key: cord-0578307-s40jt0ci
authors: Sinkala, Zachariah; Manathunga, Vajira; Fayissa, Bichaka
title: An Epidemic Compartment Model for Economic Policy Directions for Managing Future Pandemic
date: 2022-02-11
journal: nan
DOI: nan
sha: da415fe3cb5d98270dec99e2571a54f62b9b2727
doc_id: 578307
cord_uid: s40jt0ci

In this research, we develop a framework to analyze the interaction between the economy and the Covid-19 pandemic using an extension of SIR epidemic model. At the outset, we assume there are two health related investments including general medical expenditures and the other for a direct investment for controlling the pandemic. We incorporate the learning dynamics associated with the management of the virus into our model. Given that the labor force in a society depends on the state of the epidemic, we allow birth, death, and vaccination to occur in our model and assume labor force consists of the susceptible, vaccinated, and recovered individuals. We also assume parameters in our epidemic compartmental model depend on investment amount for directly controlling the epidemic, the health stock of individual representative agents in the society, and the knowledge or learning about the epidemic in the community. By controlling consumption, the general medical expenditure, and the direct investment of funds for controlling the epidemic, we optimize the utility realized by the representative individuals because of consumption. This problem is nontrivial since the disease dynamics results in a non-convex optimization problem.

COVID 19 is the most significant event for humans in this century. Compare to the impact of two world wars and several epidemics occurred in last century, this has equivalent or may be more impact on the society. The impact can be social, economical or both. Covid 19 impact on economies around the world is significant. For example, [18] summarized recent research and report on the global economic impact of the COVID 19 . However, the severity of the impact changed from one region to the other. Because of that, a slew of research articles focused on economic impact based on the region or country was published. For example, [14] discuss the social impact of COVID 19 in India, [1] discuss the impact on developing Asia, [17] put emphasis on African region , [29] discuss the COVID 19 impact on poverty in Indonesia, and [12] research on COVID 19 impact on UK economy name to few.

Many countries opt to allocate money to directly combat the virus during this time and use many mitigation policies. These prevention, intervention, and mitigation policies and their impacts were investigated in several articles. A review of current interventions were done in [23] , effects of interventions were discussed in [28] , A model for COVID 19 intervention policies in Italy were given in [7] , A model to simulate the health and economic effect of social distancing given in [26] . One of tragedy in COVID was the loss of human lives. Labor-wise, the workforce shrank significantly due to death, infection, and movement restrictions, which significantly affect the growth of the economy. We have mentioned that COVID has direct impact on labor or human capital in above. This indirectly cause the shortage in physical capital. For example current "chips shortage" [30] , and port blockage [22] discuss implication of COVID on physical capital. According to the World Bank [19] , baseline GDP of the world would fall below 2 percent from pre-pandemic baseline, for developing countries by 2.5 percent, and for industrial countries by 1.8 percent.

History has shown that humans eventually learn how to either eradicate or live with diseases. For example, smallpox, rinderpest are completely eradicated and we learned to live with AIDS using prevention methods. Similar to this, learning occurred over the time about COVID 19. At the early pandemic stage, countries opted to close schools, workplaces, and economies; however, as time passed, these economies decided to open again and learned to live with the COVID. Even though new variants such as Delta and Omicron are becoming more prevalent, many countries opt to keep open, using the knowledge accumulated over the time to handle these new variants. Therefore it is clear that we should look into learning about the disease that occurs during this time.

Learning does not have to be strictly pharmaceutical, but it can be nonpharmaceutical learning, such as wearing masks, keeping 6 feet apart from others, etc. Also, another aspect of the COVID is the introduction of the vaccine. Even though the vaccine was commonly available in developed countries, subgroups started to show resistance to getting vaccinated. Thus, vaccine hesitancy is becoming a key issue in these countries, which implies COVID 19 may become endemic over time.

