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SURVIVAL ANALYSIS OF A MULTI-STATE SEMI-MARKOV MODEL ON INFECTIOUS DISEASE CONSIDERING VARIOUS LEVELS OF SEVERITY
SujATA SUKHIJA* •
Research Scholar, Department of Mathematics, Maharshi Dayanand University, Rohtak
sukhijasujata860@gmail.com
Rajeev Kumar •
Professor, Department of Mathematics, Maharshi Dayanand University, Rohtak
profrajeevmdu@gmail.com
*Corresponding Author Abstract
The aim of the paper is to carry out survival analysis of a novel multi-state model on infectious disease considering various levels of severity using semi-Markov processes. Various levels of severity of the disease over time and transitions between these severity levels have been considered. Transition probabilities and expected waiting times are derived. Expressions for mean survival time, expected total time in home isolation, and expected total time in hospital are obtained. The analysis of the proposed model is carried out through numerical computation and plotting several graphs. Important conclusions are drawn. The modelling framework proposed here can be used to model any infectious disease irrespective of disease states. The study will be helpful in designing effective measures to control the infectious disease and selecting the appropriate intervention policies.
Keywords: Multi-state model, Markov property, Semi-Markov process, Transition probabilities
1. Introduction
Modelling infectious diseases has always been an area of interest for researchers in various fields for the sake of prevention and control of these diseases. According to World Health Organization 2019 report [17], infectious diseases are still in the top ten leading causes of death. In low income countries, six of the top ten causes of death are infectious diseases including neonatal conditions, lower respiratory infections, diarrhoeal diseases, malaria, tuberculosis, HIV/AIDS [17]. As per the above report, there was 50% drop in Disability Adjusted Life Years (DALYs) since 2000 due to infectious, maternal, perinatal and nutritional conditions.
Multi-state Markov models have been frequently used to study the progression of diseases such as cancer [7,11, 15], HIV infection [4, 20], renal disease [9] and many more. For a Markov model, it is assumed that the holding time in a state is exponentially distributed. For many real-world situations, such as time to failure and time to discover a fault, the exponential distribution may be acceptable. In fact, the exponential distributions have memoryless property. In some cases,
the memoryless property could be seen as a problematic assumption. For example, patients who respond well to a treatment are likely to respond well to the treatment in the future, violating the Markov property [5]. To overcome this limitation, semi-Markov processes came into existence.
Semi-Markov processes are very important generalizations of Markov processes. While Markov processes assume that holding time in a state is exponentially distributed, semi-Markov processes relax the assumption allowing any arbitrary distributions for holding time in a state. Semi-Markov processes were defined by Levy [12] and Smith [14]. Since then, semi-Markov process concepts have been applied to solve various problems like electronics and missile related problems, to improve reliability of various systems, for cost-benefit analysis of a system, for economic decision making problem and so on. The field of biomedical science is not an exception to this, for example, see [1, 2, 6, 8, 10, 13, 16]. Weiss and Zelen [16] applied the theory of semi-Markov processes to the construction of a stochastic model for interpreting data obtained from clinical trials on patients with acute leukemia. Kao [10] derived results for computing the mean and variance of times in transient states and times to absorption in a transient semi-Markov process. Davidov [6] developed expressions for the steady-state probabilities for regenerative semi-Markov processes. Castelli et al. [2] performed cost-effective analysis to compare the follow-up strategies in colorectal cancer study. Goshu and Dessie [8] analysed hospital data obtained from a cohort of AIDS patients who have been under antiretroviral therapy follow-up and estimated the conditional probability of transitions between two states for a finite time period. Cao et al. [1] developed a semi-Markov model to analyse the long-term cost effectiveness of heart failure management programmes. Ramezankhani et al. [13] applied a multi-state semi-Markov model to estimate the number of years of life lost due to diabetes with and without cardiovascular disease.
