QUARANTINE
SZE-BI HSU AND LIH-ING W. ROEGER
Abstract. In this article, we present the continuing work on a SARS model without quarantine by Hsu and Hsieh [SIAM J. Appl. Math., 66 (2006), 627– 647]. An ”acting basic reproductive number” ψ is used to predict the final size of the susceptible population. We find the relation among the final susceptible population size S∞, the initial susceptible population S0, and ψ. If ψ > 1, the disease will prevail and the final size of the susceptible, S∞, becomes zero; therefore, everyone in the population will be infected eventually. If ψ < 1, the disease dies out, and then S∞>0 which means part of the population will never be infected. Also, when S∞ >0, S∞ is increasing with respect to the initial susceptible population S0, and decreasing with respect to the acting basic reproductive number ψ.
1. Introduction
The following is a differential equation model for severe acute respiratory syn-drome (SARS) without quarantine proposed by Hsu and Hsieh [4].
S′ = −βIS E + I + S c 1 + a(P + R + D), E′ = βIS E + I + S c 1 + a(P + R + D)−µE, I′ = µE − (σ1+ ρ1+ γ3)I, (1) P′ = γ3I − (σ2+ ρ2)P, R′ = σ1I + σ2P, D′ = ρ1I + ρ2P.
The initial conditions are S(0) = S0 > 0, I(0) = I0 > 0, E(0) = P (0) = R(0) =
D(0) = 0, and S(t) + E(t) + I(t) + P (t) + R(t) + D(t) ≡ N = I0+ S0. The
vari-ables S(t), E(t), I(t), and P (t) are the number of susceptible individuals, infected asymptomatic individuals, infected individuals with onset of symptoms, and iso-lated probable SARS cases at time t. R(t) is the cumulative number of discharged SARS patients and D(t), the cumulative number of SARS deaths, at time t. The basic assumptions of the model (1) are the following:
(i) A SARS-infective person is infective after onset of symptoms.
(ii) An infective person can infect others unless isolated as probable case with reduced contact rate depending on the effectiveness of isolation.
(iii) A probable case is removed from isolation either by death or discharge. (iv) As people’s behavior change caused by public response to the outbreak, the
contact rate decreases with the increasing cumulated number of probable cases, deaths, and the removed.
(v) We assume homogeneous mixing population.
The parameter β is the average number of susceptible individuals infected by one infective individual per effective contact per day; the parameter a describes the effect of behavior change in reduction of contact due to cumulative numbers of probable cases, deaths, and the removed; c is the per-capita effective contact number in the absence of an outbreak; µ is the progression rate to onset of symptoms; ρi,
i = 1, 2, are the respective fatality rate of infective cases and isolated probable SARS patient; σi, i = 1, 2, are the respective discharged rate of infective cases and
isolated probable cases; γ3is the isolation rate of infectives not under quarantine.
We note that the omission of R(t) in the denominator of disease transmission term −βIS
E+I+S in the S equation is due to immunity, as many SARS patients in
Taiwan still have detectable amount of anti-bodies months after recovery [3]. The accumulated removed class R(t) does have impact on the contact number; in the S equation, c
1+a(P +R+D) term is decreasing with R(t). System (1) is a SARS
epidemic model without quarantine. To see the details of the modeling process and the related model with quarantine, please refer to [4].
Hsu and Hsieh [4] had done some analysis for this model. We will state their re-sults first. The equilibrium with the susceptible present for the system in (S, E, I, P, R, D) is (S∗ , 0, 0, 0, R∗ , D∗ ) with S∗ + R∗ + D∗
= N ; the equilibrium with no susceptible present is (0, 0, 0, 0, R#, D#) with R#+ D#= N . Define
(2) ψ = βc
(σ1+ ρ1+ γ3)(1 + aN )
.
Hsu and Hsieh [4] had obtained the following results relating to the parameter ψ. THEOREM 1. For the SARS model without quarantine (1), the solutions have the following asymptotic properties: S(t) → S∞ ≥ 0, R(t) → R∞ > 0, D(t) →
D∞> 0, I(t) → 0, E(t) → 0, and P (t) → 0 as t → ∞.
THEOREM 2. Consider system (1). The parameter ψ is defined as in (2). (i) If ψ < 1, then S(t) → S∞> 0 as t → ∞.
(ii) If ψ > 1, then S(t) → 0 as t → ∞.
