Impact of Production-Inventory Control on the
Dynamics of Epidemics
Siddhartha Paul, Jayendran Venkateswaran
Industrial Engineering and Operations Research,
Indian Institute of Technology Bombay,
Powai, Mumbai, MH 400076, India
siddhartha.paul@iitb.ac.in, jayendranQ@iitb.ac.in
Abstract
A general production inventory (PI) model is integrated with the tradi-
tional disease diffusion (SEIRS) model to understand the role of inventory
policies on vaccine pr bl idemic d, ics. An integrated PI-SEIRS
model has been described, ‘aeieais the demand to the PI component depends
on the infected population, and the recovery rate of patients depends on the
timely supply of medicines. The performance comparison of PI-SEIRS model
and the standalone SEIRS model is carried out using an illustrative influenza
epidemic data set. Results show that the supply chain or inventory effects
on the epid dynamics is significant. A given epidemic can be caused by
high infectivity parameter with sufficient supply of medicine or with a low
infectivity parameter but insufficient supply of medicine. Also, the use of a
standalone SEIRS model overestimates the disease severity, compared to the
combined inve ‘y control and epidi model.
Keywords: infectivity parameter; SEIRS model; PI-SEIRS model; inven-
tory control
1 Introduction
A severe influenza epidemic outbreak is a very real threat for any society or
country. The declaration of epidemic further can create unnecessary panic
among people and disrupt essential services, business activities, and trans-
portation services due to workforce absenteeism (Bienstock & Zenteno, 2012).
Control measures are designed by healthcare decision makers based on the
severity of the epidemic. Thus, a good estimation for epidemic severity
(disease transmissibility or infectivity) parameter is of paramount impor-
tance. Past literature (Nsoesie et al., 2013; Samsuzzoha et al., 2013; Chow-
ell et al., 2006) have used dynamic epidemic models to estimate the disease
model parameters. Now, for any vaccine preventable disease, the supply of
medicine/vaccine plays an important role in controlling the dynamics of the
epidemic (Dasaklis et al., 2012). Hence, it is necessary to incorporate the
medicine supply information into epidemic dynamics model as well to esti-
mate the disease parameters with higher precision.
There is a significant gap between epidemic modelling and supply chain
modelling literature. There is few literature (Duintjer Tebbens et al., 2010;
Thompson & Duintjer Tebbens, 2014; Chick et al., 2008) available that com-
bines these two areas. Most of the previous literature (Arinaminpathy &
McLean, 2008, 2009; Lee & Chen, 2007; Yarmand et al., 2014; Ren et al., 2013)
in planning and control of epidemic have focused primarily on resource alloca-
tion (RA) models but not on the supply chain aspects of resources (Dasaklis
et al., 2012). The procurement/ stock management of resources are of im-
portance since without ensuring the availability of resources, RA models are
pointless. Duintjer Tebbens et al. (2010), have combined a disease diffusion
and a vaccine production model to minimize the total cost of public health
and vaccine production for a suitable selection of vaccine filling flows and pro-
duction flows at each time t. Using game theoretic approach, (Chick et al.,
2008) have developed a variant of cost sharing contract between governmen-
tal healthcare sector and vaccine manufacturer to improve the overall perfor-
mance of influenza vaccine supply chain. But these models (Duintjer Tebbens
et al., 2010; Thompson & Duintjer Tebbens, 2014) are from the production
perspective while this paper addresses the inventory management perspective.
Sterman (2000) has provided a generic stock management system struc-
ture that provides the basic environment for decision making experiment in
various scenarios like, production inventory control, raw material ordering
etc. Various analyses (such as average system cost, service level, stability) on
production inventory (PI) system has already been concluded using control
theoretic approach (Venkateswaran & Son, 2007; Bijulal et al., 2011). How-
ever, these analyses on PI system have been done for time invariant demand
mean and are thus not valid for epidemic demand pattern as it follows a bell
shaped curve with a long right tail.
