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Country versus pharmaceutical company interests for hepatitis C treatment

Hepatitis C virus (HCV) is one of the leading causes of liver disease and is responsible for massive health and economic burden worldwide. The disease is asymptomatic in its early stages, but it can progress over time to fatal end-stage liver disease. Thus, the majority of individuals infected with HCV are unaware of their chronic condition. Recent treatment options for HCV can completely cure the infection but are costly. We developed a game model between a pharmaceutical company (PC) and a country striving to maximize its citizens' utility. First, the PC determines the price of HCV treatment; then, the country responds with corresponding screening and treatment strategies. We employed an analytical framework to calculate the utility of the players for each selected strategy. Calibrated to detailed HCV data from Israel, we found that the PC will gain higher revenue by offering a quantity discount rather than using standard fixed pricing per treatment, by indirectly forcing the country to conduct more screening than it desired. By contrast, risk-sharing agreements, in which the country pays only for successful treatments are beneficial for the country. Our findings underscore that policy makers worldwide should prudently consider recent offers by PCs to increase screening either directly, via covering HCV screening, or indirectly, by providing discounts following a predetermined volume of sales. More broadly, our approach is applicable in other healthcare settings where screening is essential to determine treatment strategies.


Introduction
Hepatitis C virus (HCV) is one of the leading causes of liver disease, causing massive public health and economic burden worldwide. The virus leads to chronic illness and increases the risk of liver cirrhosis, hepatocellular carcinoma, liver failure, and death. Recently, the World Health Organization (WHO) suggested that over 70 million people are chronically infected with HCV and that each year HCV-related complications account for approximately 400,000 incidents [1,2]. Disease prevalence varies widely between continents, ranging from 1.3% in the Americas to 2.9% in Africa [2][3][4], and poses a substantial burden to both developing and developed countries. In the US, it is estimated that 3.5 million people are chronically infected with HCV [5]. Moreover, each year the disease is responsible for the deaths of 20,000 individuals in the US, more than any other infectious disease prior to the COVID-19 pandemic [5].
Disease complications are, indeed, fatal, but only a small subset of individuals infected with HCV will develop endstage liver disease. After HCV enters the blood, 15% to 45% of patients clear the virus spontaneously within six months [6,7]. For those in whom HCV RNA persists in their blood for more than six months, the infection will progress into a chronic condition. The infection can be determined by screening individuals using a simple and inexpensive antibody test [8]. Although hidden inflammation progresses in the liver for those infected, most individuals are unaware of their chronic condition [9]. Inflammation progression is commonly estimated by the fibrosis progression rate of the liver and is expressed by a five-degree classification system ranging from stage F0 (no fibrosis) to cirrhosis at stage F4 [10]. Advanced checkup tests can be used to determine the fibrosis stage, but are more expensive than just determining infection [11]. A chronically infected individual may carry the virus for more than 30 years before developing any clinical symptoms; thus, only 10-20% of infections progress to end-stage liver disease [12].
A sharp decline in HCV infection has recently been observed due to advances in hygiene practices which reduced transmission through direct blood-to-blood contact as well as due to recent breakthroughs in HCV treatments. Thus, with the exception of injecting drug users that are at elevated risk to contract HCV due to the sharing of contaminated injecting equipment [13][14][15], the prevalence of HCV in the general population gradually declines. These treatments, which have been approved by the US FDA [16] and the European Medicines Agency since 2014, show low side effects and excellent efficacy in clearing the virus and halting liver deterioration. Although clinical trials have demonstrated high efficacy of the treatments, the efficacy declines with disease progression. For example, Sofosbuvir showed efficacy of 0.98, 0.98, 0.92, and 0.79 in stages F0, F1, F2, and F3, respectively [17]. Due to the dramatic decline in HCV disease, the World Health Assembly endorsed a Global Health Sector Strategy in 2016, calling for the elimination of HCV as a public health threat by 2030 by improving screening and treatment policies [18,19].
The main barrier to prompt elimination of HCV is the high price of treatment, which is unaffordable globally [20]. Pharmaceutical companies have developed and adapted differential pricing strategies to increase revenue in each country [21]. For example, while the US government paid $84,000 per course of Sofosbuvir, the price in Spain was as low as $25,000 [22]. Due to the high price of treatment, numerous studies have evaluated the cost-effectiveness of different screening policies that are specific to each country and to the individual's stage of infection [17,[23][24][25][26][27][28][29]. These health economic studies aim to balance the declining efficacy of treatment with disease progression and the understanding that for most people, natural death will occur before reaching HCV stage F4. However, although these studies underscored that the worldwide policy to combat HCV must be revised to reduce mortality, they did not optimize the frequency of checkups, which are essential to determine the fibrosis stage. Given that a decision to treat is based on the most up-to-date checkup test result regarding the fibrosis stage, rather than the actual stage itself, optimizing the frequency of checkups should be explicitly integrated into the cost-effectiveness evaluations.
The high proportion of undiagnosed individuals, the slow progression of the disease and the technological ability to diagnose and identify the HCV fibrosis stage provide a rich and broad level of strategies for policy makers to reduce the burden of HCV. These opportunities result in nontrivial pricing mechanisms for HCV treatment. Recently, various PCs have begun to provide discounts after a predetermined volume of sales. Such approaches are now widely used in Australia, France, Canada, and New Zealand [30,31]. In some countries, PCs share the risk of treatment outcome and reimburse the country if treatment is unsuccessful [32][33][34], while in other countries, they fund screening to identify more HCV-infected individuals [35,36]. Given the high burden of HCV, these strategies might be perceived as ethical and vital, as they benefit not only the PC but also society at large.
Game theory provides an analytical framework to analyze the outcome of the conflict between rational decisionmakers [37]. Several recent studies integrated game models with disease transmission models to explore the dynamics between manufacturers and decision-makers [38][39][40][41][42][43][44][45][46]. Chick et al. [41] explored the procurement of influenza vaccines in settings where production yield is uncertain and an information asymmetry exists between the manufacturer and policy makers. Özaltin et al. [40] optimized the design of a flu shot by accounting for its effect on the societal benefit by determining coverage and time availability. Using a sequential game model, Chick et al. [47] showed that production risks taken by the influenza vaccine manufacturer lead to an insufficient supply of the vaccine. The authors suggested that a global social optimum cannot be fully attained by changing the vaccine price and demonstrated that a variant of the cost-sharing contract can align incentives to achieve a social optimum. The authors further showed that due to the nonlinear health benefit of influenza vaccination, nonlinear pricing can serve as a valuable coordination mechanism. Only two studies have considered a game model for HCV policy determination. The first focuses on a unique setting of a compulsory license agreement in developing countries [38], while the latter considers different treatment cost functions to optimize the budget of a social planner in a theoretical game [48].
Therefore, we developed a game-theoretic model to assess the price of HCV treatment in consideration of the corresponding screening, checkup and treatment strategies. We employed an analytical framework calibrated to detailed HCV data from Israel to calculate the utility of the players for each strategy chosen. This study is the first to optimize treatment policy and determine comprehensive guidelines to combat HCV by considering simultaneously both screening campaigns to identify individuals with HCV infection and checkup rates for those infected. Our model provides a rational explanation for the current strategies offered by PCs to fund screening or to provide a substantial discount after a predetermined volume of sales, which results in a suboptimal outcome for the country. In addition, we present applicable mechanisms for the country to mitigate this outcome.