However, the majority of the literature studies were focused on the disease's progressive stages and omitted the transitions back to normal state. We tried to bridge this gap through our article. There were some studies which included the transitions back to normal state however their main focus was to understand the threshold dynamics of the disease, for example, see [18], [19]. Further, since infectious diseases typically necessitate isolation, such as measles, cholera, diphtheria, infectious tuberculosis, plague, smallpox, yellow fever, and viral hemorrhagic fevers [3], home isolation is one of the possible states of our model. Moreover, through our model, we estimated the expected length of stay in home isolated state which has not been reported in previous studies. Besides, it is a well known fact that elderly patients, pregnant women and patients with co-morbidities are at risk of developing severe and critical illness and transition rates would be different in each category of severity illness. Thus, it becomes necessary to consider separate states for each category of severity. Keeping this in mind, we have considered the four states as mild disease state, moderate disease state, severe disease state and critical disease state. This brings another novelty to this model.
Keeping these in mind, a novel multi-state model for infectious disease based on the theory of semi-Markov processes is proposed. Various levels of severity of the disease over time have been considered. Thus, our model included every transition that a patient who is infected might experience. As in the model, the general scenario for an infectious disease have been considered, the model can be used to study and gain insights about any infectious disease. The paper is organized as follows. The newly developed multi-state semi-Markov model is described in Section 2. Transition probabilities and expected waiting times are derived and theoretical expressions regarding mean survival time, expected total time in home isolation and expected total time in hospital are obtained in Section 3. Numerical computations are performed in Section 4. Finally, conclusions are presented in Section 5.
2. Model Formulation
A semi-Markov model is proposed considering a person having an infectious disease showing the transition between various states. There are nine states in the model in which a healthy individual has the possibility to transit (see Figure 1). Infected persons can experience a range of clinical
manifestations, from no symptoms to critical illness. Infected persons can generally be divided into categories based on the severity of their illnesses: mild illness, moderate illness, severe illness, and critical illness. In light of this, we have taken into account the corresponding four states in increasing order of illness severity. Transitions from mild illness to moderate illness, moderate illness to severe illness, and so forth are permitted since a patient is at risk of developing severe and critical illness. Keeping this in mind, we have taken into consideration the deteriorating rates from mild illness to moderate illness, moderate illness to severe illness and so on. More details are described in the notations below.
Other assumptions made in the model are as under:
(i) All normal persons are exposed to the disease.
(ii) Testing of all infected persons is done.
(iii) Clinical testing is not perfect, i.e. there may be an error in testing.
(iv) Patient is home isolated if test results are false negative while patient is hospitalised if the test results are positive.
(v) All random variables are independent of each other.
The following states are considered in the model:
50 Normal state
51 Asymptomatic state
52 Symptomatic state
53 Home isolated state
54 Mild disease state
55 Moderate disease state
56 Severe disease state
57 Critical disease state Ss Death state
The following notation is used:
P
a
A1/A2/A3
g1(t)/g2(t)/g3 (t)/g4 (t)/g5 (t)
Gi(t)/G2(t)/G3(t)/G4(t)/G5(t)
hi(t)/h2 (t)/h3(t)/h4 (t)/hs(t)
Hi (t)/H2(t)/H3 (t)/H4(t)/H5 (t)
W (t)
incidence rate testing rate
deteriorating rate from mild illness to moderate illness/moderate illness to severe illness/severe illness to critical illness probability density function of recovery time in home isolated state/mild disease state/moderate disease state/severe disease state/critical disease state cumulative distribution function of recovery time in home isolation/mild disease state/moderate disease state/severe disease state/critical disease state probability density function of time to death in home isolation/mild disease state/moderate disease state/severe disease state/critical disease state
cumulative distribution function of time
to death in home isolation/mild disease
state/moderate disease state/severe disease
state/critical disease state
probability that the patient is in home isolation
at instant t without passing through any other
state
p/q
P1/91 P2/q2
r\/T2/r3/T4
«2/^2 «2/b2/C1
«3/b3/C2 «4/^4/ C3 «5/^5
probability of an infected person to be asymptomatic/ symptomatic (p + q = 1) probability of an asymptomatically infected person to be tested false negative/positive (px + q1 = 1)
probability of an symptomatically infected person to be tested false negative/positive
(P2 + q2 = 1)
probability that a patient is diagnosed with mild illness/ moderate illness/severe illness/critical illness (ri + r2 + r3 + r4 = 1) probability that an home isolated person will recover/ move to death state («1 + b1 = 1) probability that a person with mild illness will recover to normal state/move to death/deteriorate to moderate illness («2 + b2 + C1 = 1)
probability that a person with moderate illness will recover to normal state/move to death/deteriorate to severe illness(«3 + b3 + C2 = 1)
probability that a person with severe illness will recover to normal state/move to death/deteriorate to critical illness («4 + b4 +
C3 = 1)
probability that a person with critical illness will recover to normal state/move to death state («5 + b5 = 1)
The following symbols/abbreviations are used:
SpO2 Pulse oximetry
R.R. Respiratory Rate
* Laplace Transform symbol
** Laplace Transform symbol
© Laplace Convolution symbol
© Laplace-Stieltjes Convolution symbol
Normal State Infected State
Death State
Figure 1: State Transition Diagram. Possible states which an individual may occupy are
depicted in the figure.