This research has two goals. The first goal is to bridge the gap between
supply chain and epidemic containment literature and aid healthcare deci-
sion makers in managing outbreaks more efficiently. The interaction between
epidemic outbreak and medicine stock management is a two way feedback
process, as unmet vaccine needs will generate more demands and sufficient
supply will reduce disease transmission. Hence, SD methodology is adopted
to build the integrated model. The second goal is to explore the effect of stock
management control parameters on disease dynamics and on the quality of
disease parameter estimation, since every supply chain involves various de-
lays (such as production delay, transportation delay etc.) and delays cause
additional dynamics in system variables which eventually degrade the perfor-
mance of the supply chain. For vaccine preventable diseases, poor supply of
medicines can also force the epidemic to grow further, even when the sever-
ity of the actual disease is low. Thus, it is difficult to determine whether an
epidemic has occurred due to higher infectivity or poor supply of medicine by
only looking at the disease data without analysing the corresponding supply
chain. Moreover, it is not clear as to how much of the epidemics impact is re-
duced by better supply chain management. We hypothesize that the use of a
standalone disease transmission model will over-estimate the disease severity,
as against a combined inventory control and epidemics model.
We investigate our hypothesis using data from the second wave of Spanish
flu outbreak that occurred in Sydney in 1919 (Samsuzzoha et al., 2013), for
illustration purposes only.
The rest of the paper is organised as follows: Section 2 describes a basic
disease transmission model and the proposed integrated model. In section 3
the performance of both the models are analysed, and the impact of supply
chain effects on disease dynamic quantified using the illustrative data set
of the 1919 Spanish flu . Section 4 discuss
the observations and future work.
2 Model Description
In this section, we first describe the popular SEIRS model for disease trans-
mission, and then the proposed integrated SEIRS and production inventory
(PI) models. The PI component represents the vaccine supply chain. The in-
teraction between these two models’ functions are as follows: the demands for
vaccine (in hospitals) are generated from the epidemic model and fed into the
production inventory model. Then, based on output of the production inven-
tory model (i.e., the availability of vaccines in the hospital stock), the patients
in the hospital are treated. The symbols, notations and then abbreviations as
used in this section are listed in Table 1.
2.1 Standalone SEIRS Model
A variant of the compartmental SEIRS model has been used to describe the
dynamics of influenza transmission. In a similar manner to the conventional
SD models for epidemic outbreak (e.g., SI, SIR models, discussed in (Sterman,
2000)), this model divides the total population into four groups viz. Suscep-
tible (5), Exposed (£), Infected patient under treatment (I), and Recovered
(R). Further, the mixture of population within each compartment is assumed
to be homogeneous. The stock and flow diagram of the SEIRS model is shown
in Fig.1.
A susceptible (S'}) individual may get exposed to the disease if he comes in
contact with an exposed (£) or infected (J) individual (see Equation (1)). It is
assumed that the capability of spreading the disease of an exposed individual
is only k% of an infected individual. The infectivity rate of S is governed
by the infectivity parameter y, which is defined as the product of contact
Table 1: Notations Used
Symbol Description Units
7 Infection probability (pi)x Contact rate (A) (rate) 1/day
N Total population people
P Incubation period day
TRM Time to recovery with medicine day
TR Time to recovery without medicine day
TW Waning time day
S(t) Susceptible population at time t people
E(t) Exposed population at time t people
I(t) Infected patients under treatment at time t people
R(t) Recovered population at time t people
ER(t) Exposure rate at time t people/day
AR(t) Hospital admission rate at time t people/day
RRM(t) Recovery rate with medicine at time ¢ people/day
RR(t) Recovery rate without medicine at time t people/day
WR(t) Waning rate at time t people/day
a Fractional rate of adjustment of medicine on order discrepancy (rate) 1/day
B Fractional rate of adjustment of medicine in stock discrepancy (rate) 1/day
p Smoothing factor (constant) for forecast
Ti Production lead time day
VDT Vaccine distribution time day
DCovg _ Desired stock coverage day
ss Safety stock coverage day
DS(t) D d stock of medicine at time t units
DO(t) Desired order of medicine at time t units
FD(t) Demand forecast for time period t units/day
MS(t) Available inventory or stock at time t units
MO(t) Medicine quantity on order at time t units
AdjMS(t) Adjustments for inventory discrepancy at time ¢ units/day
AdjMO(t) Adjustments for medicine on order discrepancy at time t units/da
DR(t) Medicine demand rate at time ¢ units /day
OR(t) Medicine order sent from hospital at time t units/da
PR(t) Production completion rate at time t units /day
SR(t) Shipment rate of medicine at time t units/da
MaxSR(t) Maximum Shipment rate of medicine at time t units /day
Figure 1: Stock flow diagram of disease diffusion model
rate (A, the number of people who interact per time period) and infection
probability (pi, the chance of getting infection from contacting an infected
person). Thus 7 = Ax pi. An exposed individual remains asymptomatic for an
incubation time period P (see Equation (2)). As soon as the patient develops
symptom or falls sick, he is shifted from the “2” to the “J” compartment for
treatment (see Equation (3)). Patients are assumed to recover after an average
recovery time (RM) when medicines are made available and an average time
of recovery ('R) without medicine natural burnout). It is assumed that,
TR>TRM. After recovering from the d
(this loss could be due to the mutation of the disease virus) and again becomes
susceptible to the disease after a time period of “TW” (see Equation (4)).
