Effect of Treatment on Transmission Dynamics of SARS Epidemic

Recent years have seen an increasing trend of utilizing mathematical models for the prediction of and insight in infectious diseases. These models are considered as conceptual tools to explain the behavior of disease at different scales, and allow us to understand the spread of infection in the real world and the impact of various factors on disease dynamics. The key concepts associated with mathematical modeling, such as basic and effective reproduction number, generation time, epidemic growth rates, mortality rates, transmission rates, incubation periods, heterogeneities, disease transmission routes, risk factors for diseases spread and pre-clinical infectiousness play significant roles in the epidemiological analysis and control of diseases. The process of modeling in epidemiology has, at its heart, the same underlying philosophy and goals as ecological modeling. Both endeavors share the ultimate aim of attempting to understand the prevalence and distribution of a species, together with the factors that determine incidence, spread, and persistence [1,2].

The spatio-temporal spread of infectious diseases is the most significant area of epidemic modeling. Accurate and precise mathematical models enable scientists to understand the risk factors of disease transmission and to develop workable control strategies for possible future outbreaks. There are obvious public health and/or economic benefits in understanding the infectious dynamics of diseases in humans, animals and plants. Furthermore, it is well understood how important the spatial aspect of these dynamics is to understand disease spread [3]. In the case of emerging and re-emerging outbreaks of an infectious disease, it is crucial to quantify the characteristics of a disease in order to estimate the potential threat. Accurate estimation of these characteristics relies on modified epidemiological information.

Severe Acute Respiratory Syndrome (SARS) is one of the first pandemics of the 21st century. The World Health Organization (WHO) issued a global alert for SARS on 12th March 2003, at which point there were 150 suspected cases in 7 countries. The epidemic emerged in Guangdong Province in mainland China in November 16, 2002 and, on February 26 of the following year, new reports of SARS outbreaks came from Hong Kong and Vietnam. This contagious disease spread across the world through air travel and, as of June 13, 2003, the cumulative number of probable cases of SARS worldwide, reached 8,454 with 792 deaths [4]. Hong Kong bore a large proportion of this mortality and morbidity burden. In Hong Kong alone, 1755 probable SARS cases and 302 SARS related deaths were recorded from 15 February to 31st May 2003. SARS was caused by a coronavirus called SARS-Cov. As SARS was not previously endemic in humans, it seemed likely that SARS-Cov was a virus of animals that had crossed the species barrier to humans in the recent past [5]. Having originated in animals the virus then became efficient at human-to-human transmission which helped to develop the disease around the globe. SARS was brought under control at the end of 2003 but in 2004 a few cases emerged, either by accidental release of the virus from laboratories or from infected animals. It is difficult to make predictions regarding the resurgence of SARS, but current information suggests that the greatest risk of the reemergence of the disease may derive from animal reservoirs [5]. Recently, in 2012, a group of international scientists has found a SARS-like virus retrieved from a Chinese horseshoe bat, giving weight to the theory that these bats are the ultimate source of the virus that killed more than 900 people around the globe in 2003 [6].

Although much effort has been devoted to find the cure for SARS, there are still no effective drugs or vaccines for the disease. In this situation mathematical modeling permits the quantitative assessment of the epidemic potential of SARS and the effectiveness of measures, used to control disease transmission. Donnelly et al. [7] assessed the epidemiology of SARS in Hong Kong. They estimated the key epidemiological parametric distributions using integrated data bases constructed from several sources. These data bases contain information about epidemiological, demographic and clinical variables and provided the baseline for the parameters of SARS models by Chowell et al., [8,9], Gummel et al., [10], Riley et al., [11], Yan et al., [12] and Lipstich et al. [13]. Chowell et al., [8] fitted a compartmental model for the SARS epidemic to the data from Toronto, Hong Kong and Singapore. Chowell predicted the behavior of the disease and the role of diagnosis and isolation as a control mechanism in these regions, showing the differences among the epidemic dynamics occurred in these three cities. Riley et al., [11] and Lipstich et al., [13] used dynamic models for the respective transmission dynamics of SARS in Hong Kong and Singapore. Even though these models were complex, they allowed researchers to calculate numerous epidemiologically important parameters in order to assess the potential danger of the epidemic. Wallinga and Teunis [14] developed a likelihood-based estimation procedure that infers the temporal pattern of effective reproduction numbers from an observed epidemic curve. Zhou Y et al., [4] formulated a discrete mathematical model to investigate the transmission of SARS and estimated the parameters of the model on the basis of statistical data. Numerical simulations describing the transmission process for SARS in China have been carried out. Wang W and Ruan S proposed a mathematical model to simulate the SARS outbreak in Beijing by estimating the reproduction number and other important epidemiological parameters using the available data [15]. Xia et al., analysed the pattern of SARS and predicted the course of the SARS epidemic by establishing a compartmental model using data from Guangdong and Hong Kong [16]. Yang et al., also used a compartmental model to describe the SARS epidemic in spatio-temporal dimensions to determine whether people travelling in buses and trains infect one another or not [17]. They concluded that SARS can spread through people travelling in buses and trains.