3. Analysis
Let qij(t)/Qij(t) represents probability density function/cumulative distribution function of first passage time from state Si to state Sj without visiting any other state in (0, t]. Thus, the time dependent transition probabilities are given by
q01( = PPe-pt q02(t) = q13(t) = = P1xe-xt
q14( = r1 q1 xe-xt q15(t) = r2 q1 xe-xt q16(t) = r3q1xe-xt
q17( = r4 q1 xe-xt q23(t) = P2ae-xt q24(t) = r1 q2 xe-xt
q25( = r2 q2 xe-xt q26(t) = r3 q2 xe-xt q27(t) = r4q2xe-xt
q30( = «1 §1(t) q38(t) = b1h1(t) q40(t) = «2 §2(t)
q45( = c1 A1 e-A1t q48(t) = b2 h2 (t) q50(t) = = «3 §3 (t)
q56( = c2 A2 e-A2 f q58(t) = b3 h3 (t) q60(t) = = «4 §4 (t)
q67( = C3 A3 e-A3 3 q68(t) = b4 h4 (t) q70(t) = = «5 §5 (t)
q78( = b5h5(t)
The steady state transition probabilities, pij = lim J0 qij(t) dt are obtained as
P01 = P P16 = r3 qi P26 = r3 q2 P45 = ci P60 = «4
P02 = q
P17 = r4qi P27 = ^2 P48 = b2 P67 = c3
P13 = P1 P23 = P2 P30 = «1 P50 = «3 P68 = ^4
P14 = M1 P24 = nq2 P38 = b1 P56 = c2
P70 = «5
P15 = T2q1 P25 = T2q2 P40 = «2 P58 = b3 P78 = b5
Let T denote the waiting time in state Si then the expected waiting time in state Si is given by ^i = J0°° P(Ti > t) dt. Thus, the expected waiting times are obtained as
H0 =
U2
P 1
x c1
n = - «2,§2 (0) - b2h2 (0) A1 c3
V6 = - «4§4 (0) - b4 h4 (0)
F3 = -«1g2 (0) - b1 h{ (0) F5 = g - «3§2' (0) - b3h2 (0) V7 = -«5§2' (0) - b5h*s (0)
The expected waiting time in state Si given that the next state visited is Sj, is defined as
mi
f0°° tqij(t) dt = -q2 (0). Thus, the following relations are satisfied:
m01 + m02 = U0
m13 + m14 + m15 + m16 + m17 = m23 + m24 + m25 + «26 + «27 = ^2 m30 + «38 = ^3
«40 + «45 + «48 = ^4 M50 + «56 + M58 = ^5 «60 + «67 + «68 = ^6 «70 + «78 = ^7
Theorem 1. If T0 is the mean survival time for the patient starting in state S0 then
T = N
T0 = D,
where
and
N = + poi Fi + p02^2 + poi pi3 F3 + poi pi4 F4 + p01 pi5F5 + poi pi6 F6 + poi pi7 F7 + po2 p23 F3 + po2 p24 F4 + po2 p25F5 + po2 p26 F6 + po2 p27 F7 + poi pi4 p45 F5 + poi pi5 p56F6 + poi pi6 p67F7 + po2 p24 p45F5 + po2 p25 p56F6 + po2 p26 p67F7 + poi pi4 p45 p56 F6 + po2 p24 p45 p56 F6 + poi pi5 p56 p67F7 + po2 p25 p56 p67F7 + poi pi4 p45 p56 p67 F7 + po2 p24 p45 p56 p67F7
D = i - poipi3p3o - poipi4p4o - poipi5p5o - poipi6p6o - poipi7p7o - po2p23p3o
- po2p24p4o - po2p25p5o - po2p26p6o - po2p27p7o - poi pi4p45p5o - poi pi5p56p6o
- poi pi6p67p7o - po2p24p45p5o - po2p25p56p6o - po2p26p67p7o - poipi4p45p56p6o
- poi pi5p56p67p7o - po2p24p45p56p6o - po2p25p56p67p7o - poi pi4p45p56p67p7o
- po2p24p45p56p67p7o-
Proof. Let fa (t) denote the cumulative distribution function of passage time from state Si to the absorbing state.
