key: cord-0986608-t0rhb7ee authors: Hsu, Sze-Bi; Roeger, Lih-Ing W. title: The final size of a SARS epidemic model without quarantine date: 2007-09-15 journal: Journal of Mathematical Analysis and Applications DOI: 10.1016/j.jmaa.2006.11.026 sha: c8616899b8ec8bdcdd7901deb02fb5c021056330 doc_id: 986608 cord_uid: t0rhb7ee Abstract In this article, we present the continuing work on a SARS model without quarantine by Hsu and Hsieh [Sze-Bi Hsu, 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]. 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 S 0 , 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 S 0 , and decreasing with respect to the acting basic reproductive number ψ. Severe acute respiratory syndrome (SARS) was first recognized as a global threat in mid-March 2003. The first known cases and the last case of SARS occurred in November 2002 and in July 2003. The international spread of SARS resulted in 8098 SARS cases in 26 countries, with 774 deaths [6] . It is believed that the transmission of SARS can be effectively controlled by adhering to basic public health measures-rapid case detection, case isolation, contact tracing, good infection control (hand-washing and the use of personal protective equipment), and quarantine. Hsu and Hsieh [3] had modeled the SARS outbreak in Taiwan for year 2003 using the general models that include two different levels of quarantines, level A for those who were suspected of having close contact with a suspected SARS case and level B for those who traveled from affected areas after April 28, 2003. Their model equations are as follows: where λ function is the rate of incidence of infection and is given by = βc 1 + a(P + R + D) The variables at time t are defined as the following: S-the number of susceptible individuals. E-the number of infected asymptomatic persons not under any quarantine. Q A -the number of infected asymptomatic persons under level A quarantine. Q B -the number of imported asymptomatic infected persons. I -the number of infective persons with onset of symptoms but not isolated or quarantined. P -the number of isolated probable SARS cases. D-the cumulative number of SARS deaths. R-the cumulative number of discharged SARS patients. For this model, the time unit is in days. The initial conditions are S(0) = S 0 > 0, Since the duration of the outbreak was a short time, the total population is assumed to be constant: The basic assumptions for the model are the following: (1) A SARS-infective person is infective after onset of symptoms. (2) A quarantined person is quarantined without symptoms (hence are not infective), becoming infective with reduced contact rates due to quarantine, and is isolated upon diagnosis. (3) An infective person can infect others unless isolated as probable case with reduced contact rates depending on the effectiveness of isolation. (4) A probable case is removed from isolation either by death or discharge. (5) 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. (6) Homogeneous mixing with quarantine-adjusted incidence is assumed. (7) Quarantine for level A is proportional to the number of infected asymptomatic persons. (8) Imported cases are a function of time (Q(t) = 0, 1, or 2). The parameters are defined as follows: β-the average number of susceptible individuals infected by one infective individual per effective contact per day. c-the per-capita effective contact number in the absence of an outbreak. a-the effect of behavior change in reduction of contact due to cumulative numbers of probable cases, deaths, and the removed. α A , α B , or α P -the proportional reduction in infectivity of quarantined persons due to levels A, B quarantine (before isolation) and probable cases, respectively. μ-the progression rate to onset of symptoms. q 1 -the proportion of recruitment of asymptomatic infected persons for level A quarantine. γ 1 , γ 2 -the isolation rates of Q A and Q B , respectively. γ 3 -the isolation rate of infectives not under quarantine. σ 1 , σ 2 -the respective discharged rate of infective cases and isolated probable cases. ρ 1 , ρ 2 -the respective fatality rate of infective cases and isolated probable SARS patient. For the general model of SARS (1), Hsu and Hsieh had derived the basic reproduction number and the full description of its dynamics was provided in [3] . For the worse case scenario when there are no quarantines and the probable cases are not isolated, i.e., α A = α B = α P = 0, then the model (1) becomes the following SARS model without quarantine [3] : Hsu and Hsieh [3] had done some analysis for this model. We will summarize their results. The equilibrium with the susceptible present for the system in (S, E, I, P , R, D) is The basic reproduction number R 0 for this model is If R 0 < 1, the equilibrium with the susceptible present is locally asymptotically stable and if R 0 > 1, unstable. When S * = 0 in R 0 , we define Note that 0 < ψ R 0 . If ψ > 1, then R 0 > 1 and the disease prevails. We will show that this is indeed the case. When ψ > 1, the final size S ∞ of the susceptible becomes zero. Eventually, everyone will be infected and either dies or recovers. Hsu and Hsieh [3] had obtained the following results relating to the parameter ψ . Theorem 1. For the SARS model without quarantine (2) , the solutions have the following asymptotic properties: Theorem 2. Consider system (2) . The parameter ψ is defined as in (3). 