The governing continuous time domain equations of the SEIRS disease
diffusion model are as follows:
d(S(t)) R(t) yx (kx E(t) +I(t)) x S(t)
ease, an individual loses immunity
a) (OTW N ()
UE(t)) _ yxX(kxX EO +I1C) x S$) _ El) °)
dt N P
du) _ EQ 1H _ 1) @
dt P TRM TR :
dR) _ 1), 1) _ RU)
ad TRM TR TW (4)
2.2 Description of PI-SEIRS Integrated Model
Our integrated model comprises a disease diffusion component and a stock
management component.
2.2.1 Production Inventory (PI) Model
A generic stock management model, similar to those discussed in (Sterman,
2000; Venkateswaran & Son, 2007) or class of automatic pipeline variable in-
ventory order based production control system (APVIOBPCS) (Dejonckheere
et al., 2003) is adopted for depicting the inventory management of medicines
or vaccines at the hospital, as shown in Fig. 2.
MS
- SRo+
eran st
™~ Stockout
AdiMS Ms Control AdjMS
aj
aN
4 VDT.
+ MaxSR:
Figure 2: Stock flow diagram of inventory management model
The demand is forecasted (FD) using an exponential smoothing method
with smoothing factor p, as shown in Equation (5). The hospital ordering
strategy is modelled as a generalized order-upto (G-OUT) policy (Bijulal
et al., 2011), as shown in Equation (6). The order quantity depends on
the forecasted demand F'D, the discrepancy in desired and actual orders in
pipeline (Adj MO) and the discrepancy in desired and actual stock or inven-
tory on hand (AdjMS). The discrepancy in orders in pipeline and in stock
is adjusted using the tuning parameters a (1/Ty, the reciprocal of time to
adjust MO) and 8 (1/T;, the reciprocal of time to adjust MS), respectively.
The desired orders in pipeline, as per Little’s Law, is computed as a product
of lead time L and the FD (see Equation (7)). The desired stock is also
governed by the FD and the desired coverage level and safety stock S'S (see
Equation (8)). Equations (9) and (11) are the inventory balance equations
for the orders in pipeline (MO) and (MS), respectively. The production rate
(PR) is modelled as a third order material delay to capture the mixing of
products and variabilities in production times. The delivery or sales rate SR
of medicines, Equations (12) and (13), govern the actual supply of medicines
from the stock to the hospital, subject to the actual availability. It is noted
that the ordering behaviour of the system can be tuned using the parameters
(a, 8). :
SR(t) = max{0,min{MaxSR(t), DR(t)}}
MaxSR(t) = MS(t)/VDT
a = px (DR(t)— FD(t)) (5)
OR(t) = FD(t)+a(DO(t) — MO(t)) + B(DS(t) — MS(t)) (6)
Adj MO(t) Adj M S(t)
DO(t) = Lx FD(t) (7)
DS(t) = (SS +DCovg) x FD(t) (8)
AMOO) = (nw - PRO) (9)
PR(t) = Delay3(OR(t), L) 10)
sete = (PR(t) — SR()) 11)
)
)
(
(
(
(
The demand of medicines DR depends on the number of infected patients
under treatment J‘(t), as shown in Equation (14), where V DT is the vaccine
distribution time. T ves as the linking constraint from the SEIRS model
to the PI model.