The individual in state So at t = o can reach the absorbing state at time t in two possible ways:
(i) The individual transited from state So to state Si in time t (t < t) and reached the absorbing state in t - t time.
(ii) The individual transited from state So to state S2 in time t (t < t) and reached the absorbing state in t — t time.
Thus, we obtain
<M0 = Qoi (t)©0l (t) + Q02(t)©<£2(t)
Similarly, the following equations are obtained:
<M0 = Ql3 (t)©fc (t) + Qu(t)®$i(t) + Ql5 (t)©05 (t) + Ql6(t)©06(t) + Ql7 (t) © (p7 (t) fc(t) = Q23 (t)©03 (t) + Q24(t)©&(t) + Q25 (t)©05 (t) + Q26(t)©06(t) + Q27 (t)©07 (t) ^3(t) = Q30 (t)©00 (t) + Q38(t)
&(t) = Q40 (i^Cg^o (t) + Q45(t)©05(t) + Q48 (t) &(t) = Q50 (t)©00 (t) + Q56(t)© ^6(t) + Q58 (t)
&(t) = Q60 «©<£0 (t) + Q67(t)©<M0 + Q68 (t)
<Mt) = Q70 (t)©00 (t) + Q78(t)
Taking Laplace-Stieltjes transform of the above system of equations, rearranging the terms and solving the above system of equations for ^0* (s), we obtain
fo * (s)
Ni (s) Di(s)
where
Ni (s)
o -Qoî (s) -Q*2 (s) o
o i o -Qi3 (s)
o o i -Q23 (s)
Q38 (s) o o i
Q48 (s) o o o
Q58 (s) o o o
Q68 (s) o o o
Q78 (s) o o o
o
-Qi4* (s) -Q24 (s)
0
1 o o o
o
-Qiî (s) -Q2î (s)
0
-Q4î (s)
1 o o
o
-Qi6(s)
-Q26(s)
o
0
-Q56(s)
1 o
o
-QÎ7 (s) -Q27 (s) o o
0
-Q67 (s)
1
and
D1 (s)
1 0 0
-Q30 (s)
-Q40 (s) -Q50 (s) -Q60 (s) -Q70 (s)
-Q02(s) -Q02(s)
0
-Q13 (s) -Q23 (s) 1 0 0 0 0
0
-Q14 (s) -Q24 (s) 0 1 0 0 0
0
-Q12 (s) -Q22 (s) 0
0
Q16 (s) Q26 (s) 0
0
0
Solving the above determinants, we get
0
-Q17 (s) -Q27 (s) 0
-Q42 (s) 0 0 1 -Q52 (s) 0 0 1 -Q67 (s)
1
N1(s) = Q01 (s)Q13 (s)Q38(s) + Q01 (s)Q14 (s)Q48(s) + Q02(s)Q23(s)Q38(s) + Q02(s)Q24(s)Q48(s) + Q01 (s)Q15 (s)Q58(s) + Q01 (s)Q16 (s)Q68(s) + Q01 (s)Q17 (s)Q78(s) + Q02(s)Q25(s)Q58(s) + Q02(s)Q26(s)Q68(s) + Q02(s)Q27 (s)Q78(s) + Q01 (s)Q14 (s)Q45(s)Q58 (s) + Q01 (s)Q15 (s)Q56(s)Q68(s) + Q02(s)Q24(s)Q42 (s)Q58 (s) + Q02(s)Q25(s)Q56(s)Q68(s) + Q01 (s)Q16 (s)Q67 (s)Q78 (s) + Q02(s)Q26(s)Q67 (s)Q78(s) + Q01 (s)Q14 (s)Q42 (s)Q52 (s)Q68 (s) + Q02(s)Q24(s)Q45(s)Q56(s)Q68 (s) + Q01 (s)Q12 (s)Q52 (s)Q67 (s)Q78 (s) + Q02(s)Q25(s)Q56(s)Q67 (s)Q78 (s) + Q02 (s)QS(s)Q45(s)Q56(s)Q67 (s)Q78 (s) + Q02(s)Q24(s)Q45(s)Q56(s)Q67 (s)Q78 (s)