Game model
We consider a game model composed of two players: a Pharmaceutical Company (PC) striving to maximize its revenue, and a country 1 endeavoring to maximize the social welfare of its citizens. The sequence of the game starts with the PC announcing a fixed price of treatment, p ≥ 0 , which represents the price of a full-course treatment for an individual infected with HCV. Then, the country determines which subpopulations will receive the treatment based on certain criteria. As common in worldwide guidelines [49,50], these criteria are based on the individual's age 2 and the stage of fibrosis (i.e., disease progression). The actual stage of fibrosis at each time point is unknown but can be frequently determined by checkup tests. Thus, for each individual, we distinguish between their actual fibrosis stage (denoted by f ) and their presumed stage (namely, their last diagnosis, denoted by d ), which is based on the most recent checkup test results. We define y = (y 0 , y 1 , y 2 , y 3 ) as the vector of checkup rates determined by the country, such that y d ≥ 0 represents the checkup rate for individuals diagnosed with fibrosis stage d. Similarly, we define x = (x 0 , x 1 , x 2 , x 3 , x 4 ) as a binary vector describing the country's decision of whether to treat an individual at presumed fibrosis stage d. Thus, the PC's strategy is to determine the price of treatment p , and the country's strategy is to determine both the treatment criteria x and the checkup rates y 3 . The PC's expected present-value utility is given by: where N T is the expected discounted 4 number of treated individuals.
In line with standard cost-effectiveness studies [51], the expected utility for the country comprises the overall health benefits to the population translated into financial outcomes, reduced by the overall direct and indirect costs of the disease. The health benefits comprise the overall life-years gained by the population, translated into economic costs. c H represents the economic losses attributable to HCV that arise upon reaching an end-stage liver complication. These losses may include hospitalizations and liver transplantations. Additionally, we consider the costs of checkups, c C , and treatment, p . Thus, the expected utility for the country is given by: where v is the financial value equivalent to one healthy year, multiplied by the total discounted quality-adjusted life-years earned by the entire population, 5 QALY . N H (x, y) is the expected discounted number of individuals hospitalized due to HCV complications and N C (x, y) is the expected discounted number of individuals tested to determine their fibrosis stage (an analytical formulation and solution for these values based on our HCV dynamic disease transmission model is presented in Appendix 5).
Proof As previously proven by Selten, at least one subgame perfect Nash equilibrium (p * , (x(p), y(p))) exists for any finite game [52], and can be determined through backward induction, (see also, for example, [37,53,54]). For each treatment price, p , set by the PC, the country decides on its best-response policy, defined by the vectors x and y . The optimal policies were those that maximized the country's utility in response to each price. Since the PC acts first, the equilibrium is determined by the price of treatment that maximizes the PC's utility, considering the country's bestresponse policy.■