Any treatment of infectious disease, SARS can give rise to many important questions such as, how treatment will effect disease transmission dynamics? Will it help to control the disease? Will the intensity of the disease be same or not. In order to answer such questions, we have developed an SEIJTR model for SARS in this paper. Here we investigate the transmission of SARS in the presence of treatment. A treatment class is included in the previous SEIJR model [18]. The values of the new parameters are calculated for the SARS epidemic using the data that appeared in Hong Kong 2003 [19]. This compartmental model includes susceptible, exposed, infected, diagnosed, treated and recovered classes. Diffusion has been included in the system to examine its role in transmission of the disease. The compartmental model for SARS transmission is given in section 2. The numerical scheme to solve the model is described in section 3. The stability of numerical model with and without diffusion is analysed in section 4. section 5 shows numerical simulations. Further discussions and conclusions are given in section 6.

Equations

This model of SARS [18] consists of the following system of nonlinear ordinary differential equations.

with initial conditionsS(0) = S₀, E(0) = E₀, I(0) = I₀, J(0) = J₀, T (0) = T₀ and R(0) = R₀ where S,E,I, J,T and R represent susceptible, exposed, infected, diagnosed, treated and recovered classes respectively and N denotes the total population, N = S + E + I + J + T + R. d₁, d₂, d₃, d₄, d₅ and d₆ are the diffusivity constants. The flow diagram of the SEIJTR model is given in Appendix A whereas table 1 provides description and the values of the parameters.To scale the population size in each compartment by the total population sizes by substituting s = S/N, e = E/N, i = I/N, j = J/N, t = T /N, r = R/N П=É…/N giving the system of equations (7) - (12). After simplification replacing s by S, e by E, i by I, j by J, t by T and r by R, the following dimensionless system of equations is obtained:

Where S+E+I+J+T+R=1

Table 1: Biological definition and values of parameters.

Initial and boundary conditions

The domain of all the calculations is considered as [-2,2]. Boundary and initial conditions are chosen as follows:

((S0, E₀, I₀, J₀, T₀, R₀)| S₀=0.98 Sech (5x-1), E₀ = J₀ = T₀ = R₀ = 0, I₀ = 0.02 Sech (5x-1), -2 ≤ x ≤ 2) (15)
((S0, E₀, I₀, J₀, T₀, R₀)| S₀=0.97 Sech (-5x-1)2), E₀ = J₀ = T₀ = R₀ = 0, I₀ = 0.03 exp (-5(x+1) 2), -2 ≤x ≤ 2) (16)
((S0, E₀, I₀, J₀, T₀, R₀)| S₀=0.96 Sech (15x), E₀ = J₀ = T₀ = R₀ = 0, -2 ≤ x ≤ 2 
I₀ = 0, -2 ≤ x < -0.6 and -0.6 < x < 2, I₀ = 0.04, -0.6 ≤ x ≤ 0.6 (17)
Figures 1 and 2 show the initial “population distributions" for S and I. A larger susceptible and a smaller infected proportion is concentrated towards the right half of the main domain in initial condition (i). In initial condition (ii), I has high concentration in the left half of the domain [-2,2] and population S has concentration on the right half of the domain [-2,2]. In the initial condition (iii) susceptible S exists in high concentration around the middle of domain [-2,2] with infected also around the middle but beyond the domain of S.

 

Figure 1: Initial Conditions i and ii.

 

 

Figure 2: Initial Condition iii.

