Threshold dynamics of a non-autonomous SEIRS model with quarantine and isolation
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DOI: 10.1007/s12064-011-0148-6
- Cite this article as:
- Safi, M.A., Imran, M. & Gumel, A.B. Theory Biosci. (2012) 131: 19. doi:10.1007/s12064-011-0148-6
Abstract
A model for assessing the effect of periodic fluctuations on the transmission dynamics of a communicable disease, subject to quarantine (of asymptomatic cases) and isolation (of individuals with clinical symptoms of the disease), is considered. The model, which is of a form of a non-autonomous system of non-linear differential equations, is analysed qualitatively and numerically. It is shown that the disease-free solution is globally-asymptotically stable whenever the associated basic reproduction ratio of the model is less than unity, and the disease persists in the population when the reproduction ratio exceeds unity. This study shows that adding periodicity to the autonomous quarantine/isolation model developed in Safi and Gumel (Discret Contin Dyn Syst Ser B 14:209–231, 2010) does not alter the threshold dynamics of the autonomous system with respect to the elimination or persistence of the disease in the population.
Keywords
Quarantine Isolation Periodicity Basic reproduction ratioIntroduction
It is well known that some infectious diseases, such as measles, mumps and chickenpox, exhibit periodic fluctuations in their transmission dynamics. For instance, the city of New York recorded yearly outbreaks of chickenpox and mumps, and a biennial pattern of measles outbreaks, between 1929 and 1970 (Cooke and Kaplan 1976; London and Yorke 1973). Furthermore, contact rates may vary during a time period due to a number of factors such as environmental (weather changes; emergence of insects caused by seasonal variation) and the fact that children are in school during certain months etc. (Diekmann and Heesterbeek 2000). London and Yorke (1973) showed such variations in contact rates by studying data for mumps, chickenpox and measles. Other diseases show seasonal behavior as well (see, for instance, Bacaër (2009), Bacaër and Guernaoui (2006), Cornelius (1971), Dowell (2001), Earn et al. (2002), Hethcote and Levin (1989), London and Yorke (1973)). As noted by Cooke and Kaplan (1976), since periodic fluctuation in contact rate is crucial to a number of diseases, it is instructive and theoretically evaluate the effect of such fluctuations on the transmission dynamics of the relevant diseases in a population.
During outbreaks of a communicable disease in human populations, basic public health control measures, notably quarantine (of individuals suspected of being exposed to the disease) and isolation (of individuals with clinical symptoms of the disease) are generally implemented aimed at controlling or mitigating the disease burden (measured in terms of number of new cases, hospitalization, morbidity, mortality). Over the decades, such control measures have been successfully applied to effectively combat the spread of some emerging and re-emerging diseases such as leprosy, plague, cholera, typhus, yellow fever, smallpox, diphtheria, tuberculosis, measles, ebola, pandemic influenza and, more recently, severe acute respiratory syndrome (SARS) (Chowell et al. 2004a, b; Donnelly et al. 2003; Gumel et al. 2004; Hethcote et al. 2002; Lipsitch et al. 2003; Lloyd-Smith et al. 2003; McLeod et al. 2006; Riley et al. 2003; Wang and Ruan 2004; Webb et al. 2004). However, as noted by McLeod et al. (2006), such basic control measures are gradually refined during the course of a disease outbreak (as more data and knowledge about the epidemiology and biology of the disease become available). Thus, it is reasonable to include periodicity in disease transmission models that involve the use quarantine and isolation.
The purpose of the current study is to qualitatively assess the impact of periodicity on the transmission dynamics of communicable disease in the presence of quarantine and isolation. In particular, to determine whether or not adding periodicity to the autonomous quarantine/isolation model considered in Safi and Gumel (2010) affects the dynamics of the quarantine/isolation model with respect to the elimination and persistence of the disease. To achieve this objective, a deterministic non-autonomous system of non-linear differential equations, which takes into account the aforementioned periodicity, will be designed and analyzed.