General population
In this section, we employed a dynamic model to describe the progression of HCV over time in the general population, i.e., disregarding injecting drug users (IDU) (Fig. 1). At each time t , individuals are classified into states, the model is stratified into 15 states, and a transition from one state to another occurs as a result of natural disease deterioration or a change observed following a checkup test (Fig. 1). Individuals are born at rate (t) and can be either susceptible to HCV or infected with HCV due to mother-tochild transmission at rate 7 [56]. We denote the susceptible population -individuals who might become infected in the future with HCV at time t as S(t) . We also denote the overall population at time t as N(t) , and the population infected with HCV is ∑ i∈f ,f ≥d ∑ j∈d I i,j (t) , that is, the sum of all individuals in these 15 states. Newly-infected individuals enter state I f =0,d=−1 (t) , where the disease is in its earliest stage of fibrosis and individuals are unaware of their infection. Individuals transition out of this state due to deterioration of their fibrosis stage, HCV diagnosis from general screening, or natural death at rates f , and , respectively. Accordingly, the changes in the number of undiagnosed infected individuals in a short time interval Δt is: The virus is transmitted as a result of contaminated blood via medical procedures at rate . Thus, the rate at which individuals are infected with HCV at time t , that is the force of infection, is given by: The country determines whether to treat infected individuals based on their presumed fibrosis stage. Equation 2.4 describes all the states in which the country is aware of the actual fibrosis stage of the patient, i.e. when d = f . Individuals enter these states only when their fibrosis stage is not treated per the country's policy, at rates y i , i ∈ {−1, 0, 1, ..d − 1} , the checkup rate at each diagnosed state i . For consistency, we define y −1 as the common HCV diagnosis rate [57]. Transition from these states to others at rates f and results from progression to the next stage of disease or from non-HCV death, respectively.
, ∀d = f stage of the disease, f − 1 . Transition from these states to others at rates f , y d and results from progression to the next stage of the disease, checkup, or non-HCV death, respectively.
The treated population is divided into two groups: susceptible following treatment ( ST ) representing individuals for whom treatment was effective, and infected following treatment ( IT ), representing individuals for whom treatment was not effective or individuals that were effectively treated but got re-infected. We assume that when the treatment policy starts at time t = t c , individuals who follow the criteria immediately become treated. From time t > t c , those who meet the criteria transition to the treated states at the country's chosen checkup rate. Let f represent the efficacy of the treatment when provided at stage f . An individual in stage f can be diagnosed at rates y i , i ∈ {−1,0, 1, ..., f − 1} ; then, when the fibrosis stage is treated by policy ( x f = 1 ), the individual transitions to ST f with probability f , meaning the treatment was efficient. Alternatively, the individual transitions to IT f if the treatment was inefficient. Conservatively and consistent with observations from clinical trials, retreatment following ineffective treatment is not considered. Transition from state ST f can occur at rates and because of HCV reinfection due to contaminated blood in a medical procedure or non-HCV death, respectively. We track the last fibrosis stage f , even if treatment was successful as the liver condition remains unchanged due to the very slow regression of fibrosis [58]. Transition from state IT f can occur at rates f and due to progression to the next stage of the disease and non-HCV death, respectively.
where f >0 is an indicator function taking a value one for all f >0 and zero otherwise. The initial conditions are given by l f ,d , representing the initial number of infected individuals at time t = t − c , that is, just before the treatment policy is applied in each state. Assuming that individuals are not yet treated at time t = t − c , the number of individuals at each state can be represented by: Overall, the general population model with treatment includes 25 transient states ( Fig. 1): 15 states describe HCV progression without treatment, five states describe HCV progression following an ineffective treatment, and five states describe successful treatment. Additionally, the model includes four absorbing states ( m {1, 2, 3, 4} ) to explicitly distinguish between 1) non-HCV-related death in the nontreated population, 2) HCV-related death in the nontreated population, 3) non-HCV-related death in the treated population, and 4) HCV-related death in the treated population, where 4 is the mortality rate from HCV.

Injecting drug users
We extend the model to explicitly include transmission dynamics among injecting drug users (IDU) (Fig. 2, detailed presentation of model equations is presented in Appendix 2).
In accordance with previous models [59][60][61][62], we stratify the IDU population into three mutually exclusive groups based on IDU status: 1) active IDU, 2) IDU in harm reduction (HR), and, 3) former-IDU. Active IDU refers to all individuals currently involved in intravenous drug use who are not enrolled in HR programs. When an IDU joins an HR program (for example, needle exchange and opioid replacement therapies [63]), the individual transitions to the IDU in HR group. Once the IDU completely ceases drug injection, the individual is classified as a former-IDU. Individuals transition between the IDU groups by means of drug rehabilitation or drug readdiction, which may occur with or without HR. Thus, with the exception of a direct transition from former-IDU to IDU in HR, individuals may transition between all three groups multiple times throughout their lifetime (Fig. 2).
In accordance with the basic model, newly-active IDUs enter the active IDU group at rate i,s , where they can either be susceptible to HCV S Active , infected with HCV I Active , susceptible following treatment ST Active , or infected following treatment IT Active . For those infected with HCV, the disease progresses as described in Sect. 2.2.1; thus, the I Active subgroup includes 15 transient states that explicitly track the ceptible individuals can become infected by contaminated blood via shared-drug use at rate ξ or via medical procedures at rate ρ . All subgroups can transition to the absorbing state (m = 5), representing non-HCV death. States written with a double-compound orange arrow on the top right can transition to the absorbing state (m = 6), representing HCV death. The arrows from each state depict transitions to other states actual fibrosis stage (denoted by f ) and presumed stage (denoted by d ), i.e., I Active f ,d . Similar to the general model, as treatment will not be offered again after reinfection, the ST Active and IT Active subgroups include five states each to explicitly track the five possible stages of fibrosis, i.e., IT Active f and ST Active f . As previously suggested [59,64], and due to the considerable risk of reinfection [65], treatment is not offered to active IDUs. Nonetheless, individuals in the active IDU group may have already received treatment while in the IDU in HR group or the former-IDU group.
Upon joining HR programs, IDUs transition from the active IDU group into the equivalent state of the IDU in HR group (e.g., ). Based on the country's decision of whether to treat an individual at presumed fibrosis stage d, eligible individuals transition to the matching state based on the outcome of treatment. Specifically, if the treatment is effective, individuals transition to a susceptible following treatment state ST . Recall that although the virus is cleared, the liver condition remains unchanged due to the very slow regression of fibrosis [58]; thus, individuals transition with an identical fibrosis stage (e.g., Likewise, if the treatment is ineffective, individuals transition to an infected following treatment state IT with an identical fibrosis stage (e.g., As reinfection is possible upon exposure to contaminated blood, susceptible individuals who were previously treated may transition to the infected IT state (i.e., ST HR f → IT HR f ). Transitioning from the former-IDU group follows the same logic as that for the HR group with one exception. As former-IDUs do not inject drugs, they may transition to the infected groups ( I Former and IT Former ) only due to contaminated blood transfusion via a medical procedure. As long as former-IDUs do not become re-addicted and transition to the active IDU group, they do not contribute to the transmission caused by the shared-needle exchange.
In addition to blood transfusion due to medical procedures, IDUs may become infected with HCV via contaminated needles. Inspired by Kaplan's notion of needle circulation from the moment of its introduction into a population of IDUs [66,67], we developed a nonlinear framework to model the force of infection due to needle exchange (Appendix 3). This framework explicitly evaluates the individual's risk of becoming infected with HCV as a function of the number of IDUs that previously used the needle. This needle-sharing procedure reveals a strong concave effect on the transmission following treatment (Appendix Fig. 8). The force of infection u (t) in IDUs at group u is: where (t) is the rate of infection due to needle-sharing and is the rate due to contaminated blood via medical procedures.