Operator splitting method has been used to solve the SEIJTR model. According to this technique the system of equations is divided into nonlinear reaction equations and linear diffusion equations [20]. The nonlinear reaction equations to be used for the first half-time step are given as:

The second group consists of the linear diffusion equations, to be used for the second half-time step as follows:

Applying the forward Euler scheme the non-linear equations transform to

forward Euler scheme the non-linear equations transWhere  and are the approximated values of S, E, I, J, T and R at position -2+i∆x, for i = 0,1….. and time j∆t, j = 0,1,……. and and denote their values at the first half-time step. Similarly, for the second half-time step, the linear equations transform as

The stability condition satisfied by the numerical method described above is given as:

In each case, ∆x = 0.1, d₁ = 0.025, d₂ = 0.01, d₃ = 0.001, d₄ = 0.0, d₅ = 0.0, d₆ = 0.0 and ∆t = 0.03 are used.

Disease-Free Equilibrium (DFE)

The basic reproduction number R₀ is considered to be the threshold parameter for any DFE and is defined as “the number of secondary cases which one case would produce in a completely susceptible population”. The probability of infecting a susceptible individual during one contact, duration of the infectious period, and the number of new susceptible individuals contacted per unit of time are the main factors in calculation of the reproduction number. Therefore R₀ may vary remarkably for different infectious diseases and also for the same disease in different populations. The variational matrix of the system of equations (7) - (12) at the disease-free equilibrium P₀ = (1, 0, 0, 0, 0, 0), giving:

Trace [V⁰] = -α = + qβ – γ1 – γ2 – δ – ζ + δ(θ – 1) – κ - 6П < 0,
Det [V⁰] = B × П2 × A, where B = δ(θ – 1) - γ2 – П 
and A = qβ (α+ δ + П) (γ1 – ζ – П) + βκ (lα + γ1 + ζ + П) – (κ + П) (α + δ + П) (γ1 + ζ + П)
As 0 < θ < 1 θ – 1 < 0 δ(θ – 1) < 0 δ(θ – 1) - γ2 – П) < 0, B < 0,
and for R₀ < 1, A < 0, Hence, Det [V⁰] > 0.

Where
This shows that P₀ is stable for R₀ < 1. In the same way we can illustrate the stability of the endemic point for R₀ > 1.

The variational matrix of the system of equations (7) - (12) at P* (S*, E*, I*, J*, T*, R*), is given by

a11, a12,...a66 are given in appendix B
The characteristic equation for P* (S*, E*, I*, J*, T*, R*) can be written as

where p₁, p₂, p₃, p₄, p₅, p₆ and the Routh-Hurwitz conditions are calculated on the basis of [21] and are given as:

Here, p₁, p₂, p₃, p₄, p₅ and p₆ are the points of equilibrium given as:
P₁ = (0.449403, 0.000058, 0.000043, 0.000042, 0.000122, 0.386452)
P₂ = (0.546041, 0.000048, 0.000035, 0.000034, 0.000100, 0.318624)
P₃ = (0.402087, 0.000064, 0.000061, 0.000044, 0.000131, 0.419654)
P₄ = (0.373092, 0.000066, 0.000049, 0.000047, 0.000139, 0.440013)
P₅ = (0.483878, 0.000054, 0.000033, 0.000039, 0.00012, 0.362260)
The numerical values of C1, C2, C3, C4, C5 and C6 on these points of equilibrium are given in table 5 in appendix A.

Table 5: Routh-Hurwitz criteria of equilibrium without diffusion.

Endemic equilibrium with diffusion

To calculate the small perturbations S1(x, t), E1(x, t), I1(x, t), J1(x, t), T1(x, t) and R1(x, t) the equations 7 - 12 are linearised about the point of equilibrium P*(S*, E*, I*, J*, T*, R*) as described in [22,23], giving

where a11, a12, a13…etc are the elements of the variational matrix V* calculated using the method described in [24]. We assume the existence of a Fourier series solution of equations (44) - (49), of form:

where k = , (n = 1, 2, 3,……) is the wave number for node n. Substituting the values of S1, E1, I1, J1, T1 and R1 into the equations (44) -(49), the equations are transformed into

The variational matrix V for the equations (56) - (61)

The characteristic equation for the variational matrix V is given as

where q1, q2, q3, q4 and q5 and q6 are calculated by the technique used in [24]. The Routh-Hurwitz Conditions are given in appendix B. The numerical values of Routh-Hurwitz criteria on points of equilibrium in the presence of diffusion are given in table 6 in appendix A.