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 the finding of the relation among S 0 , S ∞ , and ψ . In the classical Kermack-Mckendric SIR model, the asymptotic state or the final size S ∞ satisfies a transcendental equation [4, 5] , so does S ∞ obtained in system (2). 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. Similar results that relate the final size and the basic reproductive number R 0 can also be found in the book by Diekmann and Heesterbeek [2] . We can integrate the equations of P , R, and D in (2) from t = 0 to ∞. Since P (∞) = 0, we have and By substituting (4) into (5) and (6), we obtain Then we have R ∞ D ∞ = r. Let V = S + E + I . Then P + R + D = N − V , and system (2) can be simplified so that the two equations for S and V are Then we have dS dV = βc a(N − V ) ) . Applying the method of separation of variables and integrating both sides of the equation from t = 0 leads to where ψ is defined as in (3). Since V 0 = S 0 + E 0 + I 0 = N , and V ∞ = S ∞ , S ∞ satisfies the following: We can show that the equation in x, has only two roots at x = 0 and x = S ∞ ∈ (0, S 0 ) when ψ < 1, as seen in Fig. 1 . (10) So f is convex and hence intersects each chord twice. When x 0, Eq. (10) has a root in (0, S 0 ) and a root in (S 0 , ∞) 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 (10). But for x ∈ (0, S 0 ], The statement (ii) follows. 2 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 (2) . The parameters ψ and r are defined as in (3) and (7) , respectively. (i) If ψ < 1, then (ii) If ψ > 1, then 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 S 0 increases. That is, we can show that dS ∞ /da > 0 and dS ∞ /dS 0 > 0. Let Note that if ψ < 1, we have 0 < S ∞ < N, so that 0 < g < 1. We will need the following lemma. If ψ < 1, then Proof. From Eq. (10), we know that the function f = S (0, S 0 ) . Therefore, i.e., Using Eq. (8), we obtain When ψ < 1, S ∞ > 0. If we consider S ∞ as a function of ψ , then S ∞ is a decreasing function. Theorem 5. Consider Eq. (8). Let ψ < 1 and S ∞ be a function of the parameter ψ . Then Proof. Let S ∞ = S ∞ (ψ) and g = g(ψ) be defined as in (11). Then we have . Equation (8) is now Differentiating the equation with respect to the parameter ψ yields Applying g /g in (12) and simplifying, we have i.e., The right side of the equation is negative because 0 < g(ψ) < 1. By Lemma 1, we have The result 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. Proof. Assume that S ∞ is a function of S 0 , then g = g(S 0 ). Differentiating Eq. (8) with respect to S 0 yields The last equation comes from substituting Eq. (8) into the right-hand side. Therefore, we have In the SARS model (2), 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 be not as severe, and so the final size of the susceptible S ∞ increases. The result is the following theorem. Theorem 7. Consider Eq. (8). Let ψ < 1 and S ∞ be a function of the parameter a. Then Proof. Let S ∞ = S ∞ (a), ψ = ψ(a), and g = g(a). Then Differentiating the equation with respect to the parameter a yields Substitute g (a)/g(a) into the above equation and rearrange. We have 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 i.e., Let Then h(0+) = −∞ and h(N ) = 0. We can show that 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 which is the inequality (13). 2 We consider the final size of a SARS epidemic model without quarantine. Our model is not an integral equations model, unlike other papers [1, 4, 5] 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 S 0 , and the parameter ψ . We show that if ψ > 1, the disease stays in the population until all population is infected and recovers or dies. 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 S 0 , 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. On the final size of epidemics within herds Mathematical Epidemiology of Infectious Diseases Modeling intervention measures and severity-dependent public response during severe acute respiratory syndrome outbreak Deterministic Threshold Models in the Theory of Epidemics World Health Organization, WHO guidelines for the global surveillance of severe acute respiratory syndrome (SARS) We thank the anonymous reviewer for the helpful comments and suggestions.