Infectious diseases may bring shocking effects to economies. Therefore understanding the impact of epidemics on economies is crucial. One approach would be to model the economy's dynamics without disease and introduce disease as an exogenous shock to the model. However, this would prevent us from understanding the two-way interactions between the disease dynamics and the economy. For example, as we mentioned earlier, when the government directly allocates money to combat a deadly disease, it impacts disease transmission and other aspects of the disease. On the other hand incidence of disease negatively affect the labor force, which causes lower physical output in the physical production function. Therefore, we believe it is best to model the disease and the economic dynamics together. In this research, we model the accumulation of physical and health capital through the neo-classical growth model. COVID 19 brought renewed attention to compartmental models and their applications.We endogenize disease and use SIRV (Susceptible, Infected, Recovered, Vaccinated) model for disease dynamics.The interaction between the neo-classical growth model and the SIRV model is introduced as follows: First, we assume the labor force consists of people who are not infected. Second, we consider two parameters in the SIRV model, namely transmission rate and recovery rate; both depend on the money allocated to combat the disease, the knowledge in the society about how to control the disease, and health capital. Thus, disease and economic dynamics are needed to solve simultaneously. Another aspect of our model is dynamics of knowledge for controlling the disease.

We have mentioned that we incorporate learning dynamics into our model. The learning or knowledge production function uses direct investment to control the epidemic and existing knowledge to create more understanding about managing the outbreak. Therefore the learning by controlling function help to reduce the cost associated with disease control and improve the health stock. The process is intuitive in many ways. Any society which faces an outbreak eventually accumulates or created knowledge about how to control it, which will be used in the next epidemic.

In our model, we consider two types of health expenditure. The health capital production function produces health. In order to produce health stock, the health production function uses general health expenditure, which results in health, or more healthy people to be accurate. Healthy people are less prone to become infected. Therefore this health is required. The second form of health expenditure comes from direct allocation to combat the deadly disease, such as money to do research and find vaccines and other cures. This is also crucial when battling a virus-like COVID 19.

Several previous studies focused on modeling disease and economic dynamics together. The effect of recurring disease on an economy that follows SIS dynamics was investigated in [9] . In that paper, the authors tried to understand the best society can do to control the disease transmission while optimizing the utility gain by the consumption. The authors showed that a steady-state with zero health expenditure could be an optimal solution to the problem. However, the authors never considered a direct investment to control the epidemic or learning that occurs when controlling the disease. Also SIS model assumes no immunity from the disease compared to SIRV model we used.

SIR model with two way interaction between the economy and the disease was investigated in [11] . In this model, the authors introduced learning by controlling and direct investment to control the epidemic. However, the economy consists of a single sector with input as health capital. We have a two-sector economy where physical capital and health capital are both produced compared to this model. Also, the authors assumed that death could occur only when in infected state, but we assumed death could occur at any compartment. The paper lacks rigorous mathematical proof, which establishes necessary and sufficient conditions of optimal solutions and calculation of steady states.

The authors in [4] used the SI epidemic model and claimed that a minimal level of labor is necessary to reduce the long-term prevalence rate of the epidemic. If this minimal labor is not reached, then prevention would be temporary and may not be optimal to take. On the other hand, if enough labor exists, allocating resources to prevent the disease is feasible but not optimal.

Epidemic models and their interactions with the economy was studied in [8, 15, 5] More recently, SIR model, disease-related mortality and effect on economy was studied in [10] .

We use modified SIR model with the vaccination to describe the dynamics of the disease. The model is given in figure 1.

We assume under epidemic outbreak, direct investment, A t , to control epidemic will occur and some existing knowledge of controlling of the disease, e 0 . We assume, as time pass by learning about disease control e t , will occur where e t is the accumulated knowledge of disease [11] .

We denote the knowledge production function: learning by controlling as E(A t , e t ).

We also assume knowledge about disease control can be depreciated over time [16] .

Thus law of motion for knowledge production is given by rate δ E ∈ (0, 1). We also assume disease transmission rate β(A t , e t , h t ), recovery rate and γ(A t , e t , h t ) are depend on A t , e t and h t where h t is the health capital. In simplified assumption, we allow the transmission rate β to depend on experience in controlling the disease. In our model (2.2), we use mass action incidence principle. We denote natural birth rate by b, natural death rate as µ and assume these are constants. We also assume death due to the disease is negligible compare to the population size. As an example this model can be used in countries where epidemic death is less compared to total population. Assume that vaccination is carried out at recruitment. Thus, fraction 0 ≤ p ≤ 1 of population is vaccinated at time t and become immune to the disease, and only (1 − p) fraction of population enter into the susceptible class. Let N t denote the total population at time t. Then, we have S t + I t + R t + V t = N t . The respective dynamics of modified Susceptible-Infected-Recovered (SIR) epidemiology model with vaccination is given by:

Similarly, we get dynamics of fractions of SIR epidemiology model as follows:

From this point onward, we would suppress β(A t , e t , h t ), γ(A t , e t , h t ) and σ(A t , e t , h t ) to be just β, γ and σ respectively. Before we continue to the economic model, we would like to analyse steady states of this epidemic model. The disease free equilibrium points [21] are given by

Jacobian of the disease free equilibrium for full system is given by

Eigenvalues of the disease free Jacobian are

The local stability of the solution depends on whether the eigenvalues of the Jacobian are negative or have negative real parts [2] . Observe that if β(1 − p) < (b + γ) then disease free steady state is stable otherwise unstable. If fraction of the population vaccinated at recruitment, p, is zero, then reproduction number is R 0 = β b+γ [21] . Thus by vaccinating, we would reduce the original reproduction number by (1 − p). As mentioned in [21] , if we need the reproduction number to be less than 1, then at least 1 − 1 R 0 fraction of population need to be vaccinated otherwise the disease will invades the population. Next, endemic state equilibrium points [21] are given by

We need R 0 > 1, and (1 − p)R 0 > 1 to make sure 0 ≤ s * , i * ≤ 1 in endemic steady state. However former follow from the latter inequality. Jacobian of the endemic state

Define,

Then eigenvalues of this Jacobian given by 

Hence when (1−p)β b+γ > 1, all eigen values are negative and endemic state is stable.

Assume vaccination rate p is constant and define R vac = (1−p)R 0 . Thus the system has one stable disease free steady state if R vac < 1. Otherwise, the system has two steady states with stable endemic state and unstable disease free steady state. Hence, R vac is the bifurcation point. Following [21] , we give following diagram to illustrate forward bifurcation. 

A t , e t , h t if (1 − p)β(A, e, h) < b + γ(A,

To avoid keeping track of individual health histories, we use the framework of a large representative consumer. We assume the economy is populated by non atomic identical consumers whose consumption is same irrespective to the health status and no need to keep track of the health record of individual consumers [9] . Thus, social planner can maximize the respective optimization problem for the representative consumer. We assume social planner try to maximize following utility function

where θ is the discount factor, c t is the consumption. Social planner face following problem: how to choose consumption c t , general medical expenditure m t and direct epidemic control investment A t such that utility function is maximized. We assume general medical expenditure m t and direct epidemic control investment A t are mutually exclusive. We assume there are three production functions: one for physical goods, one for health and one for creating knowledge controlling epidemic. Physical goods are either consumed, invested again, spend in health or depreciated. The production function f of physical goods is depend on capital k t , effective labor l t . However under our model, effective labor

where δ K ∈ (0, 1), denotes the depreciation of physical capital. The health capital production function g depend on medical expenditure m t and hence we have

Here δ H ∈ (0, 1) denotes the depreciation of health capital.

We make following assumption regarding this model.

Assumption 2. Transmission rate function β(A t , h t , e t ) satisfy following conditions:

4. The second order partial derivatives, β 11 , β 22 , β 33 ≥ 0.

(the inverse of average infectious period) satisfy following conditions:

2. The first order partial derivatives γ 1 , γ 2 , γ 3 ≥ 0.

The gradient of f has positive components and the Hessian of f is negative definite.

Assumption 5. The health production function g(m t ) satisfy the following Inada conditions [13] :

Assumption 7. There exist κ 1 such that κ 1 ∈ (0, ∞) and −κ 1 ≤ dkt dt k t . [9] Assumption 8. The knowledge production for disease control, E(A t , e t ) : R 2 + → R + satisfy following conditions:

Since direct investment to control the epidemic, A t and the existing knowledge

Consider the following social planner problem(P):

and assume the assumptions 1-9.

where x t is the control vector variable, y t is the state vector variable.

In this section, we show that there exist optimal solution to the above mentioned social planer problem. Usual approach to show that existence of optimal solution is to use current Hamiltonian and then apply either Mangasarian sufficient conditions [20] or Arrow sufficient [25] conditions. Under Mangasarian sufficient conditions, the current Hamiltonian must be concave w.r.t to all state and control variables. This may not be true in our situation. For Arrow sufficient conditions, concavity with respect to the state variables of the Hamiltonian maximized with respect to control variable is required [24] . Again, this may not be true in our situation. For our problem, we rely on theorem 1 given in [6] . First we restate the theorem 1 given in [6] without proof and then show that how it can be applied to our problem.