Table 6: Routh-Hurwitz criteria of equilibrium with diffusion.

Excited mode and bifurcation value

Table 7: Bifurcation Values of α and β.

Solutions of SEIJTR model in the absence of diffusion (Case 1)

 

 

 

Figure 5: Solutions for initial condition (iii) without diffusion.

Solutions of SEIJTR model in the presence of diffusion (Case 1)

 

 

Figure 7: Solutions for initial condition (ii) with diffusion.

 

 

Figure 8: Solutions for initial condition (iii) with diffusion.

Other cases

The graphical output for Cases 2, 3, 4 and 5 is not shown because of its similarity to Case 1. The numerical results of all cases are summarized in tables 2 and 3. Here Sj, Ej, Ij and Rj for j = (i), (ii) and (iii) represent peak values of the proportion of susceptible, exposed, infected and recovered population in the domain [-2,2] in the absence (Table 2) and presence (Table 3) of diffusion for the initial population distributions (i), (ii) and (iii). The following description is based on the results given in tables 2 and 3.

 

Case	t	S(i)	S(ii)	S(iii)	E(i)	E(ii)	E(iii)	I(i)	I(ii)	I(iii)	R(i)	R(ii)	R(iii)
1	00	0.980	0.970	0.960	0.000	0.000	0.000	0.020	0.030	0.040	0.000	0.000	0.000
	05	0.034	0.039	0.009	0.406	0.455	0.393	0.324	0.313	0.322	0.024	0.014	0.026
	10	0.009	0.007	0.004	0.133	0.149	0.128	0.199	0.213	0.194	0.145	0.120	0.150
	15	0.004	0.003	0.002	0.041	0.047	0.040	0.082	0.090	0.079	0.305	0.275	0.309
	20	0.002	0.002	0.001	0.013	0.014	0.012	0.029	0.032	0.028	0.437	0.408	0.441
2	00	0.980	0.970	0.96	0.000	0.000	0.000	0.020	0.030	0.040	0.000	0.000	0.000
	05	0.053	0.070	0.009	0.423	0.479	0.409	0.324	0.309	0.322	0.021	0.012	0.024
	10	0.012	0.014	0.007	0.138	0.157	0.133	0.204	0.219	0.199	0.139	0.113	0.145
	15	0.005	0.006	0.002	0.043	0.049	0.042	0.085	0.094	0.082	0.299	0.267	0.305
	20	0.003	0.002	0.001	0.013	0.015	0.013	0.029	0.034	0.029	0.433	0.403	0.437
3	00	0.980	0.970	0.960	0.000	0.000	0.000	0.020	0.030	0.040	0.000	0.000	0.000
	05	0.034	0.039	0.009	0.407	0.373	0.394	0.456	0.352	0.372	0.019	0.011	0.022
	10	0.009	0.007	0.004	0.134	0.271	0.129	0.150	0.284	0.266	0.124	0.102	0.129
	15	0.004	0.003	0.004	0.042	0.131	0.041	0.048	0.141	0.127	0.276	0.247	0.281
	20	0.002	0.002	0.001	0.013	0.053	0.013	0.015	0.058	0.052	0.411	0.382	0.415
4	00	0.980	0.970	0.960	0.000	0.000	0.000	0.020	0.030	0.040	0.000	0.000	0.000
	05	0.008	0.037	0.008	0.217	0.439	0.383	0.191	0.315	0.322	0.025	0.016	0.028
	10	0.007	0.006	0.002	0.129	0.144	0.125	0.195	0.208	0.191	0.149	0.125	0.154
	15	0.003	0.002	0.001	0.040	0.045	0.039	0.079	0.087	0.077	0.308	0.279	0.313
	20	0.002	0.001	0.001	0.012	0.014	0.012	0.028	0.031	0.027	0.439	0.412	0.443
5	00	0.980	0.970	0.960	0.000	0.000	0.000	0.020	0.030	0.040	0.000	0.000	0.000
	05	0.035	0.039	0.009	0.406	0.456	0.393	0.284	0.279	0.281	0.028	0.016	0.031
	10	0.009	0.007	0.004	0.132	0.148	0.127	0.152	0.147	0.165	0.161	0.134	0.166
	15	0.004	0.003	0.002	0.041	0.046	0.039	0.056	0.054	0.061	0.323	0.293	0.328
	20	0.002	0.002	0.001	0.012	0.014	0.012	0.018	0.017	0.020	0.452	0.424	0.456

Table 2: Peak values of Susceptible(S), Exposed(E), Infective(I) and Recovered(R) (Without Diffusion).

