Model II: Medical Resource Is Limited
9.3 Model Formulation
9.3.1 Epidemic Compartmental Model
As shown in Fig. 9.1, we construct a compartment model for analyzing epidemic dynamics based on the 2009 H1N1 pandemic. In each affected area j,∀j ∈ J, indi- viduals are classified as: susceptible (S), exposed (E), infected (I), hospitalized (H), recovered (R), and asymptomatic and partially infectious (A). We call it SEIHR-A model. In this model, each compartment represents the corresponding health status of people with regard to the H1N1, while the connected arc shows epidemic transmis- sion between two compartments. Note that deceased individuals are not considered in our model, because H1N1 has become a seasonal flu with very low mortality rates due to the advance in medical technologies.
There are several assumptions for the proposed SEIHR-A model. First, compared to the traditional bilinear transfer rates among compartments S,E and I [14], the single-linear transfer rate assumption, except the variable transfer rate from I to H, is the foundation for modeling a mixed-integer linear programming (MIP) in Büyüktahtakın et al. [1]. Otherwise, we cannot alleviate the non-linearity of epidemic model and thus the integrated epidemic-logistics model is always non-linear. Second, we assume that individuals who are infected will go into a latent (exposed) stage, during which they may have a low level of infectivity (we define a reduction factor q for transmissibility of the exposed class in following paragraph). After that, some proportion of the exposed individuals go into the infected compartment and then to the recovered compartment through treatment. Meanwhile, some proportion of the latent individuals never develop symptoms and go directly from the latent class to the asymptomatic and partially infectious class and then to the recovered class [12]. Whatever, we assume that individuals in the recovered class receive lifelong
Table9.1Summaryoftherelevantstudies ReferencesLogisticsattributeMinimize totalnew infections ObjectiveMethodologySolution AllocationDistributionLocationMinimize unsatisfied demand
Minimize totaldeathsMinimize costIPLPMIP Rachaniotis etal.[35]XXXHeuristic Rachaniotis etal.[37]XXXHeuristic Dasaklis etal.[36]XXXSimulation Ekicietal. [38]XXXCPLEX Chenetal. [39]XXXCPLEX Renetal. [40]XXXHeuristic HeandLiu [41]XXXXMATLAB (continued)
Table9.1(continued) ReferencesLogisticsattributeMinimize totalnew infections ObjectiveMethodologySolution AllocationDistributionLocationMinimize unsatisfied demand
Minimize totaldeathsMinimize costIPLPMIP Anparasan andLejeune [42]
XXXXXCPLEX Büyüktahtakın etal.[1]XXXXXCPLEX LiuandZhao [43]XXXMATLAB Liuetal.[44]XXXCPLEX Liuand Zhang[45]XXXXCPLEX ThisstudyXXXXMATLAB
Susceptible (S) Exposed (E) Infected (I)
Asymptomatic (A)
αlj νjl βj
(1−pj)δj γ2j
γ1j
j j
pδ Hospitalized (H) Recovered (R)
Fig. 9.1 Framework of the epidemic compartment model
immunity. Third, similar to Büyüktahtakın et al. [1], we assume that transfer rate from compartment I toH is a variable parameter, which is determined by the available number of isolated wards. Correspondingly, infected persons who are not quarantined will continue to spread disease in the outside. Since the H1N1 outbreak only last for several months, natural birth/death rates are excluded from the epidemic model because they normally affect the dynamics of disease over several years.
To facilitate epidemic model formulation, we give the notations used as follows:
Parameters:
T Set of time,t = {0,1,2, . . . ,T}.
J Set of affected areas, j = {1,2, . . . ,J}.
L Set of all surrounding affected regions of area j,l= {1,2, . . . ,J}\j.
βj Transmission rate of the H1N1 in area j.
qj Reduction factor of infectiousness for the classEj. pj Proportion of symptomatic infection in area j. δj Infected rate in area j.
γ1j Recovery rate for infectious class in area j. γ2j Recovery rate for asymptomatic class in area j.
αl j Migration rate of susceptible individuals from surrounding areas to area j.
νjl Migration rate of susceptible individuals from area jto surrounding areas.
State variables:
Sj(t) Number of susceptible individuals in area jat timet. Ej(t) Number of exposed individuals in area j at timet. Ij(t) Number of infected individuals in area jat timet.
Hj(t) Number of hospitalized individuals in area jat timet.
Rj(t) Number of recovered individuals in area jat timet. Aj(t) Number of asymptomatic individuals in area j at timet.
S
j(t) Number of susceptible individuals migrate into area jat timet. Sj(t) Number of susceptible individuals travel out from areaj at timet.
Ij(t) Number of patients that can be accepted for treatment in area jat timet. Based on the above notations, we can use the following time discretized equations to demonstrate the epidemic dynamics.
Sj(t+1)=Sj(t)+S
j(t)−Sj(t)−βj[qjEj(t)+Ij(t)+Aj(t)],
∀j∈ J, t ∈T. (9.1)
Ej(t+1)=Ej(t)+βj[qjEj(t)+Ij(t)+Aj(t)] −δjEj(t),
∀j ∈ J, t ∈T. (9.2)
Ij(t+1)=Ij(t)+pjδjEj(t)−Ij(t), ∀j ∈ J, t ∈T. (9.3) Hj(t+1)=Hj(t)+Ij(t)−γ1jHj(t), ∀j ∈ J, t∈T. (9.4) Rj(t+1)=Rj(t)+γ1jHj(t)+γ2jAj(t), ∀j ∈ J, t ∈T. (9.5) Aj(t+1)=Aj(t)+(1−pj)δjEj(t)−γ2jAj(t), ∀j ∈ J, t∈T. (9.6) According to He and Liu [41], we assume that all individuals who move in and out of the area are susceptible. Therefore, following the formulation of Büyüktahtakιn et al. [1], we use Eq. (9.7) to define the immigration of susceptible individuals from all surrounding regions to areajat timet, and use Eq. (9.8) to represent the outmigration of susceptible individuals from area j to all surrounding areas at timet.
S
j(t)=
l∈L
αl jSl(t), ∀j ∈ J, t ∈T. (9.7)
Sj(t)=
l∈L
νjlSj(t), ∀j ∈ J, t ∈T. (9.8)
The above eight difference equations describe that the change rate of each com- partment is determined by the entering and exiting population in the corresponding compartment. For example, Eq. (9.6) shows that the number of asymptomatic indi- viduals at time t +1 is equal to the number of asymptomatic individuals on the previous time plus the new individuals transferring from the exposed compartment, and minus individuals who are recovered. These equations can be used to forecast the number of susceptible, exposed, infected, hospitalized, asymptomatic and recovered individuals by giving the initial values of each compartment and the corresponding parameters. The only problem need to be determined is the state variableIj(t), which means the number of infected individuals that can be accepted for treatment in area
jat timet.
As we all know, the quantity of how many infected individuals can be treated depends on the scale of emergency budget. If emergency budget is enough, all infected people can be hospitalized [44]. However, if emergency budget is limited, policymakers must consider how to effectively use the limited budget to maximize the emergency service level [45], to minimize the total number of infections and fatalities [1], or to minimize the total unsatisfied demand [41]. With the different
optimization objectives, the state variable Ij(t)will be correspondingly changed.
Considering that H1N1 is now a seasonal flu with very low mortality rates, we use the minimization of total unsatisfied demand as our optimization objective in this study. The detail of the resource allocation model is introduced in the following section.