Consider the following social planner problem(P) [6] :

whereȳ t = (ȳ 1 , · · · ,ȳ K ) is the external variable, y t = (y 1 , · · · , y K ) is the state variable,

x t = (x 1 , · · · , x C ) is the control variable and dyt dt is the derivative of y t . C, K denote number of control variables and number of state variables in the optimization problem, which are finite. Assume the following:

such that:

1. For all t, for all jȳ j t ≤ A j e a j t 2. If x t and y t satisfy the following differential constraint for all t,

then for all t, We can apply above theorem 3.1 to social planner problem given in Equation (2.7).

However, first we need to show that social planner problem (2.7) satisfy assumption A1, · · · , A5 

Since |s t |, |i t | ≤ 1 and from assumption (2), |β(·)| ≤ 1, choose

. It is clear that F (·) is continuous on R 5 + and satisfy F (y t ) ≤ẏ t Lemma 3.3. Let y t = (k t , h t , s t , i t , e t ) and x t = (c t , m t , A t ) be state and control variables given in social planner problem (2.7) . Letȳ t = (ȳ 1 , · · · ,ȳ 5 ) an external variable Then there exists a continuous function G(·) : R 5

and G j is concave with respect to x t Proof. Chooseȳ t = (1, · · · , 1) and e t ) ). We claim

It is obvious that G(·) is continuous and satisfyẏ t ≤ G(ȳ t , y t , x t ). Except g(m t ) and E(A t , e t ) all other G j are constants respect to x t , hence concave. For g(m t ), under section 2.3 assumption (5), concavity follows respect to x t . Similarly concavity of E(A t , e t ) with respect to x t follows from assumption (8) . 

for all j ∈ { 1, 2, · · · , 5}

Proof. First note thatȳ j t = 1 for all j from lemma (3.3). Now, following proof of lemma 1 (with few simple modification) given in [9] , it can be easily shown that for any a 1 , a 2 ∈ (0, θ) there exist A 1 , A 2 > 0 such that max{ 1, k t , |k t |} ≤ A 1 e a 1 t and max{ 1, h t , |ḣ t |} ≤ A 2 e a 2 t for all t, where θ denote the discount rate used in social planner problem in (2.7). Observe that s t , i t , β ≤ 1 and |ṡ t | ≤ |b| + |b| + 1. Hence for any a 3 ∈ (0, θ) there exist A 3 > 0 such that max{ 1, s t , |ṡ t |} ≤ A 3 e a 3 t for all t. Similar type arguments can be used to show that for any a 4 ∈ (0, θ) there exist A 4 > 0 such that max{ 1, i t , |i t |} ≤ A 4 e a 4 t for all t. Thus for any a 5 ∈ (0, θ) there exists constants C 0 such thatė t = E(A t , e t ) ≤ E(A, e t ) ≤ C 0 + a 5 e t . Again using similar argument given in first portion of lemma 1 in [9] , we conclude for any a 5 ∈ (0, θ) there exist A 5 > 0 such that max{ 1, e t , |ė t |} ≤ A 5 e a 5 t .

Hence the result. 

Proof. We know from assumption (4), physical production function f is increasing 1) . Now using the fact that lim kt→∞ f (k t , 1) = 0 and L'hopital's rule we conclude for any b 0 ∈ (0, θ)

Hence the result.

Before we continue observe that discount rate θ > b i given in lemma 3.5. This is because, we choose b i such that b i ∈ (0, θ). Now we are ready to show the existence of a solution 

In this section we investigate the steady states of the social planner problem at optimality. We use first order necessary condition for optimal solution to find the steady states. For SIR model with the vaccination given in Equation ( 

where λ i are co-state variables and ν i are Lagrangian multipliers. The first order necessary condition for optimal solution is given by the theorem 14.5 in [3] . First order conditions are given by,

lim t→∞ e −θt λ 1,t k t = 0, lim t→∞ e −θt λ 2,t h t = 0, lim t→∞ e −θt λ 3,t s t = 0 (4.11)

lim t→∞ e −θt λ 4,t i t = 0, lim t→∞ e −θt λ 5,t e t = 0 (4.12)

ν i ≥ 0, iν 1 = 0, mν 2 = 0, Aν 3 = 0, sν 4 = 0 (4.13)

Let x t = (c t , m t , A t ), y t = (k t , h t , s t , i t , e t ), λ t = (λ 1 , λ 2 , λ 3 , λ 4 , λ 5 ) and ν t = (ν 1 , ν 2 , ν 3 , ν 4 ).