Case	t	S(i)	S(ii)	S(iii)	E(i)	E(ii)	E(iii)	I(i)	I(ii)	I(iii)	R(i)	R(ii)	R(iii)
1	00	0.980	0.970	0.960	0.000	0.000	0.000	0.020	0.030	0.040	0.000	0.000	0.000
	05	0.00986	0.01191	0.00706	0.22313	0.29394	0.06802	0.18704	0.20155	0.05063	0.01379	0.00794	0.00613
	10	0.00108	0.00173	0.00129	0.05818	0.07859	0.01705	0.09902	0.12357	0.02796	0.08119	0.07235	0.02591
	15	0.00013	0.00067	0.00035	0.01584	0.02141	0.00467	0.03612	0.04704	0.01034	0.16517	0.16339	0.04998
	20	0.00008	0.00032	0.00017	0.00445	0.00593	0.00138	0.01168	0.01541	0.00343	0.23444	0.24169	0.07034
2	00	0.980	0.970	0.960	0.000	0.000	0.000	0.020	0.030	0.040	0.000	0.000	0.000
	05	0.01416	0.01892	0.01175	0.23344	0.31042	0.07408	0.18182	0.19248	0.04489	0.01161	0.0059	0.00552
	10	0.00171	0.00252	0.00201	0.06135	0.0835	0.01909	0.10112	0.12661	0.02878	0.07472	0.06503	0.02318
	15	0.00021	0.00096	0.00054	0.01679	0.02284	0.0053	0.03754	0.04916	0.01113	0.15645	0.15331	0.04589
	20	0.00009	0.00044	0.00023	0.00473	0.00635	0.00157	0.01226	0.01626	0.00378	0.22495	0.23083	0.06566
3	00	0.980	0.970	0.960	0.000	0.000	0.000	0.020	0.030	0.040	0.000	0.000	0.000
	05	0.00996	0.01204	0.00712	0.22348	0.29444	0.06814	0.21454	0.22479	0.05868	0.01100	0.00621	0.00499
	10	0.00109	0.00174	0.0013	0.05851	0.07904	0.01711	0.13599	0.16516	0.03819	0.06908	0.06095	0.02217
	15	0.00013	0.00068	0.00035	0.016	0.02165	0.00469	0.05891	0.07484	0.01665	0.14766	0.14526	0.04453
	20	0.00008	0.00032	0.00017	0.00451	0.00602	0.00138	0.02229	0.02882	0.00641	0.21674	0.22296	0.06451
4	00	0.980	0.970	0.960	0.000	0.000	0.000	0.020	0.030	0.040	0.000	0.000	0.000
	05	0.00759	0.00839	0.0049	0.21664	0.28347	0.06427	0.19018	0.20684	0.05379	0.01543	0.00955	0.00669
	10	0.00075	0.00132	0.00092	0.05618	0.07547	0.01587	0.09765	0.12152	0.02743	0.08576	0.07762	0.02799
	15	0.00009	0.00052	0.00025	0.01523	0.02049	0.00431	0.03522	0.04567	0.00987	0.17129	0.17053	0.05301
	20	0.00007	0.00025	0.00014	0.00427	0.00567	0.00127	0.01132	0.01487	0.00323	0.2411	0.24937	0.07383
5	00	0.980	0.970	0.960	0.0000	0.000	0.000	0.020	0.030	0.040	0.000	0.000	0.000
	05	0.00997	0.01206	0.00719	0.22333	0.29429	0.06821	0.16394	0.18103	0.043998	0.01612	0.00938	0.00704
	10	0.00109	0.00175	0.00131	0.05807	0.07846	0.01709	0.07468	0.09523	0.02127	0.09009	0.08086	0.02859
	15	0.00013	0.00068	0.00035	0.01576	0.02129	0.00468	0.02411	0.03193	0.00701	0.17671	0.17542	0.05352
	20	0.00008	0.00032	0.00017	0.00442	0.00589	0.00138	0.00712	0.00949	0.00213	0.24523	0.2531	0.07389

Table 3: Peak values of Susceptible(S), Exposed(E), Infective(I) and Recovered(R) (With Diffusion).