Model formulation
The model to be considered is that for the transmission dynamics of an infectious disease, in the presence of quarantine of exposed individuals and isolation of infected individuals with clinical symptoms of the diseases (infectious and symptomatically-infected individuals are used interchangeably in this study). It is based on splitting the total population at time t, denoted by N(t), into the sub-populations of susceptible (S), exposed (infected, but not yet show clinical symptoms of the disease; E), infected with symptoms (I), quarantined (Q), hospitalized (H) and recovered (R) individuals (it is assumed that individuals in the Q class are infected but do not display clinical symptoms of the disease).
It is worth mentioning that, although (in general) the process of quarantine also involves the isolation of susceptible individuals who are suspected of being exposed to the disease (see, for instance, Feng et al. (2007), Lipsitch (2003)), the quarantine class (Q) involves only newly-infected (asymptomatic) individuals (detected either via contact tracing of symptomatic cases or random testing). That is, in this study quarantine refers to the removal of newly-infected individuals from having contact with the general population (i.e. individuals who remain susceptible at the end of the quarantine period are not counted in the Q class). The justification for this is based on the fact that, for large total population sizes (N), the quarantine of susceptible individuals is unlikely to have a significant impact on the disease dynamics (Feng et al. 2007). It is known, for instance, that the mass quarantine implemented during the SARS outbreaks in the Greater Toronto Area of Canada only resulted in the detection of very few confirmed SARS cases (Day et al. 2006).
In (2), β(t) is the effective time-dependent contact rate, the modification parameter 0 ≤ η(t) < 1 accounts for the assumed reduction of infectiousness of quarantined and hospitalized individuals in relation to the symptomatically-infected (infectious) individuals in the I class. This study assumes that exposed individuals can transmit infection (at a assumed reduced rate β(t)η_{ E }(t), where 0 ≤ η_{ E }(t) < 1 accounts for the reduction of transmission rate of exposed individuals in relation to individuals in the I class). It should be mentioned that many disease modeling studies that include quarantine tend to assume that quarantined individuals do not transmit infection (because individuals in quarantine are typically asymptomatic; and, for some diseases such as HIV, there is positive correlation between infectiousness and viral load). This assumption is relaxed in this study by allowing for the possibility of disease transmission by individuals in quarantine. Transmission by asymptomatically-infected individuals (such as those in the E and Q classes) occurs in the context of some diseases, such as influenza.
In (3), \(\epsilon_1\) and \(\epsilon_2\) (with \(0\leq \epsilon_1,\epsilon_2\leq 1\)) are modification parameters used to measure the efficacy of quarantine and isolation in preventing quarantined and isolated individuals from having contact with the general public (thereby not partaking in the disease transmission process). If \(\epsilon_1=\epsilon_2=0, \) then quarantine and isolation are perfectly implemented (so that individuals in the quarantine and isolation classes are not part of the actively-mixing population, and do not transmit infection). This is in line with one of the six incidence function formulations (quarantine-adjusted) in Hethcote et al. (2002). Leaky quarantine and isolation is represented by the case with \(0<\epsilon_1,\epsilon_2<1. \) The case \(\epsilon_1=\epsilon_2=1\) represents the scenario when individuals in quarantine and isolation are equally likely to have contact with the general public than anyone else in the population. The vast majority of quarantine and isolation models published in the literature, such as those in Chowell et al. (2004), Feng (2007a, b, Gumel et al. (2004), Hethcote et al. (2002), McLeod et al. (2006), Mubayi et al. (2010), Safi and Gumel (2010), Webb et al. (2004), adopt the case with \(\epsilon_1=\epsilon_2=1. \) It is worth stating that quarantine is not always administered via the healthcare system. That is, it may be administered at home, and there is no guarantee that individuals in quarantine strictly adhere to the stipulated guidelines (this may be the reason for the choice of the scenario with \(\epsilon_1=\epsilon_2=1). \)
