0). The function w ↦we ^w has an inverse, defined on (− 1/e, 0), called the Lambert W function (MATLAB command lambertw). So, w = W (y), and since x = −w/s, we conclude that or, after multiplying by S ₀: Typically, I ₀ ≈0 (one individual is enough to cause an epidemic), so c ≈1 and in that case If one can measure the proportion of people who did not get sick compared to the total initial population, then one can solve for ℛ ₀. This is one way to compute ℛ ₀ from historical data.

A formula for the peak value I(t _p ) of infectives

Determining the peak value I(t _p ) is of critical importance in practice. If a proportion θ of infected individuals will need hospital care, one can then predict, early on in an infection (and assuming that the SIR model is correct), the maximum number θI(t _p ) of people who will require hospital beds (or intensive care treatment) at any given time, and thus enforce a more stringent NPI policy if this number is projected to overwhelm hospital capacity.

Let us take again the conservation law and now specialize at t = t _p , using that S(t _p ) = r. Then, Another way to write this is to use that r/S ₀ = 1/ℛ ₀, so

5.0.1 Alternative definitions of ℛ ₀

There are alternative definitions of ℛ₀ (for the SIR model) that one often encounters in the literature: ℛ₀ = βN/ν, where N is the total population size, or even ℛ₀ = β/ν. Let us quickly explain how these relate to what we are doing here. As alluded to earlier, the general definition of ℛ₀ is given in terms of what is called a "disease-free steady state" (DFSS), meaning a steady state in which there are no infected individuals. For the specific case of the SIR model, this would mean any steady state of the form S = S ₀, I = 0, and R = N −S ₀. With this definition, ℛ₀ = βS ₀ /ν, but there are many possible ℛ₀'s depending on what is the number or removed individuals at the initial time. In particular, for the equilibrium with R = 0, ℛ ₀ = βN/ν.

What about the definition ℛ ₀ = β/ν? It is often the case that one normalizes the population to fractions: . In this case, and İ _f = (1/N)(βS − ν)I = ((βN)S _f − ν)I _f , so with . Now in terms of this new β. Note that S _f and I _f are dimensionless and that has units of (1/time), while the original β had units of 1/(time × individuals), so is perhaps more elegant.

We prefer not to perform this normalization because, when there are "vital dynamics" such as immigration, emigration, births, and/or deaths, the total population N would not be constant.

Data Availability

No data is used.

Footnotes

• Keywords added. Minor typos corrected.

References

1. [1].↵

   T. P. B. Thu , P. N. Ngoc , N. M. Hai , et al. Effect of the social distancing measures on the spread of COVID-19 in 10 highly infected countries. Science of the Total Environment, 742:140430, 2020.

2. [2].↵

   A. L. Wright , K. Sonin , J. Driscoll , and J. Wilson . Poverty and economic dislocation reduce compliance with COVID-19 shelter-in-place protocols. University of Chicago, Becker Friedman Institute for Economics Working Paper, (2020-40), 2020.

3. [3].↵

   O. Coibion , Y. Gorodnichenko , and M. Weber . The cost of the COVID-19 crisis: Lockdowns, macroeconomic expectations, and consumer spending. Technical report, National Bureau of Economic Research, 2020.

4. [4].↵

   T. Andersson , A. Erlanson , D. Spiro , and R. Östling . Optimal trade-off between economic activity and health during an epidemic. arXiv preprint arXiv: 2005.07590, 2020.

5. [5].↵

   W. Kermack and A. McKendrick . A contribution to the mathematical theory of epidemics. The Royal Society Publishing, 114(772), 1927.

6. [6].↵

   D. Adam . Special report: The simulations driving the world's response to COVID-19. Nature, 580(7803):316–318, 04 2020.

7. [7].↵

   N. Ferguson , D. Laydon , G. Nedjati-Gilani , N. Imai , K. Ainslie , M. Baguelin , S. Bhatia , A. Boonyasiri , Z. Cucunubá , G. Cuomo-Dannenburg , et al. Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand. Imperial College, London. DOI: https://doi.org/10.25561/77482, 2020.

8. [8].↵

   J. Lourenco , R. Paton , M. Ghafari , M. Kraemer , C. Thompson , P. Simmonds , P. Klenerman , and S. Gupta . Fundamental principles of epidemic spread highlight the immediate need for large-scale serological surveys to assess the stage of the SARS-CoV-2 epidemic. medRxiv, 2020.

9. [9].↵

   C. Murray . Forecasting the impact of the first wave of the COVID-19 pandemic on hospital demand and deaths for the usa and european economic area countries. medRxiv, 2020.

10. [10].↵

    X. Liu , G. Hewings , S. Wang , M. Qin , X. Xiang , S. Zheng , and X. Li . Modeling the situation of COVID-19 and effects of different containment strategies in china with dynamic differential equations and parameters estimation. medRxiv, 2020.

11. [11].↵

    M. Chinazzi , J. Davis , M. Ajelli , C. Gioannini , M. Litvinova , S. Merler , A. Pastore y Piontti , K. Mu , L. Rossi , K. Sun , C. Viboud , X. Xiong , H. Yu , M. Halloran , I. Longini , and A. Vespignani . The effect of travel restrictions on the spread of the 2019 novel coronavirus (COVID-19) outbreak. Science, 368(6489):395–400, 2020.