Then steady state for social planner problem is given by a set of values (x * t , y * t , λ * t , ν * t ) which satisfyẏ t =λ t = 0. Before we continue, we make following assumption regarding 

and c * is given by 

Since b ≥ µ, we conclude that steady state value, λ * 2 = 0. This imply ν 2 = u (c) > 0. Hence, m * = 0 (and h * = 0). From Equation (4.5) we have

. Thus we conclude there exist unique k * = 0, such that

When i * = 0, Equation (4.9) reduce to λ 5 (θ + δ E − E * 2 ) = 0. Since E * 2 = θ + δ E from assumption (9), we conclude that λ * 5 = 0. When i * = 0 and using the fact ν 4 = 0, we can conclude from Equation (4.7) that λ * 3 = 0. From Equation (4.4), we conclude that ν 3 = λ 1 > 0. Hence, A * = 0. Now using the fact E(0, e) = 0 and the Equation (2.12), e * = 0. Observe that ν 1 is a non-negative free variable. Thus, we can solve for λ 4 using Equation (4.8).

exists solution m * , k * , A * , e * , h * to following set of equations and inequalities

(4.14)

Proof. Let m * , h * , s * , i * , e * , A * , k * be a solution set to above system of equations and inequalities. We claim these values satisfy Equations ( γ * +b − b β * and effective labor at endemic steady state is given by

withṡ t =i t = 0. Since s * , i * > 0, we have ν 1 = 0, ν 4 = 0. Since m * and h * satisfy Equation (4.15), from Equation (2.9) we concludeḣ t = 0. Since, k * satisfy Equation 

Thus, from equation (2.8), we conclude thatk t = 0. Since l * < 1, k * would be different from k * in disease free steady state. Since A * , e * satisfy Equation (4.17-4.18), we haveė t = 0. Therefore, if m * , h * , s * , i * , e * , A * , k * is a solution to above set of equations and inequalities, thenk t =ḣ t =ṡ t =i t =ė t = 0. Next observe that since m * , h * , s * , i * , e * , A * , k * satisfy Equation (4.16), from Equation (4.5), we conclude thaṫ

Define, 

The determinant of the matrix A is given by 

Since we assume Equation (4.14) holds, we know endemic steady state exists and s = γ * +b β * . Hence, m 53 m 64 − m 54 m 63 > 0. Thus, endemic steady state values of λ * i 's and ν * 2 , ν * 3 exists and given by B = A −1 C and λ * 1 = u (c * ). Using notations in [9] , define

where j = 1, 2, 3. Then solutions for ν 2 and ν 3 are given by Conversely suppose m * , h * , s * , i * , e * , A * , k * is solution to endemic steady state for equations given in (2.8-2.15 ) and satisfy first order conditions given in (4.2-4.13) . Then, clearly we can see that it need to satisfy the equations and inequalities given in (4.2-

We make following assumption in order to further characterize endemic steady state. Proof. First observe that, l θ,j (0, 0, 0) =ī γ j (0, 0, 0)βī − (β j (0, 0, 0)s − γ j (0, 0, 0))(θ + b)

Observe that numerator of Equation (4.32) is a quadratic polynomial in θ ∈ (0, ∞).

It is easy to show that constant term of that polynomial is negative. hence the result follows.

In next few propositions, we further analyze the endemic steady state. We denote

to be solutions to following equations: Rewrite Equation (4.34) as below.

Then observe that RHS of above equation is decreasing w.r.t θ >Ē 2 − δ E and LHS is a constant w.r.t θ. Also note that θ =Ē 2 − δ E is a vertical asymptote. Therefore, there exist a unique solutionθ 2 (b). Note that, it is possible these solutions,θ 1 (b),θ 2 (b)

to be negative.