Description of the variables and parameters of the model (4)
Description | |
---|---|
Variable | |
S(t) |
Population of susceptible individuals |
E(t) |
Population of exposed individuals |
I(t) |
Population of infected individuals with disease symptoms |
Q(t) |
Population of quarantined individuals |
H(t) |
Population of hospitalized individuals |
R(t) |
Population of recovered individuals |
Parameter | |
\(\Uppi\) |
Recruitment rate |
1/μ |
Average lifespan |
β(t) |
Effective contact rate |
η_{ E }(t) |
Modification parameter for reduced infectiousness of exposed individuals |
η(t) |
Modification parameter for reduction in infectiousness of quarantined and hospitalized individuals |
\(\epsilon_1,\epsilon_2\) |
Modification parameters for efficacy of quarantine and isolation |
κ(t) |
Progression rate from exposed to infectious class |
σ(t) |
Quarantine rate for exposed individuals |
α(t) |
Hospitalization rate for quarantined individuals |
ϕ(t) |
Hospitalization rate for infectious individuals |
ψ |
Rate of loss of infection-acquired immunity |
γ_{1}(t) |
Recovery rate for infectious individuals |
γ_{2}(t) |
Recovery rate for quarantined individuals |
γ_{3}(t) |
Recovery rate for hospitalized individuals |
δ_{1} |
Disease-induced death rate for infectious individuals |
δ_{2} |
Disease-induced death rate for hospitalized individuals |
Estimated values of the parameters of the model
Parameters |
Values (per day) |
References |
---|---|---|
\(\Uppi\) |
136 (people per day) |
Gumel et al. (2004) |
1/μ |
78 years (μ = 0.0000351 per day) |
Hong Kong Special Administrative Region (2006) |
η |
0.5 |
Assumed |
η_{ E } |
0.25 |
Assumed |
\(\epsilon_1\) |
0.5 |
Assumed |
\(\epsilon_2\) |
0.8 |
Assumed |
κ |
0.156986 |
Donnelly et al. (2003) |
σ |
0.1 |
Gumel et al. (2004) |
α |
0.156986 |
Donnelly et al. (2003) |
ϕ |
0.20619 |
Chowell et al. (2004) |
ψ |
0.5 |
Assumed |
γ_{1} |
0.03521 |
Chowell et al. (2004) |
γ_{2} |
0.042553 |
Chowell et al. (2004) |
γ_{3} |
0.042553 |
Chowell et al. (2004) |
δ_{1} |
0.04227 |
Leung et al. (2004) |
δ_{2} |
0.027855 |
Chowell et al. (2004) |
The non-autonomous model (1) is an extension of the autonomous quarantine/isolation model studied in Safi and Gumel (2010), by considering some of the parameters (namely, \(\beta,\eta,\kappa,\sigma,\phi,\gamma_1,\gamma_2, \gamma_3\;\hbox {and}\;\alpha\)) to be periodic positive continuous functions in t with period ω > 0 (unlike in the autonomous model (Safi and Gumel 2010), where all the model parameters are assumed to be constant; it is worth stating that the model in Safi and Gumel (2010) does not account for the recovery of quarantined individuals). The non-autonomous system reduces to the autonomous system in Safi and Gumel (2010) by setting \(\beta(t)=\beta,\;\eta(t)=\eta,\;\kappa(t)=\kappa,\;\phi(t)=\phi,\;\alpha(t)=\alpha,\;\gamma_1(t)=\gamma_1,\; \gamma_2(t)=0,\; \gamma_3(t)=\gamma_3\) and σ(t) = σ.
Basic properties
The basic properties of the non-autonomous model (4) (which is equivalent to system (1)) will now be studied.
Lemma 1
Stability of disease-free equilibrium (DFE)
Local stability of DFE
Although the concept of basic reproduction number has been extensively addressed (over the decades) for autonomous models for disease transmission, such a concept has not been extended to disease transmission models with periodic coefficients until very recently (see, for instance, the notable contributions of Bacaër (2007, 2009), Bacaër and Guernaoui (2006), Bacaër and Ouifki (2007), Bacaër and Abdurahman (2008), Bacaër and Ait Dads (2011) and Zhao and co-workers (2009, 2010a, b, 2008)). This article uses the methodology in Wang and Zhao (2008) to compute the reproduction number (or ratio) associated with the non-autonomous SEIRS model with quarantine and isolation, given by (4).