Model extensions
In this section, we present several practical extensions. For each extension, we first present the problem description, and briefly describe the modifications from the basic model and the evaluations of the results at equilibrium. For reproducibility, the detailed model, the analytical solutions for each extension, and the code for the simulations are available on GitHub [68].

Extension 1 -One-time screening campaign funded by the country
We examine the effect of population screening that is paid by the country on the game's outcomes. In this extension, the PC determines the treatment price. Afterward, the country simultaneously determines a one-time screening policy, relating to the number of individuals screened at time t = t c , as well as the check-up and treatment policy. This modification affects: 1) the initial number of individuals in each state, as screened individuals may transition to one of the diagnosed states, and 2) the country's utility function to account for the screening costs paid by the country. Given that a diagnosed individual is not screened again, we evaluate the expected number of identified HCV-infected individuals E(X) using a hypergeometric process, X ∼ HG (M, D, s) . A "success" indicates a newly diagnosed individual upon screening of s individuals in the undiagnosed population ( M ) containing D HCV-infected individuals. The campaign leads to an expected increase of s ⋅ D M individuals in the diagnosed population, according to the proportion of each fibrosis stage. The screening cost, C(s) , is a function of s , the number of individuals tested for HCV. In practice, as s increases, it becomes more difficult to locate and convince an additional individual to undergo screening. Following the same logic, locating the first individual is relatively easy, and the cost is based primarily on testing. Thus, we demand where c E is the fixed cost for a single HCV examination (i.e., screening).

Extension 2 -Pricing mechanisms
In this extension, we independently examine three pricing mechanisms recently observed in several developed countries. The first involves PC funding of screening, i.e., we subtract C(s) from the utility function of the PC instead of the country's. The second considers the PC sharing the financial risk associated with the treatment and charges only if the treatment is successful. Finally, we consider a nonlinear pricing strategy in which the PC implements a volumebased discount pricing mechanism. In this case, the game begins with the PC announcing simultaneously the treatment cost p and a threshold z , where all treatments up to z are provided at the corresponding price p and all treatments beyond z are provided to the country free of charge. Afterward, the country determines simultaneously a treatment, checkup and a one-time screening policy at time t = t c . This modification affects the treatment cost component in both players' utility functions, where the number of treatments that the country finances is the minimum between the actual number of treated individuals and the threshold defined by the PC.

Dataset and simulation results
We performed simulation studies parametrized based on data describing the epidemiology of HCV-infected patients in Israel to calculate the utility of the players for each strategy chosen. Israel is a good candidate for exploring the mechanism between a PC and a country. The country maintains a centralized policy, in which the Israeli Health Committee negotiates the price of treatment and covers the treatment for its citizens. Approximately 100,000 individuals (~ 1.25% of the population) are infected with HCV, and 70% are unaware of their infection [57]. It is estimated that 0.5% of the population aged 15-64 in Israel inject drugs, of which 67% are infected with HCV [69].
We conducted simulation studies using the full transmission models, including additional stratification of individuals based on gender and age group (Appendix 1). The IDU population was also stratified based on IDU status (Appendix 2), with an initial HR coverage of 40% [60,70]. The model includes epidemiological, immunological, sociodemographic, as well as economic and operational parameters, which were estimated based on the relevant literature (Table 1). Missing epidemiological parameters were evaluated by calibrating the model to data from a previous study on HCV epidemiology in Israel in 2012 [57]. Note that the disease is asymptomatic until its fatal final stage. Therefore, we assumed no difference between a healthy individual and an infected individual in terms of the quality of life-years.

Model parameters
The population was stratified into 16 age groups: 0-4 years The financial value of one life year (QALY) $40, 000 − $120, 000 Corresponds to 1-3 GDP per capita in Israel r Annual discount rate 0.03 considered injection behavior may initiate from the age of 15, in accordance with [69,73,74]; thus, the IDU population consists of the last 13 groups. The annual natural mortality rates for each age group i and gender s were parametrized based on the Israeli Central Bureau of Statistics (ICBS). In addition, we also assessed these mortality rates according to each of the u IDU types-u i,s [59]. Disease progression rates f ,i,s for disease stage f , age group i , and gender s were extracted from large retrospective cohort studies [71]. The efficacy of treatment varied widely with the stage of infection and was determined based on previous clinical trials [17]. These estimates suggest that treating at fibrosis stage 0 is very likely to be effective, with 98% efficacy, while treatment efficacy drops to 79% at fibrosis stage 3. We assume injection drug use does not affect adherence and response to treatment; thus, we assume similar treatment efficacy between IDUs and non-IDUs [75,76].
Most HCV-infected individuals in Israel are unaware of their infection. To determine the proportion of individuals with reported HCV, we used a previous study that randomly tested individuals for HCV in Maccabi Healthcare Services, a 2-million-member health maintenance organization in Israel [57]. Based on their estimates and another national report [72], approximately 30% of the population are aware of their infection. The spontaneous annual screening rate before a designated intervention policy, , was estimated based on a study that concluded that a total of 6,150 patients were identified with newly diagnosed HCV between 2003-2012, out of approximately 100,000 individuals infected with HCV, resulting in a rate of 0.006 [57]. These estimates are similar to trends observed in the US (Kershenobich et al. [77], which projected a decrease in the proportion of HCV that is undiagnosed from approximately 50% in 2009 to 80% in 2021).
Cost-effectiveness analysis (CEA) of medical intervention considers the balance between the incremental cost of the intervention and the incremental health benefits attributable to the intervention. In our model, the financial value of QALY, v , was based on the terminology suggested by the WHO [51]. Their criteria define "cost-effective" as lower than three times the annual per capita gross domestic product (GDP) and "very cost-effective" as lower than the GDP. Accordingly, in the simulations, we considered v = $40,000 and $120,000, corresponding to Israel's GDP per capita and three times its GDP per capita, respectively. All costs and effects were discounted by an annual factor of 3%, similar to previous CEA studies in Israel [78].
For screening costs, we considered the function where c E is the fixed cost for a single HCV examination (i.e., screening) and a is a parameter that reflects the change in the willingness of the population to be screened ( 0 ≤ a ≤ 1 ). This function satisfies the conditions presented in 3.1 and is based on the same rationale as the isoelastic function, which is used in the context of standard economics [79]. Note that if a = 0 , C(s) = c E ⋅ s , leading to a linear cost of screening.

