Occasionally, an infectious disease can spread to such an extent, that it significantly affects a vast number of people and indeed the majority of the population. Such a scenario is referred to as an epidemic [1] . It becomes pandemic in nature when it affects multiple countries, perhaps at the global level. The impact of an epidemic can reach the height of devastation, especially if the pathogen also leads to asymptomatic carriers [2, 3] . Obviously, such asymptomatic infections present a stark variance to symptomatic infections -wherein the symptoms of infections are clearly manifest. Henceforth, we will refer to the former as class A infections and the latter as class M infections. An infection with asymptomatic phenotypes presents remarkable consequences. It severely hinders the containment of infection and the management of the infected. These unfortunate effects have been in full display during the recent Covid-19 pandemic [4] as well as the "Spanish influenza" [5] . Revolutionary advances in medicine have been achieved between these two catastrophes separated by a century. And yet, little can evidently be done to contain the spread of such pandemics -when in full rage.
Epidemic spreading has been modelled extensively through SIRD dynamics on networks [6] . It has also been studied independently through game-theoretic approaches, many of which involve networks [7] . Herein, we adopt a game-theoretic approach towards epidemic spreading on heterogeneous networks [6] [7] [8] [9] [10] [11] . One of our prime objectives is to clearly demonstrate the effects arising from differences in asymptomatic and symptomatic phenotypes on epidemic modeling. This extremely important biological aspect has not been accorded sufficient attention in studies on epidemic spreading involv- * soumen@jcbose.ac.in ing SIRD dynamics or game-theoretic approaches on networks.
Generally, to prevent the spread of any transmissible disease, mass administration of vaccines -if available -is the most desirable option. Significant research has been conducted to scrutinize the vaccination process and its effect on the spread of an epidemic [9, 12] . However, in the absence of vaccines, asymptomatically infected individuals (especially in class A diseases), are likely to be unaware of their own condition. They can unwittingly act as spreaders or even as super-spreaders if they do not resort to proper precautionary measures [13] . As wellknown, precautionary measures in the form of frequently washing one's hands, wearing masks, and social distancing are some optimal strategies to limit the spread of many contagious diseases. In more extreme cases, lockdown may be required, which obviously presents significant economic and social costs.
Susceptible-Infected-Recovered-Deceased or SIRD dynamics specifies that every member of the population would be in one of the following four states, at any given instant of time, t. These states are susceptible, S(t), infected, I(t), recovered, R(t), and deceased, D(t) and would also be frequently referred to hereinafter as S, I, R, and D respectively [14] . Irrespective of their state, susceptible individuals and infected yet undetected individuals can choose from either of the following two strategies when facing an epidemic. These individuals can either adopt precautionary measures and act responsibly or they can remain careless and spread the disease. Any proactive person would obviously need to pay a certain "cost" in order to sustain these measures and act responsibly. In addition, mass testing would surely go a long way in identifying infected individuals. Mass testing ratio, τ , denotes the fraction of the entire population, which has been tested for a given disease. Its importance in class A diseases especially can hardly be overemphasised. A prime motivation of this work is to examine the manner in which precautionary measures and mass arXiv:2008.10387v1 [physics.soc-ph] 24 Aug 2020 testing can affect the spread of an epidemic.
In any infection, the infected individuals definitely possess the capacity to be quite contagious during the incubation period. This is true for both class A and class M diseases. Generally, most diseases are neither purely symptomatic nor purely asymptomatic. However, many diseases are predominantly symptomatic or predominantly asymptomatic. In an epidemic session, some infected individuals may show symptoms and others may not. The manifestation of symptoms may depend on the nature and severity of the disease, the immunity of the infected individuals, and sundry other factors. Herein, we distinguish between class M and class A diseases by using the parameter, σ, which is the Symptom Manifestation Ratio. Of all infected individuals, σ is the fraction of infected individuals -who clearly demonstrate the symptoms of a given disease. σ = 0 represents the scenario, where all infected individuals are asymptomatic. On the other hand, σ = 1 signifies that all infected individuals clearly exhibit symptoms of a given disease. Generally, 0 < σ < 1. The precise value of σ for a given disease can be obtained from empirical data.
The dependence of the outcome of games played on heterogeneous structures depends non-trivially on the underlying topology [7, 15] . In our simulations, a heterogeneous population structure has been considered in the form of a Barabási-Albert (BA) network, which is known to emulate many real-world scenarios [16] .
Our model is quite general and addresses both symptomatic (class M) and asymptomatic (class A) diseases. Contagious diseases can spread through several different mechanisms. These could range from being in the mere proximity of an infected individual or coming in contact with contaminated items used or even touched by the individual. It could also spread through the bodily secretions of an infected person, through sexual or other physical contact or through various vectors.
Proactive individuals will choose precautionary measures of their own accord. Examples of such precautionary measures are the maintenance of social distance, wearing masks, and frequently washing hands. These measures would surely lower the rate of the infection spreading. However, adopting these measures comes at the cost of a varying degree of restriction in one's everyday life. On the other hand, a careless individual will not choose adequate protection against infection. Such individuals are not merely a danger to many others but even to themselves. Their non-cooperative and carefree behaviour is likely to increase the rate of spread of infection.
Thus, we observe that broadly two strategies existcooperation and defection [17] . Proactive individuals are cooperators, C, who act towards preventing the spread of infection by taking proper precaution. However, this comes at a cost, c. If an individual's neighbor happens to be a cooperator, it automatically obtains a benefit, b. On the other hand, careless individuals can be considered as defectors, D. They will not spend any cost and therefore, for such defectors, c = 0. In spite of their callous behaviour -these defectors still enjoy benefit, b, due to the accommodating behaviour of cooperators in the population. Since this benefit comes at no cost, these defectors effectively act like free-riders in a population. When two cooperators, C, interact -they will be mutually benefited by each other's proactive actions. However, this benefit has been arrived at by expending a cost, c. Therefore, the actual reward enjoyed by these cooperators is (b − c).
The interaction between two defectors, D, will not lead to any benefit or cost, for either of them. Thus, this will result in zero payoffs. For the interaction between a cooperator and a defector; D will obtain temptation amounting to b, while C will derive the sucker's payoff equaling −c. True to their nature, defectors would try to avoid paying any cost. Table I displays the payoff matrix incorporating all the above interactions modeled on the Prisoner's Dilemma.
Asymptomatic individuals in class A diseases are rather dangerous because they do not display clear symptoms of infection and yet unwittingly act as spreaders or even super-spreaders. Perhaps the only way to identify such infected individuals is through mass testing. Mass testing, τ , allows the identification of both symptomatic and asymptomatic infected individuals -even the mild ones. The infected individuals who have been detected by mass testing, are kept in quarantine and denoted by I Q . Individuals in I Q do not take part in the game anymore. For simplicity and without the loss of generalisation, we presume here that they can not harm members of the susceptible population anymore. They can either recover, which depends on recovery rate, ρ, or die depending on the death rate, δ. On the other hand, individuals who are infected and yet remain undetected, I U , cause further infection in the population. I U can be of two types, I C and I D . Without medical intervention, I U individuals can also recover at a "self-recovery rate", ρ s . Recall that such self-recovery has been frequently observed during COVID-19. All the states and the possible strategies associated with each state is listed in Table II .
We consider that the rate of infection spread will depend on the strategies of individuals. In a population, S and I U individuals are likely to interact with each other causing the spread of infection. Both S and I U individuals can adopt either cooperation or defection as their strategy. S C and I C denote susceptible cooperators and infected cooperators respectively. S C and I C are both proactive and resort to appropriate precaution while interacting with others. On the other hand, S D and I D denote susceptible defectors and infected defectors respectively. Cost, c, reflects the extent of precaution undertaken by an individual while the benefit obtained from another proactive individual's precautionary actions is reflected in b, as aforementioned. c/b indicates the extent of precaution undertaken by an individual, where obviously b > c. We can also define risk as r = (1 − c b ). C will resort to precaution at some cost to lower the risk. However, c = 0 for D, implying r = 1, i.e., high risk. Let r i and r j denote the risk related to two interacting individuals i and j. If one of these individuals is infected and other one is susceptible, the probability of the susceptible individual getting infected due to the interaction is µ = (r i × r j ). This indicates that µ of S C due to interaction with
Expectedly this is low because both of them have resorted to adequate precaution. However, the probability that an S D individual gets infected due to an interaction with an I D individual is µ D,D = 1. Again, this is expectedly high because none of the individuals have resorted to undertake proper precaution. Similarly We have already discussed in detail about the importance of Symptom Manifestation Rate, σ, in Section I. For class M diseases, we have considered σ = 0.9, which is rather high. On the other hand, σ = 0.02, has been considered for class A diseases.
In our simulations, a heterogeneous population structure has been considered in the form of a Barabási-Albert (BA) network, with an average degree k . It is wellknown that BA networks possess a power-law degree distribution [16] . Each time step during both transient time and counting time sequentially incorporates the events of: (a) payoff determination, (b) strategy upgradation, and (c) state upgradation. Initially every individual in the population is susceptible. At the start of an epidemic session, some of these individuals get infected randomly. Thus, we have two states now -susceptible and infected. The population has been equally divided into cooperators, C, and defectors, D. Initially all infected individuals remain undetected i.e., I U = 0. Thus, on the basis of both strategy and state, there exist four types of individuals, namely, S C , S D , I C , and I D . Initially, prisoner's dilemma is played between C and D, irrespective of the state they belong to. They will accumulate their payoffs by interacting with each other. It is rather superfluous to consider any strategy for recovered and dead individuals and they do not take part in the game. After determination of payoffs, all cooperators and defectors randomly select a neighbor for strategy upgradation. Following Fermi's rule of strategy upgradation -an individual, i, will adopt the strategy of a randomly chosen neighbor, j, with a probability 1 1+exp[−(Πj −Πi)] . Π j and Π i denote the total payoffs accumulated by j and i respectively. Strong selection has been considered here.
After strategy upgradation, state upgradation takes place. In the previous step, S and I U individuals decide to be either C or D. In this step, infection would spread in the population. It has been considered that infection can spread through contact. Thus, susceptible neighbors of any infected individual are vulnerable. Depending on the strategy of both the interacting individuals i.e., C or D, the probability of spreading of infection will be different.
While upgrading its state, S randomly interacts with one of its neighbors. If that randomly chosen neighbor is I U , S may get infected with a probability, µ. The newly infected individuals may or may not manifest the symptoms. This would depend on the symptom manifestation rate of the infection, σ. The symptomatic individuals would be quarantined and they would not interact with the rest of the population anymore. These individuals are identified as I Q . The asymptomatic individuals remain in population and could cause further infection.
All individuals would hence undergo mass testing with probability, τ . Depending on τ , some infected individuals can also be identified as I Q . The remaining I U would remain in the population. Since they are unaware of their own situation they would unwittingly infect others. They may undergo self-recovery depending on ρ s or die without getting any medical help, depending on the value of δ. On the other hand, I Q individuals may either recover, depending on the value of ρ, or die, depending on the value of δ. Here for simplicity, the value of δ of I Q and I U has been considered to be the same. In our simulations, reinfection of recovered individuals has not been considered.
In all simulations, a transient time of 10 4 generations is considered. After this transient time, f C = S C /N , is calculated over a counting time of 10 3 generations. Every network has N nodes and the average degree is denoted by k . The overall simulations have been performed over E N networks. A definite fraction of nodes is initially assigned -at random -to act as cooperators. If f Ci denotes the initial fraction of cooperators in the population, then f Di = (1 − f Ci ) denotes the initial fraction of defectors. Apart from strategies, a definite number of nodes, i.e. individuals, are also randomly chosen to get infected. The number of individuals who are initially infected, I i , is identical to the initial number of untested individuals, I U -since they are not detected until tested. Obviously, S i = (1 − I i ) denotes the initial fraction of susceptible individuals in the population. Prisoner's Dilemma game has been played initially in the absence of any infection, i.e., between S C and S D only, which implies I i = 0 and S i = 1. In that case, f C = S C /N , depends only on c. S C dominates at lower values of c but decreases at higher values of c, as shown in Fig 1. This implies that if c increases, individuals would be quite unwilling to adopt precautionary measures and would try to defect.
A. Asymptomatic infections: class A First, we have considered class A, where the symptom manifestation ratio, σ, is very low. There are various diseases where manifestation of symptoms can not often be observed clearly, such as infections associated with Cytomegalovirus, Rhinoviruses, Salmonella or Ebola [18] [19] [20] infections. Here, we have considered that initially 2% individuals have been infected. Fig. 2 is for asymptomatic case. It can be observed that f C decreases with the increase of c. Hence, it can be concluded that individuals in the population will only cooperate for low value of c. But due to the presence of infection, low c implies higher risk. COVID19 infection can also be considered as asymptomatic infection due to a high number of asymptomatic carriers [21] . On the other hand, a higher cost is naturally unaffordable to most individuals. The standard error is smaller than the size of the data points.
Hence, alongside defectors, these cooperators are also likely to get infected at lower values of c. However, if we increase c, better protection ensures higher f C . For a BA network with k = 4, we observe that f C is higher when c = [0.25, 0.35]. Though, a higher cost implies lower risk, it also inhibits the lifestyle of S C individuals. Naturally, cooperation would be hindered at higher values of c. Hence, with increasing costs, S C individuals tend to becomes careless, implying that S D individuals would be likely to dominate. Also S D individuals would get infected at a higher rate due to the absence of proper precaution on their part. In the case of high k graphs, as shown in Fig. 2 , it has been observed the peak shifts towards the lower value of c. Generally, the maintenance of cooperation is lower when k of the graph is high. 
From Fig. 4 , it is evident that both mass testing and cost due to precautionary measures have a significant impact on f C . At higher values of c, cooperation will not be maintained in the population. But for lower values of c, the maintenance of cooperation would depend on the value of τ . In class A diseases, it is difficult to maintain cooperation at lower values of τ . This can be intuitively inferred from the fact that it is rather difficult to identify infected individuals without mass testing. Therefore, higher values of τ can lead to relatively higher cooperation. On the other hand, τ does not really have a significant impact in class M diseases.
Therefore, it is worth noting that when the mass testing ratio, τ , is low, an asymptomatic disease is likely to infect susceptible individuals more. Naturally, if the infected individuals are not identified then asymptomatic diseases have the potential to turn into pandemics. On the other hand, a symptomatic disease will not be able to spread significantly, even when τ is low. This is because I U in class M diseases, can be identified more easily via symptoms. Hence, the chances for symptomatic diseases to turn into a pandemic are lower. The initial fraction of infected individuals, I i , reflects the scenario during the initial outbreak of an epidemic. We explore the behaviour of f C with respect to I i [22] . A fraction of the population, I i , is infected right at the onset of the epidemic, before consciousness has taken root in the general population. Consciousness is expected to increase subsequently and the general populace would become aware and responsible. Fig. 5 implies that the final fraction of proactive individuals would decrease in the population at lower k . However, at higher k , the final fraction of responsible people would increase slightly in the population. In this case, the broad behaviour is notably the same for both class M and class A epidemics.
Accurate models of epidemic spreading are essential [23] [24] [25] , as has been clearly underscored during the ongoing COVID-19 pandemic. Prevalence of asymptomatic carriers complicates the detection and management of any disease to a significant extent [26] [27] [28] . Models based on SIRD dynamics or game-theoretic approaches on networks may not be able to capture the complete picture as they usually do not consider this important aspect [7, 29] . Our model explores the possible effect of various factors in epidemic spreading. Absence of vaccines makes individual and social proactiveness as extremely important factors, which can inhibit disease spreading. Also, the fraction of the population which has undergone testing is an essential factor in diseases involving asymptomatic phenotypes. In absence of high enough testing -an asymptomatic disease bears the potential to become a pandemic. We also observe that the fraction of proactive individuals varies with the initial fraction of infected individuals and the average degree of the network considerably influences this behaviour.
DN and SS contributed equally to this work. 