D1 (s) = 1 - Q01 (s)Q13(s)Q30(s) - Q01 (s)Q14(s)Q40(s) - Q02(s)Q23(s)Q30(s)
- Q02(s)Q12(s)Q52(s) - Q02(s)Q24(s)Q42(s) - Q02(s)Q12(s)Q62(s)
- Q02(s)Q22(s)Q52(s) - Q02(s)Q17(s)Q72(s) - Q02(s)Q22(s)Q62(s)
- Q02(s)Q27(s)Q72(s) - Q02(s)Q14(s)Q42(s)Q52(s)
- Q02(s)Q24(s)Q42(s)Q52(s) - Q02(s)Q12(s)Q52(s)Q62(s)
- Q02(s)Q22(s)Q52(s)Q62(s) - Q02(s)Q12(s)Q62(s)Q72(s)
- Q02(s)Q22(s)Q67(s)Q72(s) - Q02(s)Q14(s)Q42(s)Q52(s)Q62(s)
- Q02(s)Q24(s)Q42(s)Q52(s)Q62(s) - Q02(s)Q12(s)Q52(s)Q62(s)Q70(s)
- Q02(s)Q22(s)Q52(s)Q62(s)Q72(s)
- Q02(s)Q14(s)Q42(s)Q52(s)Q62(s)Q72(s)
- Q02(s)Q24(s)Q45(s)Q52(s)Q62(s)Q72(s)
Mean survival time for the patient starting in state S0 is given by
T0 = llMs-t 0
1 - <£0 2 (s)
Using the above value of <0 * (s), we obtain
T = N
T0 = D,
s
where
N = F0 + p0l Fl + p02^2 + p0l pl3 F3 + p0l pl4 F4 + p0l pl5F5 + p0l pl6 F6 + p0l pl7 F7
+ p02 p23 F3 + p02 p24 F4 + p02 p25F5 + p02 p26 F6 + p02 p27 F7 + p0l pl4 p45 F5 + p0l pl5 p56F6 + p0l pl6 p67F7 + p02 p24 p45F5 + p02 p25 p56F6 + p02 p26 p67F7 + p0l pl4 p45 p56 F6 + p02 p24 p45 p56 F6 + p0l pl5 p56 p67F7 + p02 p25 p56 p67F7 + p0l pl4 p45 p56 p67 F7 + p02 p24 p45 p56 p67F7
and
D = l - p0lpl3p30 - p0lpl4p40 - p0lpl5p50 - p0lpl6p60 - p0lpl7p70 - p02p23p30
- p02p24p40 - p02p25p50 - p02p26p60 - p02p27p70 - p0l pl4p45p50 - p0l pl5p56p60
- p0l pl6p67p70 - p02p24p45p50 - p02p25p56p60 - p02p26p67p70 - p0lpl4p45p56p60
- p0l pl5p56p67p70 - p02p24p45p56p60 - p02p25p56p67p70 - p0l pl4p45p56p67p70
- p02p24p45p56p67p70-
■
Theorem 2. Expected total time in home isolation for the patient starting in state S0 is given by
F3( p0l pl3 + p02 p23) D ,
where D has been already specified in Theorem l.