Model calibration
The only parameter calibrated in the model is the blood transfusion rate . Our calibration process is based on detailed data on HCV prevalence in Israel from 2012 stratified by age and gender. To estimate this missing parameter, we ran simulations of the full transmission model (detailed in Appendices 1 and 2) from 1950 (a few years after the establishment of the state of Israel) to 2012, with an initial condition of 1,100 patients, similar to the assumptions of a previous model [80]. We calibrated the blood transfusion infection rate that minimized the mean squared error (MSE), which is equivalent to the maximum likelihood estimator (MLE) assuming that the error is normally distributed with a mean of zero (Figs. 3A and B). The number of infected individuals in each fibrosis stage obtained by our calibrated model was also validated using partial data from the Polaris observatory website [81] and a previous study in Israel [82]. This analysis reveals that advanced fibrosis stage prevalence increases with age, in line with [81] (Appendix 4).

Simulation studies
We conducted simulations to evaluate the players' strategies and utilities at equilibrium for the basic model and the two extensions (all decision variables are detailed in Table 2). Due to transmission within the IDU population, elimination after mortality (related or unrelated to HCV) is not necessarily within reach. Thus, in typical health economics, the effectiveness of a possible policy is analyzed based on a fixed cohort in a finite horizon [83]. We consider the cohort as individuals that were part of the population during the course of 50 years. We calculated the utilities for the country and the PC (see Appendix 5) for all individuals in the cohort throughout their entire lifetime. Namely, no new births or new HCV cases were considered 50 years into the simulation, and henceforth we analytically calculate the utilities until the cohort elimination.
We calculated the country's gain by comparing each treatment policy at time t = t c to a baseline policy, in which no treatment is considered. Thus, we defined the country's gain as the difference between the country's utility and the baseline utility. Using grid search for each set of decision variables (Table 2), the optimal policy chosen by the country for each treatment cost and quantity discount threshold is the one that maximizes its gain. Then, using backward induction, given each set of the country's decision variables for a specific treatment price, we calculated the PC's utility function. For the basic model and all the extensions, we analyzed the PC's utility and country gain for each treatment price. The equilibrium point is the point that maximizes the PC's utility, given the country's optimal decision. From a practical standpoint, if there is more than one equilibrium, we choose the equilibrium with the lowest treatment price.

Basic model
In these settings, the PC chooses the fixed treatment price per course. Afterward, the country determines its optimal policy for treatment in terms of fibrosis stage, age and checkup rate per diagnosed disease stage. Since the treatment coverage policy selects groups (by stage of disease and age group) that vary in size, the change in the utility of the PC is not linear (Fig. 4A).
We found that compared to no treatment, treatment is highly beneficial to PC and the country (Fig. 4). Assuming a willingness to pay of v=$40,000 per QALY gained, which corresponds to 1 GDP per capita, the price of treatment is expected to be $22,000 ( Fig. 4A and B). The country's best response is to provide treatment to individuals above the age of 15 that are diagnosed with HCV at fibrosis stages

Extension 1 -One-time screening funded by the country
In this extension, the PC chooses the fixed treatment price per course. Afterward, the country determines simultaneously the number of screened individuals in a one-time

Extension 2 -Pricing mechanisms
In this extension, we examine three pricing mechanisms observed in several developed countries: One-time screening funded by the PC If the PC funds the screening, the PC achieves a superior outcome compared to a scenario in which the country funds the screening. This superior outcome is achieved because in equilibrium the treatment price rises to $22,200 (instead of $10,600), and as the number of treated individuals increases compared to a scenario where the country funds the screening. Counterintuitively, this case leads to a suboptimal outcome for the country as it forces the country to conduct more screening than it would have desired, lowering its gain compared to the latter. (Fig. 6).

Performance-based mechanism (Risk-sharing)
In this case, the PC reimburses the country when treatment is unsuccessful. In other words, the country pays only for year v set to set to 0 and $40,000, respectively. The PC increases its utility by funding screening, thereby forcing the country to conduct more screening than it would have desired, lowering the country's gain. Alternatively, performance-based agreements provide the country with additional flexibility to postpone treatment as treatment efficacy decreases with disease progression. They lead the PC to lower the price to increase the country's motivation for screening and early treatment successful treatments. We found that this kind of mechanism is beneficial for the country. As treatment efficacy decreases with disease progression, performance-based agreements provide the country with additional flexibility to postpone treatment. In turn, it leads the PC to slightly lower the price ($10,400 instead of $10,600) to encourage the country to increase screenings and treat early (Fig. 6).
Quantity discount In this case, the PC chooses the fixed treatment price per course and applies a discount-forquantity policy, allowing the country free treatment for all patients above a certain threshold. Afterward, the country decides its optimal policy for screening and treatment. When the price of treatment is low, the PC chooses a very high threshold value to avoid loss. As the price of treatment increases, the PC lowers the threshold, incentivizing the country to increase the volume of treatments. The points at which the PC reduces its threshold value are the points at which the dashed purple line breaks (Fig. 7). At equilibrium, the PC sets a threshold of 15,000 individuals and a treatment price of $47,200. In contrast to the basic case and the first extension, where increasing the price of treatment causes the country to treat fewer patients, in this extension, the country's policy The PC increases its utility by applying a discount-for-quantity pricing mechanism. The utility of the country decreases even compared to that of the basic model remains consistent: treat the entire diagnosed population. Since the country treats every diagnosed patient, no checkups are done. Our simulations show, for each of the values of a , the country's policy remains constant. Thus, increasing the treatment price affects only this parameter in the country's utility function; therefore, a symmetrical graph is obtained between the PC and the country's utility graphs ( Fig. 7A and B).
The equilibrium point of the discount-for-quantity pricing mechanism achieves a suboptimal outcome for the country in terms of overall gain and a superior outcome for the PC in terms of utility compared to those achieved under a fixed pricing mechanism. This outcome is even less favorable than not conducting any screening at all (as shown in the basic model). Although there are treatment costs that can improve outcomes for both the PC and the country, they do not belong to the subgame perfect Nash equilibrium.