The above two theorems state that the asymptotic dynamics are actually global so that we may write the equilibrium with the susceptible present or the equilibrium without the susceptible present to be (S∞, 0, 0, 0, R∞, D∞) and (0, 0, 0, 0, R∞, D∞)
depending on the parameter ψ. We will show that although ψ is not a basic reproductive number, it acts like one. If ψ > 1, the disease will prevail. When ψ > 1, the final size S∞of the susceptible becomes zero. Eventually, everyone will
be infected and either dies or recovers. In the following section, we will give details of finding the relation among S0, S∞, and ψ.
In the classical Kermack-Mckendric SIR model, the asymptotic state S∞satisfies
a transcendental equation [5, 6], so does S∞obtained in system (1). Diekmann et.
al [1] also found the final size of epidemics in a closed population. Their model is described by a nonlinear Volterra integral equation of convolution type, just as the general Kermack-McKendrick model. Similarly results that relate the final size and the basic reproductive number R0can also be found in the book by Diekmann and
2. Main Results
We can integrate the equations of P , R, and D in (1) from t = 0 to ∞. Since P (∞) = 0, we have (3) γ3 Z ∞ 0 I(t)dt = (σ2+ ρ2) Z ∞ 0 P (t)dt, (4) R∞= σ1 Z ∞ 0 I(t)dt + σ2 Z ∞ 0 P (t)dt, and (5) D∞= ρ1 Z ∞ 0 I(t)dt + ρ2 Z ∞ 0 P (t)dt. By substituting (3) into (4) and (5), we obtain
R∞= σ1+ σ2 γ3 (σ2+ ρ2) Z ∞ 0 I(t)dt, and D∞= (ρ1+ ρ2 γ3 (σ2+ ρ2) ) Z ∞ 0 I(t)dt. Let (6) r = σ1+ σ2 γ3 (σ2+ρ2) ρ1+ ρ2(σ2γ+ρ3 2) . Then we have R∞ D∞ = r.
Let V = S + E + I. Then P + R + D = N − V , and system (1) can be simplified so that the two equations for S and V are
S′ = −βIS V c 1 + a(N − V ), V′ = −(ρ1+ σ1+ γ3)I. Then we have dS dV = βc ρ1+ σ1+ γ3 S V (1 + a(N − V )).
Applying the method of separation of variables and integrating both sides of the equation from t = 0 leads to
ln S S0 = ψ ln V V0(1 + a(N − V )) ,
where ψ is defined as in (2). Since V0 = S0+ E0+ I0 = N , and V∞ = S∞, S∞
satisfies the following
(7) S∞ S0 = S ∞ N (1 + aN − aS∞) ψ . We can show that the equation in x
(8) x S0 = x N (1 + aN − ax) ψ
THEOREM 3. Consider the equation in (8).
(i) If ψ < 1 this equation has a root at zero and a unique positive root x ∈ (0, S0).
(ii) If ψ ≥ 1 this equation has a root at zero and no roots in (0, S0].
PROOF. (i) ψ < 1. The roots of this equation are x = 0 and the roots of
(9) N (1 + aN − ax) = S 1 ψ 0 x 1−1 ψ. Write f (x) = S 1 ψ 0 x 1−1 ψ, then d 2 f dx2 = 1 ψ2(1 − ψ)S 1 ψ 0 x −1−1 ψ > 0. So f is convex and
hence intersects each chord twice. When x ≥ 0 equation (9) has a root in (0, S0)
and a root in (S0, ∞) so cannot have any more positive roots. The statement (i)
follows in this case.
(ii) ψ ≥ 1. The roots of this equation are x = 0 and the roots of (9). But for x ∈ (0, S0] N (1 + aN − ax) ≥ N (1 + a(N − S0)) > S0≥S 1 ψ 0 x 1−1 ψ.
The statement (ii) follows.
0 50 100 150 200 250 300 175 180 185 190 195 200 Time S 0 50 100 150 200 0 0.05 0.1 0.15 0.2 x
Figure 1. The asymptotic example for system (1). The top fig-ure shows the susceptible population approaching S∞ = 175.60
when ψ = 0.6768 < 1. The lower figure shows the graph of the function g(x) = x
N(1+aN −ax)
ψ − x
S0 and its two roots, 0 and
S∞ = 175.60. We obtain the results by choosing the parameters
β = 0.3 (person)(contact)−1(day)−1 , c = 2 (contact)(person)−1,
a = 0.0013 (person)−1, µ = 0.14 (day)−1, σ
1= σ2= 0.2 (day)−1,
ρ1= ρ2= 0.1 (day)−1, γ3= 0.4 (day)−1, and the initial conditions
In case (i) of Theorem 2, since S∞+ R∞+ D∞= N , and R∞ and D∞ satisfy
R∞/D∞ = r, we can find all three values, S∞, R∞, and D∞. In case (ii) of
Theorem 2, all solutions approach the equilibrium (0, 0, 0, 0, R∞, D∞), that is to say
that eventually everyone gets infected and recovers or dies. Since R∞+ D∞= N ,
it is also easy to find R∞ and D∞. Therefore, we have the following results.