Proof. Let ty (t) denote the probability that the patient is in home isolation at instant t, given that the patient entered state Si at t = 0. Proceeding on similar lines shown in Theorem l, we obtained the following recursive relations:
ty (t tyl (t ty (t ty (t ty (t ty (t ty (t $7 (t where
W3(t) = l - alGl(t) - blHl(t)
q0l (t)©tyl (t)+ q02(t)©ty2 (t)
ql3 (t)©ty (t) + ql4(t)©ty4 (t) + ql5 (t)©ty (t) + ql6(t)©ty (t) + ql7 (t)©^7 (t)
q23 (t)©ty3 (t) + q24(t)©ty4 (t) + q25 (t)©ty (t) + q26(t)©ty6 (t) + q27 (t)©^7 (t)
W3 (t)+ q30(t)©ty0 (t)
q40 (t)©ty0 (t)+ q45(t)©ty5 (t)
q50 (t)©ty0 (t)+ q56(t)©ty6 (t)
q60 (t)©ty0 (t)+ q67(t)©ty7 (t)
q70 (t)©ty0 (t)
Taking Laplace transform of the above system of equations and solving for ty* (s), we obtain
N2(s)
(s) =
Dl (s)'
where
N2(s) = W3* (s)(q0l (s)ql3 (s) + q02(s)q23 (s))
and Dl (s) has been already specified in Theorem l.
Expected total time in home isolation for the patient starting in state S0 is given by
!• TO
J ty0(t) dt = li'ms^0 (s)
= F3( p0l pl3 + p02 p23) D ,
where D has been already specified in Theorem l. ■
Theorem 3. Expected total time in hospital for the patient starting in state S0 is given by
D (p0l pl4 F4 + p0l pl5 F5 + p0l pl6F6 + p0l pl7F7 + p02 p24 F4 + p02 p25 F5 + p02 p26^6
+ p02 p27 F7 + p0l pl4 p45 F5 + p0l pl5 p56 F6 + p0l pl6 p67 F7 + p02 p24 p45 F5 + p02 p25 p56 F6 + p02 p26 p67 F7 + p0l pl4 p45 p56F6 + p02 p24 p45 p56 F6 + p0l pl5 p56 p67 F7 + p02 p25 p56 p67^7 + p0l pl4 p45 p56 p67F7 + p02 p24 p45 p56 p67^7 ),
where D has been already specified in Theorem l.
Proof. Let Xi (t) denote the probability that the patient is in hospital at instant t, given that the patient entered state Si at t = 0. Proceeding on similar lines shown in Theorem l, we obtained
the following recursive relations. X0 (t) = q0l (t)©Xl(t) + q02(t)©X2 (t)
Xl (t) = ql3 (t)©X3(t) + ql4(t)©X4 (t) + ql5 (t)©X5(t) + ql6(t)©X6 (t) + ql7 (t)©X7(t) X2 (t) = q23 (t)©X3(t) + q24(t)©X4 (t) + q25 (t)©X5(t) + q26(t)©X6 (t) + q27 (t)©X7(t) X3 (t) = q30 (t)©X0(t)
X4 (t) = W4(t) + q40(t)©X0 (t) + q45 (t)©X5(t) X5 (t) = W5(t) + q50(t)©X0 (t) + q56 (t)©X6(t) X6 (t) = W6(t) + q60(t)©X0 (t) + q67 (t)©X7(t)
X7 (t) = W7(t)+ q70(t)©X0 (t) where
W4(t) = l - a2G2(t) - cl(l - e-Alf) - b2H2(t) W5(t) = l - a3G3(t) - c2(l - e-A2f) - b3H3(t) W6(t) = l - a4G4(t) - c3(l - e-A3f) - b4H4(t) W7(t) = l - a5G5(t) - b5H5(t)
Taking Laplace transform of the above system of equations and solving for X0(s), we obtain
X0 (s) = ^,
where
N3 (s) = W4* (s)(q0l (s)ql4(s) + q02 (s)q24(s)) + W5* (s)(q0l (s)q£; (s) + q02(s)q25 (s) + q0l (s)q*4(s)q45 (s) + q02(s)q24 (s)q45 (s)) + W6* (s)(q0l(s)q*6 (s) + q02(s)q26 (s) + q0l(s)q*5(s)q56(s) + q02(s)q25(s)q56(s) + q0l(s)q*4(s)q45(s)q56(s) + q02 (s)q24(s)q45 (s)q56 (s)) + W* (s)(q0l (s)q*7 (s)+ q02(s)q27 (s) + q0l(s)q*6(s)q67(s) + q02(s)q26(s)q67(s) + q0l(s)q*5(s)q56(s)q67(s) + q02(s)q25(s)q56(s)q67(s) + q0l(s)q*4(s)q45(s)q56(s)q67(s) + q02(s)q24(s)q45(s)q56(s)q67(s))
and Dl (s) has been already specified in Theorem l.