We defineθ max (b) = max(θ 1 (b),θ 2 (b), 0) andθ min (b) = min(θ 1 (b),θ 2 (b)) . Observe that from Equation (4.15),

Since g (m * ) > 0, g (m * ) < 0 from assumption (5), we conclude that ∂g (m * ) ∂h < 0. Hence g (m * (h)) decrease as h increases. Next, observe that at endemic steady state, 

Since f 12 > 0, f 11 < 0, ∂l * ∂A > 0, we have ∂k * ∂A > 0. Similar proof apply w.r.t e and h. Thus, k * (A, e, h) increases as A, e, h increases. Now, we want to see the behaviour of f 2 (k * , l * ) as A, e, h increases.

Next,from Equation (4.16), 

is the unique steady state solution to the system given in Equations (4.14-4.24) . In otherwords, when θ >θ max there exist unique endemic steady state without health expenditure and without direct investment to control the epidemic.

Proof. Since b ∈ [µ, (1 − p)β − γ) we know, we are in endemic state. Let A * = 0, e * = 0, h * = 0, m * = 0, k * = k, l * = l and θ >θ max (b). We want to claim that this satisfy equations and inequalities in (4.14-4.24). Note that only conditions we need to check are inequalities (4.20) and (4.23) . Observe that, 3 (0, 0, 0)f 2 (k, l)g (0). Hence inequality (4.20) is satisfied. Also note that,

Hence inequality (4.23) is satisfied. Hence A * = 0, e * = 0, h * = 0, m * = 0, k * = k, l * = l is a solution to endemic steady state when θ >θ max (b). The solution become unique from proposition (4.5). Let θ >θ max and A * , e * , h * be a solution to the system given (4.14-4.24). Then,

Thus, we conclude ν 2 > 0 and ν 3 > 0. Hence A * = 0, m * = 0, which imply, e * = 0, h * = 0. This would also imply l * = l and k * = k. Hence solution is unique.

there exists a unique endemic steady state with zero health expenditure and nonzero epidemic control investment with solution determined by the following system of equations 

We also know that,

Hence only possibility is

However, this is a contradiction to the definition ofθ 2 (b) given in Equations (4.34).

Therefore A * > 0 and A * is determined by the Equation (4.41). The uniqueness of the solution follows from implicit mapping Theorem. 

Proof. omitted

We have found solutions for steady states and we know these solutions satisfy first order necessary conditions for optimality given in Equations (4.2-4.13 [3] . Define x t = (k t , h t , s t , i t , e t ) as state variables, z t = (c t , m t , A t ) as control variable set and λ t = (λ 1,t , · · · , λ 5,t ). Then Hamiltonian is given by

with first order and transversality conditions given as,

lim t→∞ e −θt λ 1,t k t = 0, lim t→∞ e −θt λ 2,t h t = 0, lim t→∞ e −θt λ 3,t s t = 0 (5.11)

lim t→∞ e −θt λ 4,t i t = 0, lim t→∞ e −θt λ 5,t e t = 0 (5.12) 

is non negative or non-positive. Therefore, Mangasarian condition is not easy to apply. 

Observe that physical production function f (k, l), health production function g(m)

and knowledge production E(A, e) all follows neoclassical growth theory [3] . Hence, the steady state solution given in in the proposition (4.1) is optimal (see Chapter 15 in [3] ). Now consider the endemic steady state without health expenditure and without direct investment to control the epidemic. Then, current value Hamiltonian is given by

From first order necessary conditions,

Since E 1 > 0 by assumptions and using Equation (5.19) , we conclude that lim t→∞ e −θt λ 5,t = 0

Thus, 

Hence the result.

Since endemic steady state solutions with positive health expenditure and positive direct investment to control the epidemic satisfy first order necessary conditions, we have following result.

where 0 < ψ 2 < 1, ψ 1 ≥ 0, ψ 3 ≥ 0 [9] . Our third choice, we used transmission rate function given in [11] β(A, e, h) = β 0 e −ηAeh , (6.5) where η ≥ 0. Hereβ = β 0 or β 1 + β 0 . If there is no learning by controlling effect of the disease then we set η = 0, in which case the transmission rate is a constant.

We assume that the recovery rate depend on accumulative knowledge of the disease, health capital and capital to control the disease. we study the following examples: 

We numerically vary the social planner's discount rate and true to see the affects of the endemic steady state of our model with learning by controlling versus the model without learning by controlling. The discount rate, θ reflects the patience of the social planner, with a higher discount rate reflecting a less patient planner.