Discussion
We developed a game-theoretic model between a PC and a country to determine the optimal price of HCV treatment and the corresponding screening, checkup, and treatment strategies. Calibrated with detailed data from Israel, we found that with the standard fixed treatment pricing, applying a screening policy is beneficial to both the PC and the country. However, if the PC offers discounts for quantities or funding for screening, it will result in a suboptimal outcome for the country. Our simulations suggest that at equilibrium, a quantity discount or funding for screening force the country, either indirectly or directly, to conduct more screening and treat more individuals than it would have desired otherwise, lowering the country's gain.
By contrast, the country will gain from a mechanism in which the PC is required to cover the cost of unsuccessful treatments. One would expect that in such a mechanism, the PC would simply increase the price to balance their additional loss due to the inefficiency of the treatment. However, our analysis indicated the opposite: as the efficacy of the treatment declines with disease progression, covering the price of unsuccessful treatment by the PC drives the country to postpone treatments. To avoid a potential postponement, the PC reduces the price of treatment in order to increase the country's motivation for screening and early treatment. Our work provides a rational explanation for the current agreements between countries and PCs [32][33][34]84], including the PC's ethically questionable funding of screening programs [35,36].
These key findings were strengthened when we extended the model to account for IDUs explicitly. The logic behind our findings is that, due to the strong concave effect of transmission through needle exchange (Appendix Fig. 8), the contribution of treating an IDU to decrease transmission is only marginal. Namely, to substantially reduce transmission due to needle exchange, a drastic behavioral change is required. These findings are in line with the results of several recent studies [85][86][87]. Moreover, IDUs who are treated and successfully clear the virus might return to needle-sharing behavior and are at a high risk of becoming reinfected with HCV. Finally, the life expectancy of IDUs is shorter than the non-IDUs: they are more likely to die from other causes unrelated to HCV due to their high-risk lifestyle. Thus, from an economic perspective, treating IDUs is less beneficial because fewer QALYs are saved. Consequently, the lower economic benefit reduces a country's willingness to pay for treatment, which further enhances the gap between the interests of the country and the PC.
Our simulations further indicated that even if the entire IDUs population is screened for HCV, and all eligible individuals (i.e., IDUs in HR or former IDUs) are treated, HCV will not be eliminated. Thus, we believe that treatment alone is not likely to reach the WHO targets and achieve viral elimination [18]. Multiple interventions, including treatment, scaled-up HR, and needle exchange programs that are jointly conducted, should be examined. The effectiveness and cost-effectiveness of such strategies can be more accurately evaluated using network-based models (See, for example, [88,89]). HCV prevalence among IDUs. The needle-sharing procedure reveals a strong concave effect for IDU transmission. The contribution of treating an IDU to decrease transmission is only marginal due to the nonlinearity of transmission through needle exchange Our model has several potential limitations. First, the findings obtained from the simulations are based on data that are specific to Israel and thus might not apply to other countries. Specifically, disease prevalence varies widely in different regions of the world [90,91], as does the proportion of HCV-infected individuals who are undiagnosed. In addition, there is considerable uncertainty regarding IDU risk behavior, in particular, the risk of becoming reinfected for those who joined an HR program. Nevertheless, given the wide range of parameter values considered, we expect that the observed trends are broadly the same for most developed countries. From a broader perspective, our model provides a general framework that can be applied to many other healthcare settings where screening is essential to determine treatment strategies. Second, our model suggests favoring treatment for younger age groups. The economic justification for that observation is clear. Younger individuals have more years to lose and have a higher probability of reaching end-stage cirrhosis later in their lives. This result might raise an ethical question of agebased discrimination. While our model provides insights from a health economics perspective, public health policy makers may consider a wide range of factors when developing policies.
In addition, our model does not provide for screening focused sub-population groups other than IDUs. Clearly, the number of HCV patients among IDU is disproportionate, and thus, we tested screening policies that are based on IDU group. However, as results show, as treating IDUs is less beneficial for the country because fewer QALYs are saved, such focused screenings would only strengthen our key findings. Additional stratification for scanning is possible (e.g., based on ethnicity or age groups), but may raise additional ethical questions. Nevertheless, results show that the country achieves an inferior outcome even when it does not fund screening, and thus, although such policies would make the screening process more effective, we believe this will not change our main findings.
Our model does not consider future improvements in technologies that are likely to occur with time. A previous model highlighted that such improvements in technology can affect medical decision-making [92]. We expect that any improvement in technology would lower the checkup cost, providing the country with additional flexibility to postpone treatment, thereby intensifying our main results. In this context, it should be noted that the game model between the country and PC examined in this study does not include the management of negotiations between the parties nor any risk-sharing approaches examined in other studies [93,94].
In conclusion, HCV is one of the leading causes of liver disease, causing massive public health and economic burden worldwide. In the US, it is estimated that 3.5 million people are chronically infected with HCV and that each year the disease is responsible for the deaths of 20,000 individuals -more than any other infectious disease before the COVID-19 pandemic [5]. Thus, the PC and the country appear to share a similar interest -treating as many HCV-infected individuals as possible. Moreover, as observed in several countries, allowing the PC to cover the cost of screening to identify more HCV-infected individuals or subsidizing treatment via discounts for quantities appears to benefit the public good. However, our work shows that the opposite is the case, namely, pharmaceutical companies, and not society at large, benefit, as they indirectly impose on the country to conduct more screening and treat more individuals than would be optimal for society. Counterintuitively, the results from our game model emphasize that allowing the PC to cover the cost of screening to identify infected patients might not be ethical and should be carefully evaluated by policy makers.
The logic behind our finding is that the clock is ticking differently for the PC and the country. HCV is a chronic illness that progresses slowly, and most HCV-infected individuals, in particular high-risk groups such as IDUs, will die naturally before reaching end-stage cirrhosis. Thus, given the inferior footing caused by discount-for-quantity treatment pricing, credible strategies that should be considered by the country include delaying screening and requiring the PC to cover the price of unsuccessful treatments. If such strategies are considered, the PC can be expected to respond by lowering the price to encourage the country to conduct screening as soon as possible. Eventually, a new equilibrium point that will mitigate the losses to society will be reached.