DR(t) = I'(t)/VDT (14)
2.2.2 Disease Diffusion Component
The SEIRS model (sce Section 2.1) is integrated with a production inventory
(PI) model of the hospital to explore the impact of supply chain on epidemic
dynamics. In the integrated model, the recovery rate with medicine (RRM"*)
depends on the flow of medicine from the stock management model. That
is, RRM’ = max{0, min{I‘(#), MS(t)}}
Ti
governing the SEIRS arn tert of the integrated model i: shown in Equa-
tions (15) - (18). The variables of the PI-SEIRS model is inguished from
those of the stand-alone SEIRS model using superscript i, while the mean-
ing of the notations remains the same as defined in Table 1. In addition to
the above, we made the following assumptions: the hospital was under stable
condition before the occurrence of the epidemic outbreak, and each epidemic
patient required one unit of drug to recover. All other general assumptions of
the compartmental epidemic models (Sterman, 2000) hold true.
. The updated differential equations
asi) _ RO x (kx BW) +P) x S')
da OTWe Ne (15)
aE (t)) _ Yx(kx EW +O) xsi) BO) (16)
dt Nt Pt
d(I*(t)) _ EX(t) — max{0,min{/‘(t), MS(@}} IO) (17)
dt ps TRM* rh
d(Ri(t)) _ max{0, min{I‘(#), MS(t)}} 4: T(t) Rt) (18)
dt TRM* TR TW?
3 Experiments and Analyses
In this section we study, using simulation, the appropriateness of standalone
SEIRS model and the integrated PI-SEIRS model in modelling the epidemic
dynamics. The second wave of the 1919 Spanish Flu data set (Samsuzzoha
et al., 2013) or raw data is used for illustration. The SEIRS model and
PI-SEIRS model are both independently calibrated for the raw data, using
least square regression method, so as to minimize the root mean square error
(RMSE). Thus the objective function can be written as shown in Equation
(19), where p is the set of all parameters, I’ is the set of ranges of corresponding
parameters, and T is the total run time.
Jno (Ht) - FY’ at
RMSE = mip \| > — (19)
The general simulation settings, and simulation-based optimisation method
employed is described in (Section 3.1). The SEIRS model results are discussed
in (Section 3.2). (Section 3.3) demonstrates the effect of stock control param-
eters on the final epidemic size using PI-SEIRS model. Finally, the PI-SEIRS
model results are discussed in (Section 3.4)
3.1 Simulation Settings and Optimisation
Simulation models for SEIRS and PI-SEIRS were built using Anylogic® 6.
RK4 integration method with time step of 0.001 was used for integration
and simulation was run for 70 days (I). For PI-SEIRS’s stock management
component, we chose L = 3 days (Bijulal et al., 2011), VDT = 1 day, SS =0.1
day (chosen arbitrarily but lesser system inventory preferred), p =1 (to give
higher weightage to current demand). For the SEIRS model and the epidemic
component of PI-SEIRS model, we have taken k = 0.85 (chosen arbitrarily),
TW = 365 days (Samsuzzoha et al., 2013), Ip = 79, and Ro = 0. Note that
the influence of Waning Rate (WR(t)) in this illustration is expected to be
negligible due to large Waning Time (TW). For MO,MS, and FD initial
values are chosen as DO, DS and DR respectively.
8
Calibration of the models using least square regression was carried out
using the optimisation solver OptQuest®, a commercial tool. This simulation
based optimisation works as follows. The optimisation solver (i.e. OptQuest)
chooses a set of parameter values and passes it on to the simulation (SD)
model. The model runs the simulation, evaluates the objective function (i.e.