THEOREM 4. Consider system (1). The parameters ψ and r are defined as in (2) and (6) respectively. (i) If ψ < 1 then S∞> 0, R∞= r(N − S∞) 1 + r , and D∞= N − S∞ 1 + r . (ii) If ψ > 1, then S∞= 0, R∞= rN 1 + r, and D∞= N 1 + r.
If ψ < 1, then S∞ > 0, we can show that S∞ decreases as the parameter ψ
increases, dS∞/dψ < 0. We can also show that S∞ increases as the parameter a
increases or as S0increases. That is, we can show that dS∞/da > 0 and dS∞/dS0>
0. Let
(10) g = S∞
N (1 + aN − aS∞)
.
Note that if ψ < 1, we have 0 < S∞ < N , so that 0 < g < 1. We will need the
following lemma.
LEMMA 1. Consider the equation (7). If ψ < 1, then 1 − ψ(1 + aN )
1 + aN − aS∞
> 0.
PROOF. From equation (9), we know that the function f = S
1 ψ
0 x1−
1
ψ is convex
and there is only one positive root of equation (9), S∞, in (0, S0). Therefore,
f′ (S∞) < d dx(N (1 + aN − ax)) x=S ∞= −aN, i.e., 1 ψ−1 S0 S∞ ψ1 > aN, Using equation (7), we obtain
1 ψ −1 N (1 + aN − aS∞) S∞ > aN ⇐⇒(1 − ψ)(1 + aN − aS∞) > aψS∞ ⇐⇒1 + aN − aS∞−ψ(1 + aN ) > 0 ⇐⇒1 − ψ(1 + aN ) 1 + aN − aS∞ > 0. When ψ < 1, S∞ > 0. If we consider S∞ as a function of ψ, than S∞ is a
THEOREM 5. Consider the equation (7). Let ψ < 1 and S∞ be a function of
the parameter ψ. Then
dS∞
dψ < 0.
PROOF. Let S∞= S∞(ψ) and g = g(ψ) be defined as in (10). Then we have
(11) g ′ (ψ) g(ψ) = S′ ∞(ψ)(1 + aN ) S∞(ψ)(1 + aN − aS∞(ψ)) . Equation (7) is now S∞(ψ) S0 = g(ψ)ψ.
Differentiate the equation with respect to the parameter ψ yields S′ ∞(ψ) S0 = g(ψ)ψ ln g(ψ) + ψg ′ (ψ) g(ψ) = S∞(ψ) S0 ln g(ψ) + ψg ′ (ψ) g(ψ) Apply g′
/g in (11) and simplify, we have S′ ∞= S∞ln g(ψ) + S ′ ∞ ψ(1 + aN ) 1 + aN − aS∞ , i.e., 1 − ψ(1 + aN ) 1 + aN − aS∞ S′ ∞= S∞ln g(ψ).
The right side of the equation is negative because 0 < g(ψ) < 1. By Lemma 1, we have S′
∞(ψ) < 0.
The results in this theorem does not surprise us, since we have mentioned in the introduction that the parameter ψ is acting like a basic reproductive number. Therefore, increasing ψ should also increase the epidemic and hence it decreases the final size of the susceptible. Similarly, the following theorem says that S∞increases
with the initial number of the susceptible.
THEOREM 6. Consider the equation (7). Let ψ < 1 and S∞ be a function of
S0. Then
dS∞
dS0
> 0.
PROOF. Assume that S∞is a function of S0, then g = g(S0). Differentiate the
equation (7) with respect to S0yields
1 S2 0 (S0S∞′ (S0) − S∞) = ψ S∞ N (1 + aN − aS∞) ψ−1 1 + aN N (1 + aN − aS∞)2 S′ ∞(S0) = 1 S0 ψ(1 + aN ) 1 + aN − aS∞ S′ ∞(S0).