Expected total time in hospital for the patient starting in state S0 is given by
!• TO
Jo X0(t) dt = Ii'ms^0X0(s) = lims-° D^M
Using the above value of N3(s) and simplifying we get the required result. ■
4. Numerical Computations
Numerical computations for the mean survival time, expected total time in home isolation, and expected total time in hospital have been performed. For illustrating our model results, the waiting time distributions are assumed as exponentials, as follows: gi(t) ~ exp(Yi) and hi(t) ~ exp(^i) where i=l,2,...,5.
The severity levels of the disease are defined as under. mild illness SpO2 > 94% on room air and no shortness of breath
moderate illness 90% < SpO2 < 94% on room air or 24 <R.R.< 30 breaths
per minute
severe illness SpO2 < 90% on room air or R.R.>30 breaths per minute
critical illness Respiratory failure or septic shock or multiple organ dys-
function or requires life sustaining treatment
In addition, the following values for parameters are assumed:
f=0.03l/day, a=0.5/day, Yl=0.074/day, Y2=0.07l/day, Y3=0.055/day,
Y4=0.034/day, Y5=0.023/day, Al=0.l2/day, A2=0.l5/day, A3=0.20/day,
¿l=0.00ll/day, ¿2=0.00l2/day, ¿3=0.00l5/day, ¿4=0.00l8/day, ¿5=0.0020/day,
p=0.7, q=0.3, pl=0.74, ql=0.26, p2=0.l8, q2=0.82, rl=0.83, r2=0.07, r3=0.06, r4=0.04, al=0.98, bl=0.02,
a2=0.85, b2=0.05, cl=0.l0, a3=0.75, b3=0.08, c2=0.l7, a4=0.65, b4=0.l5, c3=0.20, a5=0.l, b5=0.9.
For the above values of parameters, we obtained mean survival time, expected total time in home isolation and expected total time in hospital and analysed how these parameters vary corresponding to perturbations in transmission rates, deteriorating rates, and death rates. The results obtained are depicted below. Figure 2-6 forecasts how variations in the transmission rate and death rates will affect the mean survival time. Figure 7 illustrates how the expected total time in home isolation changes as the recovery rate and death rate vary. Figure 8-ll predicts how perturbations in the recovery rates and deteriorating rates will affect the expected total time in hospital.
0.34
f
0.34
f
Figure 2: A plot of T0 with varied f and 5l. The behaviour of the mean survival time (T0) with respect to the transmission rate (f)for different values of death rate (5l) is demonstrated. From the figure, it can be seen that the mean survival time (T0) decreases as the transmission rate (f) increases and gives lower values for higher values of death rate (¿l).
Figure 3: A p/ot of T0 with varied f and ¿2. The behaviour of the mean survival time (T0) with respect to the transmission rate (f)for different values of death rate (¿2) is demonstrated. From the figure, it can be seen that the mean survival time (T0) decreases as the transmission rate (f) increases and gives lower values for higher values of death rate (¿2).
1040
1000
5 =0.0009 1
1035
995
5 =0.0010
1
1030
5 =0.0011 1
990
1025
985
1020
980
1015
975
1010
970
1005
965
1000
960
995
990
955
0.32
0.35
J.36
0.32
0.33
0.35
0.36
Figure 4: A Plot of T0 with varied p and 53. The behaviour of the rne«n surviv«l tirne with resPect to the tr«nsmission mte (p) for different v«lues ofde«th r«te (53) is demonstr«ted. From the fi§ure, it c«n be seen th«t the rne«n surviv«l tirne (T0) decre«ses «s the tr«nsmis-sion r«te (p) incre«ses «nd §ives lower v«lues for hi§her v«lues of de«th r«te (53).