RMSE), and returns the same to the optimisa
intelligently chooses the next set of parameters for evaluation by the sim-
ulation model. This iterative scheme continues until a pre-defined stopping
criteria is met, and the best result obtained is reported. OptQuest is based on
scatter search, tabu search and neural networks. Since OptQuest internally
uses a probabilistic scheme to search the solution space, multiple runs of the
simulation-based optimisation might converge to different best solutions. In
our experiments, the simulation-based optimization was run for 5500 itera-
tions (stopping criteria).
ion solver. The solver, then
3.2 SEIRS Model Calibration with Raw Data Set
The standalone SEIRS model was calibrated using the raw data set of the
second wave of 1919 Spanish Flu by tuning the parameter set p = {So, Ep,
y, P, TRM, TR}. Based on the characteristic of a disease, some basic pa-
rameters (related to length of stay (LOS)) of the epidemic model can be
confirmed either from the healthcare experts’ opinion or from past data. Un-
fortunately, for the 1919 Flu outbreak, most of the LOS paramete
TRM, TR) are unknown. Hence, they were also included as a decision param-
eter in our model. The range of input parameters used are Sp € [3000, 5000},
Ep € [20,90], y € [0,1], P € [1,4], TRM € [4,6], TR € [6,8].
Ten different experimental run were performed for simulation-based op-
timization to minimize RMSE and the results are presented in Table 2.
Based on the results, it is observed that the average RMSE is 13.057 with a
95% confidence interval half width of 0.009, and the corresponding infectivity
parameter y € (0.307 + 0.013). The 95% confidence of the other parame-
ters are P € (1.99 + 0.242), TRM € (5.06 + 0.268), TR € (7.04 + 0.194),
So € (4024 + 275.846), Eo € (46 + 6.508).
3.3 Impact of Inventory on Final Epidemic Size
In this section, the effect of inventory control on the final epidemic size is
analysed using the PI-SEIRS model. The final epidemic size (R..) is defined
as the total number of people who were infected during the outbreak, as shown
below:
,
Rx | I(t)dt (20)
In order to show the impact of ordering policies on Ro, a total of 961
(a, 8) pairs were generated combining 31 values for each of a and 8 between
Table 2: Results of calibration of SEIRS model
Experiment RMSE x P TRM TR SS £o
1 13.059 0.302 2.1 5 7.1 3951 49
2 13.064 0.298 2.1 56 66 3777 48
3 13.066 0.297 2.2 5.2 7 3798 51
4 13.043 0.328 1.5 5.1 6.8 4312 32
5 13.028 0.349 1.3 43 7.4 4964 29
6 13.065 0.297 2.1 5.5 69 3716 47
7 13.057 0.308 2.1 4.7 69 4234 52
8 13.066 0.297 2.2 5.2 7 3798 51
9 13.057 0.305 1.9 5.1 7.5 3858 42
10 13.069 0.293 24 49 7.2 3832 58
0 to 3 with step size 0.1. The parameters (5), Ej, 7', P', TRM', TR’) of the
epidemic component of the PI-SEIRS model is assumed to be the same as
the results obtained for Experiment 1, Table 2, for all (a, 3) combinations.
Simulations were carried out for all (a,) pairs, the R,. calculated and a
contour plot of the same plotted in Fig. 3. The figure shows that the ordering
policies (a, 3) affect Roo significantly. Moreover, the set of optimal (a, 3) lies
in a < @ region (i.e., the region shown by 2375), where § is high and a is
small.
3.0F oa
nN
uw
he
o
‘a
So
-
a
Fractional adjustment rate for MO, a
=
a
0.5 1.0 1,5 2.0 2.5 3.0
Fractional adjustment rate for MS, 8
0.0E¢
0.0
Figure 3: Contour plot of R,, with 7 = 0.302
Figure 4 compares the change in R.. over time for selected values of (a, 3),
and the standalone SEIRS model. It is again noted that the epidemics com-
10
ponent of PI-SEIRS model and the SEIRS model parameters are exactly the
same, with y = 0.302. SEIRS results in lowest Ro since it assumes infinite
inventory and instantaneous supply of medicine, thus giving a lower bound
on Ro. Similarly, the worse case (giving an upper bound on R.<) is obtained
by substituting a = 0,3 = 0 (i.e. inventory discrepancies are not adjusted) in
the PI-SEIRS model.
— 2500+
2
a
3
& 2005
o
N
® 1500-4
2
5 a I—— (a=0, B=0)
$ 10004 fb (0-0, B=0)
ce j F--- (a=1, B=1)
BS 5004 (a=1, B=2)
i (a=0.5, B=3)
[=> SEIRS Model
0 T T T T 7 T
0 10 20 30. 40 50 60 70
Days
Figure 4: Comparison of R. of SEIRS model and PI-SEIRS model for
different ordering policies with y = 0.302.