12. [12].↵

    H. Tian , Y. Liu , Y. Li , C. Wu , B. Chen , M. Kraemer , B. Li , J. Cai , B. Xu , Q. Yang , B. Wang , P. Yang , Y. Cui , Y. Song , P. Zheng , Q. Wang , O. Bjornstad , R. Yang , B. Grenfell , O. Pybus , and C. Dye . The impact of transmission control measures during the first 50 days of the COVID-19 epidemic in china. medRxiv, 2020.

13. [13].↵

    A. Kucharski , T. Russell , C. Diamond , Y. Liu , J. Edmunds , S. Funk , R. Eggo , F. Sun , M. Jit , J. Munday , N. Davies , A. Gimma , K. Van Zandvoort , H. Gibbs , J. Hellewell , C. Jarvis , S. Clifford , B. Quilty , N. Bosse , S. Abbott , P. Klepac , and S. Flasche . Early dynamics of transmission and control of COVID-19: a mathematical modelling study. Lancet Infect Dis, 20(5):553–558, 05 2020.

14. [14].↵

    S. Park , D. Cornforth , J. Dushoff , and J. Weitz . The time scale of asymptomatic transmission affects estimates of epidemic potential in the COVID-19 outbreak. medRxiv, 2020.

15. [15].↵

    F. Di Lauro , I. Z. Kiss , and J. C. Miller . Optimal timing of one-shot interventions for epidemic control. PLoS Comput Biol, 17(3):e1008763, 03 2021.

16. [16].↵

    D.H. Morris , F.W. Rossine , J.B. Plotkin , and S. A. Levin . Optimal, near-optimal, and robust epidemic control. arXiv, 2004.02209, 2021.

17. [17].↵

    P. A. Bliman and M. Duprez . How best can finite-time social distancing reduce epidemic final size? J Theor Biol, 511:110557, Feb 2021.

18. [18].↵

    E.A. Hernandez-Vargas , G. Giordano , E. D. Sontag , J. G. Chase , H. Chang , and A. Astolfi . First special section on systems and control research efforts against covid-19 and future pandemics. Annual Reviews in Control, 50:343–344, 2020.

19. [19].↵
20. [20].↵

    G. Giordano , F. Blanchini , R. Bruno , P. Colaneri , A. Di Filippo , A. Di Matteo , and M. Colaneri . Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy. Nat. Med., 26:855860, 2020.

21. [21].↵

    M. Robinson and N. I. Stilianakis . A model for the emergence of drug resistance in the presence of asymptomatic infections. Mathematical Biosciences, 243(2):163–177, 2013.

22. [22].↵

    M. Grunnill . An exploration of the role of asymptomatic infections in the epidemiology of dengue viruses through susceptible, asymptomatic, infected and recovered (sair) models. Journal of Theoretical Biology, 439:195–204, 2018.

23. [23].↵

    E. Franco . A feedback SIR (fSIR) model highlights advantages and limitations of infection-based social distancing. arXiv preprint arXiv: 2004.13216, 2020.

24. [24].↵

    M. G. Pedersen and M. Meneghini . Quantifying undetected COVID-19 cases and effects of containment measures in Italy. ResearchGate Preprint (online 21 March 2020) DOI, 10, 2020.

25. [25].↵

    J.L. Gevertz , J.M. Greene , C Hixahuary Sanchez Tapia , and E D Sontag . A novel COVID-19 epidemiological model with explicit susceptible and asymptomatic isolation compartments reveals unexpected consequences of timing social distancing. Journal of Theoretical Biology, 510:110539, 2020.

26. [26].↵

    M.A. Al-Radhawi , M. Sadeghi , and E.D. Sontag . Long-term regulation of prolonged epidemic outbreaks in large populations via adaptive control: a singular perturbation approach. 2021. Submitted (preprint in http://arxiv.org/abs/2103.08488).

27. [27].↵

    S. Federico and G. Ferrari . Taming the spread of an epidemic by lockdown policies. Center for Mathematical Economics Working Papers 639, Center for Mathematical Economics, Bielefeld University, July 2020.

28. [28].↵
29. [29].↵

    E.D. Sontag . Mathematical Control Theory. Deterministic Finite-Dimensional Systems, volume 6 of Texts in Applied Mathematics. Springer-Verlag, New York, second edition, 1998.

30. [30].↵

    O. Diekmann , J. Heesterbeek , and J. Metz . On the definition and the computation of the basic reproduction ratio r0 in models for infectious diseases in heterogeneous populations. J. Math. Biol., 28, 1990.

31. [31].↵

    O. Diekmann , J. Heesterbeek , and M. Roberts . The construction of next-generation matrices for compartmental epidemic models. J. R. Soc. Interface. DOI: 10.1098/rsif.2009.0386, 7, 2010.

Sontag Notes on Systems Biology Solutions

Source: https://www.medrxiv.org/content/10.1101/2021.04.11.21255289v3.full

Share :