The last equation comes from substituting equation (7) into the right hand side. Therefore, we have 1 − ψ(1 + aN ) 1 + aN − aS∞ S′ ∞(S0) = S∞ S0 > 0. Use Lemma 1, we have S′
In the SARS model (1), the parameter a describes the effect of behavior change in reduction of contact due to the cumulative number of probable cases. If a increases, number of contacts will be reduced, the epidemic will not as severe, and so the final size of the susceptible S∞ increases. The result is the following theorem.
THEOREM 7. Consider the equation (7). Let ψ < 1 and S∞ be a function of
the parameter a. Then
dS∞
da > 0.
PROOF. Let S∞= S∞(a), ψ = ψ(a), and g = g(a). Then
g′ (a) g(a) = S′ ∞(1 + aN ) − S∞(N − S∞) S∞(1 + aN − aS∞) and ψ ′ (a) ψ(a) = − N 1 + aN. Then equation (7) is now
S∞(a)
S0
= g(a)ψ(a).
Differentiate the equation with respect to the parameter a yields S′ ∞(a) S0 = g(a)ψ(a) ψ′
(a) ln g(a) + ψ(a)g
′ (a) g(a) =S∞(a) S0 ψ′
(a) ln g(a) + ψ(a)g
′ (a) g(a) . Substitute g′
(a)/g(a) into the above equation and rearrange. We have 1 − ψ(1 + aN ) 1 + aN − aS∞ S′ ∞(a) = S∞ψ ψ′ ψ ln g − N − S∞ 1 + aN − aS∞ .
If we can show that the right hand side of the above equation is positive, and since by Lemma 1, the factor in front of S′
∞(a) is positive, then we should obtain the
result that S′
∞(a) > 0. Therefore, we need to show that
ψ′ ψ ln g − N − S∞ 1 + aN − aS∞ > 0. i.e., (12) N 1 + aN ln S∞ N (1 + aN − aS∞) + N − S∞ 1 + aN − aS∞ < 0. Let h(x) = N 1 + aNln x N (1 + aN − ax)+ N − x 1 + aN − ax. Then h(0+) = −∞ and h(N ) = 0. We can show that
h′
(x) = (1 + aN )(N − x)
x(1 + aN − ax)2 > 0, for all x ∈ (0, N ).
The function h(x) is increasing on (0, N ) from −∞ to 0, and therefore h(S∞) < 0
3. Conclusion
We consider the final size of a SARS epidemic model without quarantine. Our model is not an integral equations model, unlike other papers [1, 5, 6] of similar results which describe the epidemics using integral equations.
We use an ”acting basic reproductive number” ψ to predict the final size of the epidemics, and there is a relation among the final size of the susceptible S∞, the
initial susceptible S0, and the parameter ψ. We show that if ψ > 1, the disease stays
in the population until all population are infected and recover or die. If ψ < 1, then the final size of the susceptible is greater than zero, S∞> 0. S∞decreases with the
parameter ψ, increases with the initial size of the susceptible S0, and increases with
the parameter a which is the measure of the effect of behavior change in reduction of contact due to the cumulation number or probable cases.
Acknowledgement. We thank two anonymous reviewers for their helpful com-ments that improved the paper. S.B. Hsu is supported by NSC of Taiwan under grant 95-2115-M-007-008. The research was done partially while Lih-Ing Roeger was visiting the National Center for Theoretical Sciences at Tsing-Hua University in Taiwan.
References
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[2] O. Diekmann and J. A. P. Heesterbeek. Mathematical Epidemiology of Infectious Diseases. John Wiley and Sons Ltd., New York, 2000.
[3] Y.-H. Hsieh, C.-C. King, C. W. S. Chen, M.-S. Ho, J.-Y. Lee, F.-C. Liu, Y.-C. Wu, J.-S. Wu Quarantine for SARS, Taiwan. Emerging Infectious Diseases, 11(2): 278–282, 2005. [4] Sze-Bi Hsu and Ying-Hen Hsieh. Modeling Intervention Measures and Severity-dependent
Public Response During Severe Acute Respiratory Syndrome Outbreak, SIAM J. Appl. Math., 66 (2006), 627–647.
[5] J. D. Murray. Mathematical Biology. Springer-Verlag, New York, 1989.
[6] P. E. Waltman. Deterministic Threshold Models in the Theory of Epidemics. Lecture Notes in Biomathematics, Vol 1, Springer-Verlag, New York, 1974.
Sze-Bi Hsu, Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan
E-mail address: sbhsu@math.nthu.edu.tw
Lih-Ing W. Roeger, Department of Mathematics and Statistics, Box 41042, Texas Tech University, Lubbock, TX 79409, Tel: 806-742-2580, Fax: 806-742-1112