The percentage (%) increase in the final Roo value at different settings
of (a, 8), relative to that obtained using SEIRS is computed using Equation
(21), and tabulated in Table 3.
— Roo, SEIRS 1
Paea—* .sE1Rs) x 100 (21)
SEIRS
These results clearly show that inventory control aspects does influence the
dynamics of the epidemics, with a net increase in Roo of 1.693% to 8.208%. A
‘natural’ order setting of (a = 3 = 1), where discrepancies in order and dis-
crepancies in inventory are adjusted as per their exact shortages, also results
in a 3.676% increase in Roo.
3.4 PI-SEIRS Model Calibration with Raw Data
Set
The integrated PI-SEIRS model is calibrated using the raw data set of the
second wave of 1919 Flu by tuning the parameter set p = {5}, Ej, y', P',
TRM', TR’, a, 3}. The range of used input parameters are $4 € [3000, 5000],
1
Table 3: Increment of Rx (in %) from lower bound (Ro.,serrs) based on
different ordering policies.
Ordering Policy (a,) Increment of R, (in %)
(0.5,3) 1.693
(1,2) 2.782
(1,1) 3.676
(0,0) 8.208
Ey € [20,90], y' € [0,1], P* € [1,4], TRM* € [4,6], TR’ € [6,8], and 0 <
a, 8 <2. The root mean square error RMSE" is minimized, with objective
as shown below.
RMSE’ = ; {Holfo= m0)" (22)
ie
min ~
(Si, Bi, P?.TRM* TR o.,8)eC T
Ten different experimental run were performed for simulation-based opti-
mization to minimize RMSE‘ and the results presented in Table 4. Based
on the results, it is observed that the average RMSE’ is 13.044 with a 95%
confidence interval half width of 0.026, and the corresponding infectivity pa-
rameter y' € (0.267 + 0.003). The 95% confidence of the other parameters
are P’ € (3.34 + 0.227), TRM"’ € (4.69 + 0.240), TR’ € (7.34 + 0.268),
S§ € (3520 + 93.628), Ej € (78 + 6.036). a, 8 take only two (0 & 0.2) and
four (1.1, 1.2, 1.9, 2) values respectively. Based on the results from Table 4 it
can be seen that multiple combinations of (a, 3,7’) exist that can give a close
fit to the raw data, thus highlighting the importance of the medicine supply
chain on epidemic dynamics.
Table 4: Results of calibration of PI-SEIRS model
Experiment RMSE’ vy P! TRM' TRi Si EX a 8B
1 13.053 0.265 3.4 5.1 6.8 3,384 78 0.2 2
2 13.01 0.273 3.2 4.5 6.9 3,783 76 0 1.2
3 13.069 0.265 3.2 5.1 7.2 3,394 74 0.2 2
4 13.109 0.267 3 5 7.2 3,566 70 0 11
5 13.016 0.264 3.7) 4.3 7.6 3,526 89 0.2 1.9
6 13.033 0.265 3.5 4.6 74 3466 82 0.2 2
7 13.089 0.273 2.8 5 7.3 3,577 62 0 1.2
8 13.025 0.271 3.2 4.6 7.2 3,659 76 0.2 2
9 12.998 0.261 3.8 4.2 8 3464 90 O 12
10 13.035 0.262 3.6 4.5 7.8 3,379 82 0 1.2
A comparison of the results from the SEIRS model (Table 2) and the
PI-SEIRS model (Table 4) reveals the following. The use of the PI-SEIRS
model provides a comparable, if not better fit to the raw data with an average
RMSE of 13.044, which is slightly lower than 13.057 obtained for the SEIRS
model. However, for the PI-SEIRS model, the average infectivity parameter
7’ is found to be 0.267 which is significantly lesser than the average of 0.307
obtained using the SEIRS model. This supports our hypothesis that a given
epidemic/disease dynamics can be caused by high infectivity parameter with
infinite or sufficient supply of medicine (SEIRS model) or a lower infectivity
parameter with insufficient supply of medicine (PI-SEIRS model).