Appendix 1 -Transmission model -Non-IDU population
In the full transmission model, we generalize the set of difference equations that describe HCV disease progression, by the stage of the disease (f , d) , age group . We assume that f ≥ d , meaning an individual cannot be diagnosed in a more advanced stage than his actual stage. For the general population, we account for two main transmission routes of HCV: 1) mother-to-infant transmission at birth and 2) contaminated blood dose via medical procedures. We denote S i,s (t) , the number of non-IDU susceptible individuals per age group i and gender s . Entry into this state can result from new births either from an HCVinfected mother that did not transmit the disease or from a non-HCV mother. Another transition into this state can result from growth from the previous age group, i − 1 , at rate i−1 . Transition out of this state can result from growth to the subsequent age group at rate i , disease infection via medical procedures , drug addiction i,s or natural death i,s . Hence, the change with time in the number of susceptible individuals, stratified by age group and gender is: where i,s is the rate of blood donation per age group i and gender s. We assume that all newly infected individuals are not diagnosed and are at fibrosis stage F0; thus, entry to state I 0,−1 can occur due to infection i,s (t) or aging. Transition out of this state can result from growth to the subsequent age group at rate i , next stage of the disease f ,i,s , the discovery of the disease , drug addiction i,s or natural death i,s . As the patient has not been diagnosed at this state, no treatment is provided. The change with time in the number of undiagnosed infected individuals, stratified by age group and gender is: where i=1 is an indicator function taking a value one for the first age group, i = 1 and zero otherwise. We define X as a decision matrix describing the country's decision whether to treat an individual at presumed fibrosis stage d and age group i . Equation (A5) describes all the diagnosed states where d = f . Only when X f ,i = 0 , individuals at fibrosis stage f and age group i are not treated, and can enter these states at rates y i , i ∈ {−1, 0, 1, ..d − 1} as a result of stage diagnosis (checkup). Similar to equation (A4), entry to these states derive from aging. Transition out of these states to others can result from growth to the next age group, next stage of the disease, drug addiction or natural death, respectively.
Equation (A6) describes all other states (where d < f ). For consistency, we marked 4,i,s as the mortality rate from HCV in group i and gender s and y −1 as the common HCV diagnosis rate [57].
The treated population is divided into two groups: susceptible following treatment ( ST f ,i,s ) representing individuals for whom treatment was effective, and infected following treatment ( IT f ,i,s ), representing individuals for whom treatment was not effective or individuals that were effectively treated but got re-infected. Let f represent the efficacy of the treatment at stage f . An individual in stage f can be diagnosed at rates y i , i ∈ {−1,0, 1, ..., f − 1} ; then, when X f ,i = 1 (the fibrosis stage and age group are treated by policy), the individual can transition to ST f ,i,s if treatment is efficient with probability f . Alternatively, the individual transitions to IT f ,i,s , if treatment is inefficient. Transition out of state ST f ,i,s can result from growth to the subsequent age group at rate i , disease reinfection via medical procedures , drug addiction i,s or natural death i,s . Transition out of state IT f ,i,s can result from growth to the subsequent age group at rate i , next stage of the disease f ,i,s , drug addiction i,s or natural death i,s . The changes with time in the number of treated individuals, stratified by age group and gender is: The initial conditions are given by l f ,d,i,s , the initial number of infected individuals in each state, leading to equation (A9): (A5) The rate at which individuals in age group i are infected with HCV at time t , that is the force of infection, is given by: where s is the proportion of infected individuals in gender s that had not been spontaneously cleared from the disease.
and due to the considerable risk of reinfection [65], treatment is not offered to active IDUs. Nonetheless, individuals in the active IDU group may have already received treatment while in the IDU in HR group or the former-IDU group.
Newly-active IDUs enter the active IDU group from the Non-IDU population at rate i,s . IDUs who cease with drug-injection behavior remain in the IDU transmission model and are classified as former-IDUs. Other entries to the susceptible states result from growth from the previous age group, i − 1 at rate i−1 or from a change in injection behavior at rate . Transition out of these states can result from growth to the subsequent age group at rate i , disease infection u i,s , change in injection behavior , or natural death u i,s : where u=IDU is an indicator function taking a value one for all u = IDU and zero otherwise. Recall that 0 = 0 . We assume that all newly infected individuals are not diagnosed and are at fibrosis stage F0; thus, entry to the state I u 0,−1,i,s can occur due to infection at rate u i,s , as a result of contaminated blood either via medical procedures or via shared injections (Appendix 3). Other entries to the infected subgroups derive from initiation of drug-use at rate , aging at rate i−1 and change in injection behavior . Transition out of the infected subgroups to others at rates i , f ,i,s , , and u i,s , can result from growth to the next age group, next stage of the disease, the discovery of the disease, change in injection behavior, or natural death, respectively.
We define X as a decision matrix describing the country's decision whether to treat an individual at presumed Equation (B4) describes all other states (where d < f ). Transition into these states occur at rates , i−1 and f −1,i,s from initiation of drug use, aging, or from disease progression, respectively. Transition out of these states occur at rates i , f ,i,s , y d , and i,s can result from growth to the next age group, next stage of the disease, stage diagnosis (checkup), change in injection behavior, or non-HCV death, respectively. For consistency, we marked 4,i,s as the mortality rate from HCV in group i and gender s and y −1 as the common HCV diagnosis rate [57].
Assuming homogenous mixing, the probability of becoming infected given injecting with a needle ordered Z is: where g is the probability of becoming infected given exposure to a contaminated needle. Let W be a random variable representing the number of times a needle is used (see, for example, Hopkins [98] and Judd et al. [99]). Note that Pr Z (z;t|w) = 1 w ; thus, the rate of becoming infected by using a contaminated needle p(t) is: where m is the rate of shared injections, and s is the probability that the individual will not spontaneously clear the virus. The force of infection u i,s (t) in IDU group u consists of the rate of becoming infected through contaminated blood either by via a contaminated needle or via medical procedures: where u∈{IDU,HR} is an indicator function taking a value one for all u ∈ {IDU, HR} and zero otherwise. Figure 8 9 Appendix 4 -Simulation analysis to estimate initial number of infected individuals We calculate the PC utility using absorbing Markov Chain's expected-time-to-absorption formula [97]. Let an absorbing Markov chain with k states, out of which q are transient and k − q are absorbing states. The transition matrix P can be written as: where Q (x, y) is a qxq matrix, describing the probability of transitioning from some transient state to another in one step, R describes the probability of transitioning from some transient state to some absorbing state, and Ĩ r is a (k − q)x(k − q) identity matrix. ' ∼ ' represents the omission of financial discounting, which is presented next. Multiplying each transition rate by Δt , such that Δt → 0 , and scaling each step of the difference equations to 1 unit defines a probability matrix, in line with the differential equations we described in Sect. 2.2.2, as: According to [97], the expected time spent in each nonabsorbing state, H (x, y) , is given by: In addition, the probability of being absorbed in each absorbing state from any transient state, Ã (x, y) , is given by: To explicitly track treated individuals at their actual time of treatment, we aggregate all the treatment states ( ST f and IT f ) into two absorbing states, resulting in an additional Markov chain consisting of 4 absorbing states: two representing death ( m = 1, 2 ) and two representing treatment. Solving similarly, the expected number of individuals treated against HCV from time t > t c , Ñ T 2 (x, y) , is the dot product of a transposed vector representing the initial number of individuals in each pre-treatment transient state after treatment policy x is initially applied, I T x, t = t c + Δt , and the j th column of the B matrix, which is the equivalent of matrix Ã (x, y) in the other Markov process. This matrix represents the probabilities to transition from each pre-treatment transient state to the absorbing states representing treatment: Q(x, y) = Thus, the total number of individuals treated against HCV over an infinite horizon is composed of both the individuals that meet the treatment criteria when the treatment policy starts at time = t c Ñ T 1 (x, y) , and those discovered from time t > t c Ñ T 2 (x, y) : In order to take into account financial discounting, we add to the transition matrix a dummy absorbing state, named "Capitalization". Dividing each transition rate by the discount value,1 + r , so that in each time step we only transfer the discounted individuals, and the remainder is transferred to the "Capitalization" state at a rate of 1 − 1 1+r . The P and Q matrices will be set accordingly to: and the adjusted expected time spent in each non-absorbing state, H(x, y) , is given by: In addition, the discounted probability of being absorbed in each absorbing state from any transient state, A(x, y) , is given by: Consequently, we can calculate the total discounted number of individuals treated against HCV over an infinite horizon: As the country's utility also relies on the following discounted parameters over an infinite horizon, given policy: 1) N H (x, y) -expected number of individuals hospitalized due to HCV infections, 2) N C (x, y) -the total expected number of times that all individuals were checked to determine their (E5) N T 2 (x, y) = I T x, t = t c + Δt ⋅ B (x, y) * ,ST +B(x, y) * ,IT .
(E6)  where G T t = t − c is a 1x25 vector specifying the number of individuals in each transient state immediately before the treatment policy is initially applied: A(x, y) is a 25x4 matrix specifying the discounted probability of dying from non-HCV and HCVrelated deaths for nontreated and treated individuals, starting from each transient state . H(x, y) is a 25x25 matrix specifying the expected time spent in each transient state adjusted to the present value. Thus, yielding equations E16 and E17.
Considering disease progression in the utilities of both players, the discounted outcome for the PC is: and the utility function of the country,U COUNTRY (x, y) , is given by the following closed-form: A(x, y) * ,m x j I i,j t = t c + I T x, t = t c + Δt ⋅ B(x, y) * ,ST + B(x, y) * ,IT ,

Extended population including injecting drug users
Due to transmission, elimination after mortality (related or unrelated to HCV) is not necessarily within reach. Thus, in typical health economics, the effectiveness of a possible policy is analyzed based on a fixed cohort in a finite horizon [83]. We consider the cohort as all individuals, infected and not infected, that were a part of the population until time t d . Thus, no new cases are considered from this time on, but we continue following all individuals in the cohort throughout their lifetime. Therefore, we can model the process as a Markov model until the elimination of the cohort. Note, as a result of frequent checkups, individuals in the cohort may be treated also after time t d .
Proposition If no new HCV cases occur after time t d , the total discounted number of individuals treated from time t ≥ t c is given by: where u denotes the injection behavior of individuals, i.e., active, IDU in HR, former or non-IDU. I T x, t = t + d is the number of individuals in each transient state at time t d and D(x, y) * ,ST + D(x, y) * ,IT is a closed-form specifying the proportion of all individuals -IDU and non-IDU -who will be treated for HCV, adjusted to time t d . Namely, the solution from time t d is similar to the one presented in E11. Note as