Figure 6: A Plot of T0 with v«ried p «nd 55. The be-h«viour of the «e«n surviv«l ti«e (T0) with resPect to the tr«ns«ission r«te (p) for different v«lues ofde«th r«te (55) is demonstr«ted. From the fi§ure, it c«n be seen th«t the rne«n surviv«l ti«e (T0 ) decre«ses «s the tr«ns«is-sion r«te (p) incre«ses «nd §ives lower v«lues for hi§her v«lues of de«th r«te (55).
Figure 5: A Plot of T0 with v«ried p «nd 54. The be-h«viour of the me«n surviv«l tirne f^) with resPect to the tr«ns«ission r«te (p) for different v«lues of de«th r«te (54) is demonstr«ted. From the fi§ure, it c«n be seen th«t the me«n surviv«l time decre«ses «s the tr«nsmis-sion r«te (p) incre«ses «nd §ives lower v«lues for hi§her v«lues of de«th r«te (54).
Figure 7: A plot of expected total time in home isolation with varied Yi and Sj. The behaviour of the expected total time in home isolation with respect to the recovery rate (y\) for different values of death rate (S\) is demonstrated. From the figure, it can be seen that the expected total time in home isolation decreases as the recovery rate (yi) increases and gives lower values for higher values of death rate (S\).
Figure 8: A plot of expected total time in hospital with varied Y2 and Al. The behaviour of the expected tota/ time in hospita/ with respect to the recovery rate (72) for different values of deteriorating rate (Al) is demonstrated. From the figure, it can be seen that the expected to-ta/ time in hospital decreases as the recovery rate (72) increases and gives lower values for higher values of deteriorating rate (Al).
Figure 9: A plot of expected total time in hospital with varied 73 and A2. The behaviour of the expected tota/ time in hospita/ with respect to the recovery rate (73) for different values of deteriorating rate (A2) is demonstrated. From the figure, it can be seen that the expected to-ta/ time in hospita/ decreases as the recovery rate (73) increases and gives lower values for higher va/ues of deteriorating rate (A2).
628
626.5
626
626
625.5
625
620
- 624.5
■» 624
623.5
623
614
622.5
J.Ob
0.055
J.06
J.065
0.075
0.08
0.085
Y
Y
Figure 10: A p/ot of expected total time in hospital with varied 74 and A3. The behaviour of the expected tota/ time in hospita/ with respect to the recovery rate (74) for different values of deteriorating rate (A3) is demonstrated. From the figure, it can be seen that the expected total time in hospital decreases as the recovery rate (74) increases and gives lower values for higher values of deteriorating rate (A3).
0.028 Y 5
Figure 11: A p/ot of expected tota/ time in hospita/ with varied 75 and ¿5. The behaviour of the expected tota/ time in hospita/ with respect to the recovery rate (75) for different values of death rate (¿5) is demonstrated. From the figure, it can be seen that the expected total time in hospital decreases as the recovery rate (75) increases and gives /ower va/ues for higher values of death rate (¿5).
635
627
<5 =0.0019 5
326.5
5 =0.0020 5
626
5 =0.0021 5
630
325.5
^ 625
324.5
625
624
323.5
620
623
622.5
622
615
0.035
0.04
0.045
0.02
0.022
0.024
0.026
0.03
0.032
0.034
0.036
Y
5. Conclusion
Designing prevention strategies and infection control policies can be benefitted using mathematical models of infectious diseases. On the basis of the idea of semi-Markov process, a new framework for modelling infectious diseases have been presented. The analysis of the model aids in examining the effects of various parameters on various system measures. According to the analysis presented,
it is concluded that the mean survival time declines as the disease's transmission rate rises and has lower values for greater values of death rate. The expected total time in home isolation reduces with rising recovery rates and has lower values for higher death rates. The expected total time in hospital decreases as the recovery rate increases and gives lower values for higher values of deteriorating rate. Through this article, the use and significance of semi-Markov models in understanding infectious diseases trends is demonstrated. This study may be helpful in selecting the optimal intervention tactics and creating effective infection control measures.
DecLArAtioN of Competing Interest
None.
AckNowLedgemeNt
The author Sujata Sukhija delightedly acknowledges Human Resource Development Group of Council of Scientific & Industrial Research (CSIR), India, for providing fellowship through file number 09/382(0258)/2020-EMR-I.
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