Fig. 5 plots and compares the dynamics of the infected population as ob-
tained using calibrated SEIRS model and PI-SEIRS, with the raw data. As
illustrated, multiple configurations result in very similar epidemics behaviour.
The corresponding dynamics of two key stock management component vari-
ables, MS and MO are also shown in Fig. 6.
In this section, the epidemic component of PI-SEIRS model is calibrated
with raw data within a given range of ordering policies (a, 3) to explore the
impact of medicine supply chain on y'.
250
— (a=0, B=1.1, y'=0.267)
- --- (a=0.2, B=2, y'=0.265)
= 77 SEIRS (y=0.302)
e | AY ROR SEIRS (=0.308)
J ------ Raw Dat
g 150 aw Data
no}
a
G 1004
& P
=
50
0 T T t T T T —
0 10 20 30 40 50 60 70
Days
Figure 5: Infected data comparison from SEIRS and PI-SEIRS model
13
MS (Units)
MO (Units)
(a) (b)
Figure 6: Inventory MS, MO dynamics of calibrated PI-SEIRS model
3.4.1 Effect of Improved Medicine Supply on Epidemic Dy-
namics
In this section, we highlight the scope of reduction in epidemic peaks through
better inventory/ supply chain management. We compared the differences
in infected curves for (a, 3,7') tuple scenario and y' with infinite inventory
scenario. The results of this section are based on the data corresponding to 4!"
& 6” row of Table 4. For example, for 7 with infinite inventory scenario, we
simply substitute the values of 7‘ and other epidemic component parameters
(ie., $§, £5, P',TRM',TR’) of the PI-SEIRS model in the SEIRS model.
Similarly, the plots of infected population for (a, 8,7‘) tuple scenario are
obtained from the PI-SEIRS model, substituting all the parameters mentioned
above.
Fig. 7(a) shows that with sufficient supply of medicine an epidemic peak
can be reduced by 8.303% while in Fig. 7(b), it is 5.536%. The reason for
such difference in reduction size may be because in Fig. 7(b), we are already
using better inventory policy (see Fig. 3) compared to Fig. 7(a), and hence
increasing the supply of medicine further would reduce the peak of epidemic
a little. We require more analysis on the selection of control parameters to
further improve the system.
4 Conclusions and Future Work
An integrated PI-SEIRS model has been proposed which combines the general
production inventory model with an influenza disease diffusion model. The
performance of the PI-SEIRS model has been analysed using an illustrative
data set based on the 1919 Spanish flu. It is noted that these results are ex-
pected to be valid for any other disease epidemic data. Experimental results
show that the inventory control parameters (a and 3) significantly affect in-
fectivity parameter estimation and disease dynamics. Further, it is seen that
14
1D
[---- Infinite Supply
Infected (people)
Infected (people)
(a) 7 = 0.267, Exp. 4 from Table 4 (b) 7 = 0.265, Exp. 6 from Table 4
Figure 7: Comparison of Infected population of PI-SEIRS and SEIRS
Model with (a) and (b)
for vaccine preventable disease epidemic, a standalone epidemic model will
lead to over estimation of infectivity parameter (see Fig. 5; y' = 0.265, 0.267
and ¥ = 0.302, 0.308 resulted in the same infected curve). Hence, the integra-
tion of epidemic model and stock management model (PI-SEIRS) is important
for vaccine preventable diseases for estimating unknown epidemic model pa-
rameters. We conclude that epidemic infected data alone is not sufficient to
estimate the disease severity (infectivity) parameter indeed we also need to
consider supply chain operations information to improve accuracy. The im-
plications of these results are that better inventory management techniques
can help in reducing and controlling epidemic dynamics.