In the first case, we consider the model without disease control. In this case, the epidemiological variables are invariant to changes in the discount rate. The economic variables behave just as expected in a standard growth model, with a higher discount rate leading to a lower steady-state health capital stock and physical capital stock and therefore lower output and consumption.

The second case, we consider the model with disease. In this case, the epidemiological variables vary significantly with respect to the discount rate. Since changes in the discount rate affect learning by controlling and thus the disease transmission rate.

The result of this is that a less patient planner chooses higher total and susceptible populations and lower infected and recovered populations. In terms of the economic variables, the stock of disease controlling experience varies dramatically with the discount rate. A less patient planner devotes far more resources to disease control. Because of the positive effect on the working population, this increased disease control does not come at the expense of output or consumption. Rather, output and consumption are both increasing in the discount rate. A final noteworthy aspect of the model with disease control is that, as the discount rate changes, general health investment and capital do not move monotonically. As the discount rate increases, they both initially fall and then rise. We found that this may not be a good assumption once we allow for endogenous changes in health investment and disease control. 

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The impact of covid-19 outbreak on poverty: An estimation for indonesia

An analysis on the crisis of "chips shortage" in automobile industry--based on the double influence of covid-19 and trade friction

Thus, this is correspond to neoclassical steady states and hence the solution given in the proposition (4.6) is optimal in a local neighborhood where m = 0, A = 0, h = 0, e = 0. Now consider the endemic state with either positive health expenditure or positive direct investment to control epidemic. We follows the proof given in [9] with few modifications.Assumption 11. For all l ∈ (0, 1) with f 1 (k, l) > b − d + δ there exist a maximum sustainable capital stock,k, for the given l. This maximum sustainable capital stock can be obtained by solving f (k, l), and λ t = (λ 1,t , · · · , λ 5,t ) denote a path with positive health expenditure and positive direct investment to control the epidemic which satisfies necessary condition given in Equations (5.2-5.10 ) along with the transversality conditions given in (5.11-5.12 ) andinitial condition x 0 = (k 0 , h 0 , s 0 , i 0 , e 0 ). Then this path is a locally optimal for the social planner problem given in (2.7) Proof. Now suppose there exist another path x t , z t , λ t with same initial condition x 0 .

Notice that,from concavity of u(·) and g(·),Now using concavity of β, γ and E,for last inequality we used first order conditions. Therefore the first term of the Equa-Now consider the second term of the Equation (5.17),Therefore the second and third term of the Equation 5.17 can be simplify as,where x * 0 = x 0 from the initial conditions. Now we want to claim lim t→∞ e −θt λ t (x * t − x t ) = 0. Using assumption (11) and similar argument given in proposition 4 in [9] , we conclude that lim t→∞ e −θt λ 1,t = lim t→∞ e −θt λ 2,t = lim t→∞ e −θt λ 3,t = lim t→∞ e −θt λ 4,t = 0 (5.19)Hence, Corollary 5.2. The endemic steady state with positive health expenditure and positive direct investment to control the epidemic is locally optimal., and λ t = (λ 1,t , · · · , λ 5,t ) denote a path with positive health expenditure but zero direct investment to control the epidemic which satisfies necessary condition given in Equations (5.2-5.10 ) along with the transversality conditions given in (5.11-5.12 ) and initial con-Then this path is a locally optimal for the social planner problem given in (2.7) Proof. omitted, and λ t = (λ 1,t , · · · , λ 5,t ) denote a path with zero health expenditure and positive direct investment to control the epidemic which satisfies necessary condition given in Equations (5.2-5.10 ) along with the transversality conditions given in (5.11-5.12 ) and initial condition x 0 = (k 0 , h 0 , s 0 , i 0 , e 0 ). Then this path is a locally optimal for the social planner problem given in (2.7) Proof. omitted

In this section, we present numerical simulations of the model in Python. To implement our model, we specify simple standard production function forms that meet the assumptions of the model, the transmission rate function form and the recovery rate function form. We choose a Cobb-Douglas form for the production function of physical goods:f (k, l) = k ψ l 1−ψ , (6.1)where 0 < ψ < 1. For healthy production function g, we choose g(m) = ψ 3 (m + ψ 1 ) ψ 2 − ψ 4 ψ ψ 2 1 , (6.2)