Lemma 2
The DFE of the model (4), given by (6), is locally-asymptotically stable if \(\mathcal{R}_0<1,\) and unstable if \(\mathcal{R}_0>1.\)
To compute the reproduction ratio \(\mathcal{R}_{0},\) associated with the model (4), the following result will be used.
Theorem 1
- (i)
If ρ(W(ω, λ)) = 1 has a positive solution λ_{0}, then λ_{0} is an eigenvalue of L, and hence \(\mathcal{R}_0>0; \)
- (ii)
If \(\mathcal{R}_0>0, \) then \(\lambda=\mathcal{R}_0\) is the unique solution of ρ(W(ω, λ)) = 1;
- (iii)
\(\mathcal{R}_0=0, \) if and only if ρ(W(ω, λ)) < 1 for all \(\lambda>0.\)
- (a)
First of all, for a given value of λ, the matrix W(ω, λ) is numerically computed using a standard numerical integrator (such as the forward-Euler or Runge-Kutta finite-difference method (Kincaid and Cheney 1991));
- (b)
Then, the spectral radius ρ(W(λ)) is calculated;
- (c)
Let f(λ) = ρ(W(λ)) − 1. Then, a root finding method (such as the bisection method (Kincaid and Cheney 1991)) is used to find the zero of f.
The epidemiological implication of the result in Lemma 2 is that the disease can be eliminated from the community (when \(\mathcal{R}_0<1\)) if the initial sizes of the sub-populations of the model are in the basin of attraction of the DFE (\(\mathcal{E}_0\)). To ensure that disease elimination is independent of the initial sizes of the sub-populations of the model, it is necessary to show that the DFE is GAS if \(\mathcal{R}_0<1. \) This is explored below.
Global stability of DFE
Theorem 2
The DFE of the model (4), given by (6), is GAS in \(\mathcal{D}\) whenever \(\mathcal{R}_0<1.\)
The following result can be proved for the system (4) using persistence theory (see, for instance, Liu et al. (2009), Zhang and Zhao 2007, Zhao 2003)):
Theorem 3
- (i)
For each value of β_{0}, the model is run 5000 times, and the transient solutions are removed by discarding the first 4900 iterates;
- (ii)
An arbitrary point (typically the first local maximum) is picked out of the remaining 100 iterates;
- (iii)
A time period of 12 days is arbitrarily selected;
- (iv)
The fixed-points of the Poincar\(\acute{\hbox {e}}\) map are then plotted, starting from the first local maximum.
Conclusions
A deterministic non-autonomous model for assessing the impact of quarantine (of asymptomatic cases) and isolation (of symptomatic cases) on curtailing the spread of a communicable disease is presented. The model is simulated using a reasonable set of parameter values (consistent with the 2002/2003 SARS outbreaks). The study shows that the associated disease-free solution is globally-asymptotically stable whenever the reproduction threshold is less than unity. The disease persists in the population if the threshold exceeds unity. The study shows that adding periodicity to the corresponding autonomous model in Safi and Gumel (2010) does not alter its qualitative dynamics with respect to the elimination and persistence of the disease.
Acknowledgements
One of the authors (ABG) acknowledges, with thanks, the support in part of the Natural Science and Engineering Research Council (NSERC) and Mathematics of Information Technology and Complex Systems (MITACS) of Canada. MAS gratefully acknowledges the support of the University of Manitoba Graduate Fellowship. The authors are grateful to N. Bacaër, L. Liu and X.-Q. Zhao for helpful discussions on the computation of reproduction ratio for non-autonomous systems. The authors are very grateful to the anonymous reviewers for their very constructive comments.