Some limitations of the work: included the assumption that the production
lead time and infectivity parameter as deterministic constants, but in reality
they may be stochastic. We have also not considered perishability iss ora
capacity constraint of medicine ordering. Future work can be carried out in
several directions. Additional empirical/ simulation/ field studies are needed
to quantify the magnitude of inventory aspects in managing the impact of vac-
cine preventable disease epidemics. Specific questions arising thereof include:
is there scenario where the primary reason for the epidemics occurrence is the
mismanagement of the supply chain? If so, what kind of inventory ordering
policies should be adopted? The current work has used a general PI model,
which is designed to operate under stable demand conditions. Thus, in the
case of epidemics driven demand pattern, further work is needed to develop
a more suitable forecasting and PI model. In the standalone SEIRS model,
the time to recover with medicine (TRM) parameter can be seen as a sum of
average waiting time for medicine and recovery time after getting medicine.
In future, one can undertake the explicit (separate) modelling of these two
delays within the PI-SEIRS model framework. Further, the cost aspects (of
the medicine supply chain), the service level aspects (of providing treatment
by hospitals), and resource constraints (at the hospitals) can be studied in
conjunction to understand the interaction effects.
References
Arinaminpathy, N., & McLean, A. (2008). Antiviral treatment for the control
of pandemic influenza: some logistical constraints. Journal of the Royal
Society Interface, 5, 545-553.
Arinaminpathy, N., & McLean, A. (2009). Logistics of control for an influenza
pandemic. Epidemics, 1, 83-88.
Bienstock, D., & Zenteno, A. C. (2012). Models for managing the impact of
an epidemic. Technical Report Working paper, Department of Industrial
Engineering and Operations Research, Columbia University. Version Sat.
Bijulal, D., Venkateswaran, J., & Hemachandra, N. (2011). Service levels,
system cost and stability of production-inventory control systems. Inter-
national Journal of Production Research, 49, 7085-7105.
Chick, S. E., Mamani, H., & Simchi-Levi, D. (2008). Supply chain coordina-
tion and influenza vaccination. Operations Research, 56, 1493-1506.
Chowell, G., Ammon, C., Hengartner, N., & Hyman, J. (2006). Transmission
dynamics of the great influenza pandemic of 1918 in geneva, switzerland:
assessing the effects of hypothetical interventions. Journal of Theoretical
Biology, 241, 193-204.
Dasaklis, T. K., Pappis, C. P., & Rachaniotis, N. P. (2012). Epidemics control
and logistics operations: A review. International Journal of Production
Economics, 139, 393-410.
Dejonckheere, J., Disney, S. M., Lambrecht, M. R., & Towill, D. R. (2003).
Measuring and avoiding the bullwhip effect: A control theoretic approach.
European Journal of Operational Research, 147, 567-590.
Duintjer Tebbens, R. J., Pallansch, M. A., Alexander, J. P., & Thompson,
K. M. (2010). Optimal vaccine stockpile design for an eradicated disease:
application to polio. Vaccine, 28, 4312-4327.
Lee, V. J., & Chen, M. I. (2007). Effectiveness of neuraminidase inhibitors
for preventing staff absenteeism during pandemic influenza. Emerging in-
fectious diseases, 13, 449.
Nsoesie, E. O., Beckman, R. J., Shashaani, S., Nagaraj, K. §., & Marathe,
M. V. (2013). A simulation optimization approach to epidemic forecasting.
PloS one, 8, e67164.
16
Ren, Y., Ordénez, F., & Wu, S. (2013). Optimal resource allocation response
to a smallpox outbreak. Computers & Industrial Engineering, 66, 325-337.
Samsuzzoha, M., Singh, M., & Lucy, D. (2013). Parameter estimation of
influenza epidemic model. Applied Math tics and Comp ion, 220,
616-629.
Sterman, J. D. (2000). Business dynamics: systems thinking and modelii
for a complex world volume 19. Irwin/McGraw-Hill Boston.
Thompson, K. M., & Duintjer Tebbens, R. J. (2014). Framework for optimal
global vaccine stockpile design for vaccine-preventable diseases: Application
to measles and cholera vaccines as contrasting examples. Risk Analysis, .
Venkateswaran, J., & Son, Y.-J. (2007). Effect of information update fre-
quency on the stability of production-inventory control systems. Interna-
tional Journal of Production Economics, 106, 171-190.
Yarmand, H., Ivy, J. S., Denton, B., & Lloyd, A. L. (2014). Optimal two-phase
vaccine allocation to geographically different regions under uncertainty. Eu-
ropean Journal of Operational Research, 233, 208-219.
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