 

Moving from Case 1 to Case 2, there is a decrease in the transmission coefficient from β = 0.242 to β = 0.182, as shown in table 7 (Appendix A). As a result, the susceptible population proportion show an increase in peak values for initial population distributions (i) - (iii) with and without diffusion. There is quite significant increase in the first ten days of SARS as compared to Case 1. The proportions of population exposed also show a significant increase from t = 10 to t = 15 days as compared to Case 1. Infection grows in last ten days of disease without diffusion but with diffusion infective population show an increase as compared to Case 1 from t = 10 to t = 20 days. There is moderate decrease in the peak value of the recovered population proportion with all initial conditions.

 

In Case 3, the rate of progression from infective to diagnosed, α is decreased from α = 0.238 to α = 0.179 as compared to Case 1, while keeping values of β same. There is no change in the susceptible population proportion in the absence of diffusion but, with diffusion a small increase appears at t = 5 and t = 10 days of disease as compared to Case 1. Initial conditions (i) and (iii) show a small increase in the proportion of exposed both with and without diffusion while initial condition (ii) shows a decrease at t = 5 days and great increase in next t = 10 days without diffusion and small increase with diffusion. In Case 3 infected proportion values are higher than for Case 1 both with and without diffusion. Peak values of the recovered population proportion are lower for Case 3 than Case 1.

 

In Case 4, there is an increase in the transmission coefficient, β, from β = 0.242 to β = 0.303 as compared to Case 1. As a result, peak susceptible proportion values in Case 4 decrease in comparison to Case 1 for initial population distributions (i) - (iii), with and without diffusion. There is also a decrease in exposed population proportion as a result of the reduction in the proportion of susceptible under all initial conditions. In particular in the absence of diffusion it occurs in the first five days of SARS, where population distribution with initial condition (i) shows significant decrease in susceptible population as compared to initial conditions (ii) and (iii) while in the presence of diffusion this decrease can be observed on all days. A decrease in proportion of infected population has been observed in the emergence of SARS as compared to Case 1, in the absence of diffusion. This decrease is quite significant in case of initial condition (i). With diffusion in the system, the peak values of infected proportion are higher for Case 4 than Case 1 under all initial conditions. Only very small increase in recovered proportions from Case 1 to Case 4 has been observed under all three conditions (i), (ii) and (iii), without diffusion. With diffusion in the system, initial conditions (i) and (ii) reflect a significant increase in recovery from Case 1 to Case 4, while initial condition (iii) shows a slight increase. In comparison to Case 2, it has been observed that there are lower proportions of susceptible and exposed in Case 4. Also in comparison to Case 2, proportion of infective is slightly less with diffusion as compared to without diffusion in the system during the first five days of disease. As compared to Case 2, there is an increase in the recovered population both with and without diffusion in the system.

 

In Case 5, as compared to Case 1 the rate of progression from infective to diagnosed, α is increased from α = 0.238 to α = 0.298. In the absence of diffusion this has not affected susceptible and exposed proportion values. With the inclusion of diffusion in the system, a small increase is observed in the susceptible and exposed population proportion in the early stage of disease. A significant reduction in the proportion of infected individuals has been noticed both with and without diffusion, showing that if infection is diagnosed earlier, the population move to diagnosed compartment quickly for treatment. A large increase in the recovered population proportion is observed both with and without diffusion as compared to Case 1. In comparison to Case 3, lower proportion of exposed and infective is observed in Case 5. Proportion of the population recovered both with and without diffusion, has been observed to be higher in Case 5 as compared to Case 3.

Discussion and Conclusion

 

Figure 9: The flow diagram of SEIJTR model.

Table 4: Basic Reproduction Number R₀

One of the authors, Afia Naheed thanks Swinburne University of Technology for the Postgraduate Research Award.

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Endemic Equilibrium without Diffusion:

The Routh-Hurwitz Conditions with Diffusion are given as: