Who is the infector? Epidemic models with symptomatic and asymptomatic cases
Ka Yin Leung Pieter Trapman Tom Britton
q-bio.PE math.DS

New measurement of the 1S-3S transition frequency of hydrogen: 
contribution to the proton charge radius puzzle
    Jocelyne Guéna
    December 30, 2023
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What role do asymptomatically infected individuals play in the transmission dynamics? There are many diseases, such as norovirus and influenza, where some infected hosts show symptoms of the disease while others are asymptomatically infected, i.e. do not show any symptoms. The current paper considers a class of epidemic models following an SEIR (Susceptible → Exposed → Infectious → Recovered) structure that allows for both symptomatic and asymptomatic cases. The following question is addressed: what fraction ρ of those individuals getting infected are infected by symptomatic (asymptomatic) cases? This is a more complicated question than the related question for the beginning of the epidemic: what fraction of the expected number of secondary cases of a typical newly infected individual, i.e. what fraction of the basic reproduction number R_0, is caused by symptomatic  individuals? The latter fraction only depends on the type-specific reproduction numbers, while the former fraction ρ also depends on timing and hence on the probabilistic distributions of latent and infectious periods of the two types (not only their means). Bounds on ρ are derived for the situation where these distributions (and even their means) are unknown. Special attention is given to the class of Markov models and the class of continuous-time Reed-Frost models as two classes of distribution functions. We show how these two classes of models can exhibit very different behaviour. 




  
Keywords: two-type SEIR epidemic; final size; type of infector; continuous-time Reed-Frost models; Markov models 

[1]Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden.
 Email: {tom.britton, kayin.leung, ptrapman}@math.su.se



§ INTRODUCTION

For many known human infectious diseases we have, to some extent, an idea about their clinical features, e.g. the typical incubation period and common symptoms associated to them. Despite this, there can be large heterogeneity between infected hosts in a population. Some infected individuals may never show symptoms of disease (i.e. are asymptomatic) while others do (i.e. are symptomatic). Suppose that we can categorize infected individuals as either asymptomatic or symptomatic, and that some significant fraction of the infected host population is asymptomatic. Then, even if asymptomatics make up a large part of the infected host population, can we say how important their role is in the transmission dynamics? 

In order to answer such a question, one needs to define the meaning of `importance'. A natural approach is to consider the beginning of the epidemic, in particular the basic reproduction number R_0. This approach was taken by <cit.>, where they considered the (weighted) contribution θ of infectious individuals without symptoms (either asymptomatically infected or prior to symptom onset) to the basic reproduction number. The idea is that if θ is small, i.e. most of the secondary cases in the beginning of an epidemic are caused by symptomatic individuals, then control measures such as isolation of such (in principle detectable) individuals would greatly reduce spreading whereas this approach may not be successful if θ is large.

In case of an emerging infectious disease such as the recent 2013-2015 Ebola outbreak, one will mostly be interested in (controlling) the beginning of the outbreak. In such cases, we want to understand epidemiological quantities such as R_0 and θ. However, in case of infectious diseases such as seasonal influenza in the general population, it is quite usual that major outbreaks occur. In particular, the infectious disease is present in the population beyond its initial phase of the outbreak. In the current paper we are interested in the fraction ρ of the final epidemic size that were infected by symptomatic cases. Understanding the contribution of symptomatic cases to the final size leads to better insights into possible transmission routes, but also how things would change when e.g. removing a fraction of symptomatic individuals. 

We consider a simple class of epidemic models that we call the symptom-response SEIR epidemic models, defined in Section <ref>. This class of models has an SEIR (Susceptible → Exposed → Infectious → Recovered) structure. The time that an individual spends in either the exposed or infectious period can take on some general distributions. In particular, these distributions need not necessarily be exponentially distributed or be the same for symptomatic and asymptomatic cases. We can use the standard toolbox for analyzing mathematical models for infectious disease dynamics to derive e.g. the basic reproduction number and the final size. In fact, these and other characteristics have been studied before, e.g. in <cit.>, which we will make use of. 

Characterizing the fraction ρ turns out to be much more complicated than the related fraction θ that only deals with the beginning of the outbreak on a generation basis. Whereas θ only depends on the type-specific reproduction numbers (i.e. the expected number of secondary cases generated by a newly infected individual that is either symptomatic or asymptomatic), the fraction ρ also depends on the mean latent and infectious periods of the two types as well as their probabilistic distributions. The general characterization of ρ, given in Section <ref>, does not provide much insight even though it may be computed numerically when all distributions and parameters are given. Because of this, and the fact that distributions of latent and infectious periods are rarely fully known, it is of interest to give upper and lower bounds for ρ. In a `twin' paper <cit.> we use probabilistic tools to derive bounds for ρ for the setting where these distributions (including their means) are unknown (but note that <cit.> is more general than determining the upper- and lower bound for the symptom-response SEIR model). In Section <ref> of the current paper, we present these bounds and explain the heuristics in deriving them. Moreover, we are interested in two special classes of the model, namely the class of Markov models with exponentially distributed latent and infectious periods and the class of continuous-time Reed-Frost models with constant latent and infectious periods. We show that the choice of distribution functions can have significant qualitative differences on the time-evolution of the epidemic by comparing the class of Markov models with the Reed-Frost models having identical means. Furthermore, in Section 4 we consider different scenarios in order to gain insights into ρ through numerical investigation and to illustrate numerically the qualitative differences between Markov models and Reed-Frost models. 




§ MODEL DEFINITION




 §.§ The stochastic symptom-response SEIR epidemic model

The stochastic symptom-response SEIR epidemic model is defined as follows (in our twin paper <cit.> we study a more general form of the current model). We consider a closed population of homogeneously mixing individuals, all being equally susceptible. We let n denote the number of individuals in the population. The specific feature of the model is that infected individuals can either become symptomatic or asymptomatic. In order not to confuse Symptomatic with Susceptible, we use subscript d to denote symptomatic individuals, i.e. the individuals that show signs of the disease. Individuals are initially susceptible (S). If an individual gets infected it becomes symptomatic with probability p_d and asymptomatic with the remaining probability p_a=1-p_d. These events are independent between individuals and also independent of whom the individual was infected by (see <cit.> for a situation where the response does depend on the infector).

We let S(t) denote the number of individuals in the population that are Susceptible at time t. If the individual becomes symptomatic, it first has a random latency period L_d (and we let E(t) with a subscript a or d denote the number of asymptomatic and symptomatic individuals in the population at time t that are infected and Exposed but not yet infectious) followed by a random infectious period ι_d (where we let I(t) with subscripts a and d denote the number of Infectious individuals that are asymptomatic and symptomatic). Symptomatic individuals have infectious contacts at rate λ_d during the infectious period, each time with a uniformly chosen individual in the population. Infectious contacts with susceptible individuals result in the latter getting infected. Other infectious contacts have no effect on the epidemic (hence the term infectious contacts). The situation for asymptomatic cases is analogous with latency period L_a, infectious period ι_a, and infectious contact rate λ_a. When the infectious period of an individual terminates, the individual recovers and becomes immune (where R(t) with subscript a and d denotes the number Recovered individuals that were asymptomatic or symptomatic) for the rest of the epidemic. Note that S(t)+E_d(t)+E_a(t)+I_d(t)+I_a(t)+R_d(t)+R_a(t)=n. The epidemic starts with a small number of infectious individuals and all other individuals susceptible. The end of the epidemic is at time T, where T is the first time that there are no more latent or infectious individuals around. At time T there are only susceptible and recovered individuals. Let Z_d=R_d(T) and Z_a=R_a(T) be the final number (previously) symptomatic and asymptomatic cases, respectively (so S(T)=n-(Z_d+Z_a)). The overall number of individuals infected is denoted by Z=Z_d+Z_a. We are mainly interested in large population sizes and hence study the situation where n→∞. Moreover, we let S̅(t)=S(t)/n and we define other population quantities decorated with a bar over it in an analogous fashion. The limit of Z_d/Z as n→∞ conditioning on Z→∞ is simply the probability p_d that a newly infected individual becomes a symptomatic case, and Z_d/n→ p_dz, where z is the large population limit fraction of the population getting infected in case of a major outbreak (and is characterised by the final size equation (<ref>) below).

Note that we have not specified anything about symptoms and when they appear. In fact, we could equally well model a situation with mildly infected and severely infected (or two types 1 and 2 for that matter; see also <cit.>). Having symptomatic cases in mind, the contact rate λ_d should reflect both the more likely situation that symptomatic cases have a higher viral load as compared to asymptomatic cases (thus leading to higher transmission probability upon contact), and the fact that symptomatic cases may reduce their number of social contacts and thus meet fewer individuals. Since the infectious contact rate reflect both components, it need not necessarily be that λ_d>λ_a. 

Note that if there are no latency periods (L_d≡ L_a≡ 0) the model reduces to an SIR model. Furthermore, we give special attention to two distributions for latent and infectious period when studying S(E)IR models. The first class of distributions is where both latency and infectious periods follow exponential distributions, giving rise to Markov models. In the deterministic setting this corresponds to the model being described by ordinary differential equations (ODE). The second class of distributions is where these periods are non-random, i.e. fixed and the same for all symptomatic (asymptomatic) individuals. We refer to this class of models as continuous-time Reed-Frost epidemic models, since the events to infect different individuals are then independent. In the deterministic setting such models can be described with integral equations.



  §.§.§ Basic reproduction numbers and the final size

If we do not distinguish symptomatic and asymptomatic cases, the model defined in Section <ref> falls under the general framework of epidemic models defined in  <cit.>, and treated specifically in <cit.>. The fundamental quantity is the distribution of C, the (random) number of infectious contacts that an infected individual has during its infectious period. To this end, let the random variables C_d and C_a denote the number of infectious contacts for symptomatic and asymptomatic individuals, respectively. 

The final size distribution is completely determined by the distribution of C (which in turn is specified by C_d, C_a and p_d). For the symptom-response SEIR model, C follows a mixed Poisson distribution: MixPo(λ_d1_dι_d+λ_a(1-1_d) ι_a) (all random variables being independent and not necessarily exponentially distributed), where 1_d is 1 with probability p_d and 0 otherwise. Note that the latency periods L_a and L_d have no effect on the number of infections – they only affect the timing of the outbreak.   

The basic reproduction number denotes the expected number of infectious contacts and equals 

    R_0=(C)=p_d(C_d) + (1-p_d)(C_a) = p_dλ_d(ι_d) + (1-p_d)λ_a(ι_a)= p_dR_0,d+(1-p_d)R_0,a,

where we call R_0,d=λ_d(ι_d) and R_0,a=λ_a(ι_a) the type-specific reproduction numbers for newly infected symptomatic and asymptomatic individuals, respectively. We denote the fraction of R_0 that is caused by symptomatic individuals by θ_d, so

    θ_d=p_dR_0,d/R_0.

(Note that <cit.> consider the quantity θ=1-θ_d for a related but different model.) 

Next, we make use of known results on the final size Z from <cit.>. First of all, if R_0>1, then the final fraction infected Z̅=Z/n converges to a two-point distribution with probability mass at 0 (small outbreaks) and at another point z, where z is the unique positive solution to the equation 

    1-z=e^-R_0z
 
(major outbreaks). So, if R_0≤ 1 only minor outbreaks are possible. The probability for a major outbreak depends more delicately on the distribution of C, and not only on its mean (e.g. <cit.>). Second of all, in case of a minor outbreak, Z has a limiting distribution described by the finite part of the distribution of the total progeny of a branching process with offspring distribution C. Moreover, there is a central limit theorem for Z̅ in case of a major outbreak. 

We use these results on the final size in the present paper. However, our focus is different: we are interested in the number of cases that got infected by symptomatic and asymptomatic individuals, respectively. Here, we let Y_d and Y_a denote the number of individuals that got infected by a symptomatic and asymptomatic case, respectively (so Z=Y_a+Y_d). We are interested in the law of large number limit ρ_d that denotes the fraction of infected individuals that get infected by symptomatic cases, given a major outbreak. We hence seek the limit ρ_d of Y_d/Z as n→∞ conditioning on Z→∞ (note the symmetry in asymptomatic and symptomatic cases, ρ_a+ρ_d=1, so we can just as well switch the roles of a and d). We turn to this in Section <ref>. 



 §.§ Deterministic symptom-response epidemic models

As with nearly all epidemic models, the current model may be approximated by a deterministic epidemic model when the population size is large enough and when the number of individuals in the different compartments are large enough. More specifically, when n is large, (S̅(t), E̅_a(t), E̅_s(t), I̅_a(t), I̅_d(t), R̅_a(t), R̅_d(t)) may be approximated by a deterministic model on the part of the time axis where all fractions are not too close to 0. The latter means that the approximation does not work in the beginning nor at the end of an outbreak (when the number of latent and infectious individuals are moderate to small), since these settings imply that the corresponding fractions are close to 0. If the epidemic is initiated by a small positive fraction of infected people (either latent or infectious and symptomatic or asymptomatic), the starting phase is removed. In that case the approximation applies on any fixed time interval [0,t]. 

The statement that the stochastic model may be approximated by a deterministic counterpart really should mean that there is a law of large number theorem that proves that, as n→∞, the stochastic process for the fractions converge in probability, uniformly on bounded intervals, to the deterministic process. It is not the purpose of the present paper to prove such results for the current model. Instead we assume this to hold and therefore work interchangeably with the deterministic and stochastic setting. The law of large numbers should be proven using population process theory (e.g. Ethier and Kurtz, 2006) in ways similar to what has been done for related models, cf. <cit.>, Section 3.4.

In the deterministic formulation of the model, we let the population size n→∞ and consider expected fractions of the population in different disease states. The corresponding population quantities are denoted by a small letter, i.e. s(t) is the expected fraction of the population that is susceptible at t, and similarly for other population quantities.  

We do the bookkeeping using the age τ since infection, taking into account asymptomatic and symptomatic cases. Let 

    π_a(τ)    P(L_a≤τ≤ L_a+ι_a)

i.e. π_a(τ) denotes the probability that an asymptomatic individual is infectious at age since infection τ. Furthermore, let F_a(t) denote the force of infection from asymptomatic individuals at time t (and similarly we have π_d(τ) and F_d(g) for symptomatics). So, if i_a(t) and i_d(t) denote the fraction of individuals that are asymptomatic infectious and symptomatic infectious respectively, then F_a(t)=λ_ai_a(t) and F_d(t)=λ_d i_d(t). Furthermore,

    ṡ=-(F_a+F_d)s.

From (<ref>) it follows that the expected incidence at time t-τ is (F_a(t-τ)+F_d(t-τ))s(t-τ). Of this quantity, a fraction p_a=1-p_d is asymptomatic. An asymptomatic individual that was infected time t-τ ago has expected infectivity λ_aπ_a(τ) at time t. Then, by integrating over all possible ages since infection τ≥0, we find an expression for F_a(t) (and similarly for F_d(t)):

    F_a(t)   =∫_0^∞λ_aπ_a(τ) (1-p_d)(F_a(t-τ)+F_d(t-τ))s(t-τ)dτ
    
    F_d(t)   =∫_0^∞λ_dπ_d(τ) p_d(F_a(t-τ)+F_d(t-τ))s(t-τ)dτ.

Together, (<ref>) and (<ref>) form a closed system. From this system, one can obtain e.g. the mean fraction e_A(t) of the population that is asymptomatically infected and still in the latent period. To this end, we simply note that an asymptomatic individual that was infected time t-τ ago is in its latent period with probability P(L_a>τ). Therefore, the fraction of exposed asymptomatic-to-be individuals is

    e_a(t)= ∫_0^∞ P(L_a>τ) (1-p_d) (F_a(t-τ)+F_d(t-τ))s(t-τ)dτ,

and similarly one can recover the formulas for the other population quantities. 



  §.§.§ Markov models

Often in the modelling community, ordinary differential equations (ODE) are used to describe deterministic compartmental disease models. Implicitly this assumes that an individual spends an exponentially distributed amount of time in each disease compartment, giving rise to the special case of Markov models. Assume that L_k∼exp(α_k), ι_k∼exp(γ_k), k=a,d. Then, by working out the probability π_k, k=a,d, to be infectious at age-since infection τ:

    π_k(τ)   =α_k/γ_k-α_k(e^-α_kτ-e^-γ_kτ).

In ODE formulation, the dynamics of the model are described by

    s'   = -(λ_a i_a+λ_d i_d) s
    
    e_k'   = p_k(λ_a i_a+λ_d i_d) s-α_k e_k
    
    i_k'   = α_k e_k-γ_k i_k
    
    r_k'   = γ_k i_k,

k=a,d, with p_a=1-p_d. Note that consistency requires 1=s(t)+e_a(t)+i_a(t)+r_a(t)+e_d(t)+i_d(t)+r_d(t). As mentioned earlier, the forces of infection satisfy F_a(t)=λ_a i_a(t) and F_d(t)=λ_d i_d(t). Finally, note that the type-specific reproduction numbers R_0,a and R_0,d reduce to R_0,a=λ_a/γ_a and R_0,d=λ_d/γ_d (exactly as one would expect from the interpretation). 



  §.§.§ Continuous time Reed-Frost models

Another special class of models is when each individual spends a deterministic amount of time in each disease compartment. We refer to this class of models as continuous-time Reed-Frost models. This class of models plays an essential role in both this paper and the twin paper <cit.>. Assume that L_k≡ℓ_k, and ι_k≡ x_k, k=a,d, with ℓ_k, x_k nonnegative constants (so all symptomatics have equal lengths of latent and infectious periods, as do all asymptomatics). Then 

    π_k(τ)   = 1_(ℓ_k, ℓ_k+x_k)(τ),

k=a,d. This yields the following renewal equations for the forces of infection F_k:

    F_k(t)   =∫_t-(ℓ_k+x_k)^t-ℓ_kp_kλ_k(F_a(τ)+F_d(τ))s(τ)dτ,

k=a,d and p_a=1-p_d. By differentiating with respect to t, we can reformulate (<ref>) as delay differential equations (DDE), and we end up with a system of three DDE for s, F_a, and F_d:

    s(t)'   =-(F_a(t)+F_d(t))s(t)
    
    F_k(t)'   =p_kλ_k{(F_a(t-ℓ_k)+F_d(t-ℓ_k))s(t-ℓ_k)
       = -[F_a(t-(ℓ_k+x_k))+F_d(t-(ℓ_k+x_k))]s(t-(ℓ_k+x_k))},

k=a,d and p_a=1-p_d. For given values of ℓ_a, ℓ_d, x_a, and x_d it is straightforward to numerically simulate s, F_a, and F_d. 

Finally, note that type-specific reproduction numbers R_0,a and R_0,d reduce to R_0,a=λ_a x_a and R_0,d=λ_d x_d.



 §.§ Initial conditions

So far, we have not specified initial conditions, i.e. how infection is introduced into the susceptible population at the beginning of the epidemic. In general, in a stochastic formulation, one of two options is chosen. There is at time t=0, either a positive fraction of infectious individuals, or a fixed number of infectious individuals, as the population size n→∞. In other words, if we let μ_n=n^-1m_n, then, in the first case, μ_n→μ as n→∞, with μ>0 constant. In the second case, m_n=m is fixed (and therefore μ_n→0 as n→∞). In general, the initial conditions can matter for the possible courses that the epidemic can take, see e.g. <cit.>. Implicitly, for the basic reproduction number and the final size of the epidemic, we assume the second setting of a fixed number m_n=m (but approximately the same results apply if μ>0 but very small). 

As we discuss in Section <ref>, the stochastic model of Section <ref> can be approximated by a deterministic epidemic model when the population size n is large enough. In the deterministic formulation we implicitly assume an infinite population size and work with mean fractions of the population. In particular, if we define the initial condition at t=0, there is a strictly positive fraction of infected individuals in the population at the beginning of the epidemic. This corresponds to an infinite number of infecteds, no matter how small this fraction is. One way to obtain the setting of an epidemic with only few infectious individuals at the beginning is by starting with a small positive fraction of infected individuals at time t=0 (chosen in the right way), and then to let this fraction tend to zero while also letting time t→-∞. This is how we formulated (<ref>)-(<ref>) without worrying too much about initial conditions. In the `far past' t=-∞, the population is completely susceptible. If R_0>1 and we let t→+∞, then an epidemic outbreak will occur with final size characterized by (<ref>) (in case of a major outbreak). It is not the purpose of this paper to make these arguments fully rigorous. See also <cit.>. 

Together with the `far past' conditions, the deterministic model allow for the derivation of R_0 and the final size. By doing so, we find (<ref>) for R_0 and (<ref>) for the final size. In particular, the interpretation of R_0 and the final size is the same as in the stochastic reasoning. In fact, the law of large numbers results from <cit.> show that there is convergence of these epidemiological quantities of the stochastic process to the deterministic process as population size n→∞. 




§ WHO IS THE INFECTOR?



 §.§ Timing matters

Now that we have set up the model framework in Section <ref>, we can ask: who is the infector? We assume that a major outbreak occurs and the final number of infected individuals Z→∞. In particular, R_0>1. In the remainder of the text we take the deterministic viewpoint, while making use of probabilistic arguments. Assume that the final fraction of infected individuals Z̅ converges to z, where z is the nontrivial solution of (<ref>). Recall from Section <ref> that we let ρ_a=1-ρ_d and ρ_d denotes the fraction of z that had a symptomatic infector. 

At time t, the rate at which new infections are caused by symptomatics is F_d(t)s(t), and the rate at which new infections are caused by asymptomatics is F_a(t)s(t). The fraction of individuals infected by symptomatic individuals is obtained by integrating over the entire epidemic. In particular, 

    ρ_d=∫_-∞^∞ F_d(t)s(t)dt/∫_-∞^∞ (F_a(t)+F_d(t))s(t)dt = ∫_-∞^∞ F_d(t)s(t)dt/z.

Note that indeed z=∫_-∞^∞ (F_a(t)+F_d(t))s(t)dt (integrate both left- and right-hand side of (<ref>) over all time and observe that s(∞)=1-z, s(-∞)=1). In principle, this characterizes ρ_d. A different characterization for ρ_d involving the Malthusian parameter and (generally implicit) limiting random variables of branching processes can be found in <cit.>. However, in general, both the expression (<ref>) for ρ_d and the alternative expression of <cit.> are not very informative and we have not managed to make either of them more explicit in this general setting. 

Due to the issues with obtaining general results for ρ_d mentioned above, we instead derive general bounds for ρ_d in Section <ref>. First, we obtain explicit expressions of ρ_d for the very special case that π_a(τ)=π_d(τ) for all τ, i.e. asymptomatic and symptomatic cases have the same probability to be infectious at any given age-since-infection. From (<ref>), we find that 

    F_a(t)=(1-p_d)λ_a/p_dλ_dF_d(t).

Therefore, 

    z=∫_-∞^∞ (F_a(t)+F_d(t))s(t)dt=(1+(1-p_d)λ_a/p_dλ_d)∫_-∞^∞ F_d(t)s(t)dt,
 
leading to

    ρ_d = p_dλ_d/p_dλ_d+(1-p_d)λ_a =p_dR_0,d/R_0,

where the second equality follows from (ι_a)=(ι_d). Note that (<ref>) is monotonically increasing in both p_d and λ_d. Furthermore, the fraction ρ_d only depends on the type-specific reproduction numbers R_0,a and R_0,d and the probability p_d. In fact, (<ref>) is equal to the quantity θ_d (see (<ref>)) that indicates the importance of symptomatics at the beginning of an epidemic. 

Additionally, if also λ_d=λ=λ_a, the expression (<ref>) reduces even further to 

    ρ_d=p_d.
 




 §.§ General upper- and lower bounds on ρ_d

Two features of ρ_d seem obvious, but we have not been able to prove them since ρ_d depends on the distributions of the latent and infectious periods in a rather complicated way (see (<ref>)). Instead we conjecture the two features regarding ρ_d and explain why we believe them to be true. First, we believe that ρ_d is increasing in p_d (and hence decreasing in p_a=1-p_d). This means that, keeping all other parameters fixed, the fraction infected by symptomatic individuals increases with the probability that an infected individual becomes a symptomatic case. Secondly, we believe that ρ_d is increasing in λ_d (and decreasing in λ_a). Note that in the special case that latent and infectious periods of symptomatics and asymptomatics are identical, i.e. π_a(τ)=π_d(τ) for all τ, we find from (<ref>) that these two statements regarding monotonicity of ρ_d hold true. 

Note that as p_d or λ_d changes, the overall fraction infected z also changes. We believe that when one keeps R_0 fixed (and therefore also z) as well as R_0,d, the statements regarding monotonicity of ρ_d in p_d and λ_d remain true. If we change λ_d while keeping R_0 and R_0,d=λ_d(ι_d) fixed, we also need to change the mean infectious period (ι_d). In the case that we change p_d, also R_0,a changes. 

Even though we believe the monotonicity of ρ_d in p_d and λ_d to hold, there is little qualitative insight that we can obtain from (<ref>). Therefore, we derive explicit ranges within which ρ_d will certainly lie, when keeping the reproduction numbers R_0, R_0,d, and R_0,a fixed (and consequently also p_d=1-p_a), in particular since in many situations the exact distributions of latent and infectious periods are not known in detail. 

Now we provide an intuitive argument for our beliefs on the monotonicity of ρ_d and the extreme model that provides us an upper bound ρ_d^+ (and for symmetry reasons also a lower bound ρ_d^-). Consider one newly infected symptomatic individual. Suppose we shorten the latent period of this individual to zero and we make its infectious pressure very intense (by  decreasing the infectious period and at the same time increasing its infectious contact rate so that the product remains constant), then this individual can cause infections earlier on in the epidemic when there are more susceptible individuals. So more contacts will result in infections, i.e. this symptomatic individual will cause more infections. By assuming that the new latent and infectious periods are deterministic, we ensure that the symptomatic individual actually cause more infections rather than only making it more probable (which would be the case with e.g. the Markov assumption). Moreover, increasing the latent period of an asymptomatic individual will imply that it can only start infecting others later on in the epidemic. Note that by assuming that asymptomatics also have very intense infectious pressure, this also implies that the symptomatic secondary cases they generate become infectious earlier on in the epidemic as well. It seems reasonable that by applying the changes in latent and infectious periods to all individuals in the population, i.e. if we consider an extreme case of the class of continuous-time Reed-Frost models where L_a is long relative to L_d, ι_a, and ι_d, and very short ι_a and ι_d combined with very large λ_a and λ_d, the extreme model will lead to an upper bound in ρ_d for all models with fixed R_0, R_0,d, and R_0,a. Of course, all of this is by no means a proof (for the proof as well as the precise assumptions we need for the latent and infectious periods we refer to <cit.>), but we hope that this convinces the reader why ρ_d^+ is the upper bound (and/or encourages the reader to read <cit.>). 

The proof in <cit.> uses an epidemic graph construction that couples susceptibility processes with backward branching processes. The general idea in the proof presented in <cit.> is that there is a shortest (directed) path from the individual that is infectious at the beginning of the epidemic to the individual under consideration. Only the type of the neighbour with an edge pointing towards the individual under consideration determines whether it got infected by an asymptomatic or symptomatic case. We can determine this by considering the backward branching process of the individual under consideration. This branching process has an offspring distribution that is Poisson distributed with mean p_d R_0,d. The offspring of an individual u are those individuals that have an edge pointing towards u in the epidemic graph, i.e. those individuals that could potentially infect u if they become infected themselves. The calculations in <cit.> provides us with the following upper bound ρ_d^+ (see also <cit.>):

    ρ_d^+=η_d/z+(1-η_d/z)(2-η_d-z)p_dR_0,d/2.

Here, 0≤η_d<1 is the largest solution to

    1-η_d=e^-p_dR_0,dη_d,

which is strictly between 0 and 1 if and only if p_dR_0,d>1 and zero otherwise. Moreover, 1-η_d can be interpreted as the probability that a randomly chosen individual is susceptible at the end of epidemic when only symptomatic cases can reproduce (set λ_a=0). We discuss η_d in more detail in Section <ref>.

The lower bound ρ_d^- corresponding to (<ref>) is obtained by interchanging the roles of symptomatic and asymptomatic cases, i.e. 

    ρ_d^-=1-ρ_a^+
 
where

    ρ_a^+=η_a/z+(1-η_a/z)(2-η_a-z)(1-p_d)R_0,a/2.

and 0≤η_a<1 is the largest solution to 

    1-η_a=e^-(1-p_d)R_0,aη_a

(which is strictly between 0 and 1 if and only if (1-p_d)R_0,a>1 and zero otherwise). Note that (<ref>) and (<ref>) are not explicit since z, η_a, and η_d are only implicitly characterized through equations (<ref>), (<ref>), and (<ref>), respectively. Since ρ_d^+ and ρ_d^- are upper- and lower bounds for all distributions of the latent and infectious periods with fixed R_0, R_0,a, and R_0,d (with some technical restrictions, see <cit.> for details), we find that ρ_d^- ≤θ_d ≤ρ_d^+. Indeed, ρ_d=θ_d in the special case that π_a(τ)=π_d(τ). 




 §.§ The epidemic curve for extreme settings

In this section we discuss the epidemic curve for the extreme setting where the mean latent period (L_a) is long relative to (L_d), (ι_a), and (ι_d). To characterize the curve, we revisit (<ref>) for η_d. Suppose that p_dR_0,d>1, then symptomatic individuals themselves can sustain an epidemic (without asymptomatic cases reproducing) and 0<η_d<1 is unique and strictly positive. In fact, if (L_a) is relatively long, asymptomatic cases will initially not reproduce, and we are in the setting of an epidemic outbreak generated by symptomatic individuals only. (Or, from a stochastic viewpoint, there is a positive probability that this occurs, and we condition on this happening.) 

What happens after the epidemic `symptomatic' outbreak? This very much depends on the distributions for L_a, L_d, ι_a, and ι_d! In fact, we discuss the epidemic curves for the class of continuous time Reed-Frost models and Markov models in Sections <ref> and <ref> below. The upshot is that these two classes of models exhibit very different behaviour. Note that, when p_d R_0,d<1, there is no initial `symptomatic' outbreak, and the epidemic curve takes the form of a `standard' curve. 



  §.§.§ The continuous-time Reed-Frost model

In the continuous-time Reed-Frost model, latent and infectious periods are deterministic. Therefore, we know with certainty that newly infected asymptomatic individuals will  initially remain in their latent period and do not transmit. There is a `symptomatic' outbreak until there are no more infectious symptomatics left. At this point, when the `first symptomatic wave' of the epidemic has occurred, the probability that a randomly chosen individual is susceptible is given by 1-η_d. Note that this allows us to interpret the fraction η_d/z in (<ref>) as the probability that an individual was infected in the first wave of the epidemic, given that it got infected. 

After the first wave, the epidemic continues when the asymptomatic cases of the first outbreak become infectious. Additionally, there will be newly infected symptomatic cases. Together these infectives generate a second wave of infections (where the asymptomatic cases generated in the second wave are not yet infectious). The same reasoning as before yields the probability that an individual escapes infection in the second wave, given that it escaped infection in the first wave, is (1-η_2)/(1-η_d). Here η_2 denotes the fraction of infectives after the second wave (so η_1:=η_d would be consistent notation). Such an individual needs to escape infection from the (1-p_d)η_d asymptomatic individuals of the first wave as well as all the symptomatically infected individuals from the second wave. Hence 0<η_2<1 is the unique solution to

    1-η_2/1-η_d=e^-((1-p_d) R_0,aη_d+p_dR_0,d(η_2-η_d)).

Note that, contrary to the first wave, the second wave is not an epidemic outbreak per se. Indeed, it starts with a significant fraction (1-p_d)η_d of latently infected individuals. The susceptible population is of size 1-η_d, and even if at the end of the second wave we have a fraction η_2 infected individuals, we may have R_0(1-η_d)<1.

By the same reasoning, we find that 

    1-η_k/1-η_k-1=e^-((1-p_d) R_0,a (η_k-1-η_k-2)+p_dR_0,d(η_k-η_k-1)),

for k≥3, where 0<η_k<1 is the fraction of individuals infected in one of the first k waves (and η_1=η_d). So, we find that the fraction of susceptibles depletes over time to 1-z in an infinite number of waves that are characterized by 1-η_d, 1-η_2, 1-η_3,… This qualitative multi-wave process is illustrated in Figs. <ref> and <ref> of Section <ref>.




  §.§.§ The Markov model

The crucial difference between the continuous-time Reed-Frost models and the Markov models is that we now only know that on average newly infected asymptomatic individuals will remain in their latent period during the `symptomatic outbreak' since (L_a) is relatively long. 
However, due to the memoryless property of the exponential distribution, there is a constant probability per unit of time that an asymptomatic case in the latent stage becomes infectious, regardless of how long ago that individual got infected (in contrast to the continuous-time Reed-Frost of Section <ref> with deterministic latent periods). 

Given that a major outbreak occurs, at first, the probability that any of the asymptomatic cases become infectious in the `symptomatic outbreak' is small enough to neglect. Therefore, as in Section <ref>, the epidemic starts with a symptomatic outbreak that is characterized by η_d satisfying (<ref>). But, after the first `symptomatic' wave, there is a significant fraction (1-p_d)η_d of asymptomatic individuals in the latent stage. These all have a constant probability per unit of time to become infectious (whereas in Section <ref> these asymptomatics became infectious more or less at the same time). As a result, after the first (symptomatic) wave, both symptomatic and asymptomatic cases can transmit infection to the remaining susceptible population. The susceptible fraction slowly decreases from 1-η_d down to the fraction 1-z. The time scale at which this occurs is of the same order as (L_a). In particular, the epidemic curve can be approximated by a two-wave process rather than the multi-wave process of Section <ref>. This qualitative two-wave process is illustrated in Fig. <ref> of Section <ref>. 






§ NUMERICAL ILLUSTRATION

We consider different scenarios for the parameter values to illustrate the analytical results of this paper. We use the system of ODE (<ref>) and the system of DDE (<ref>) for the numerical simulations in this section. 



 §.§ Epidemic curves: Markov models vs. continuous-time Reed-Frost models

We choose parameter values to illustrate (i) the extreme behaviour described in Sections <ref> and <ref> and (ii) the difference between the Markov (M) assumption and the Reed-Frost (RF) assumption in temporal behaviour. We let R_0=2, R_0,d=2.5, p_d=0.5 (consequently R_0,a=1.5). Note that the final sizes z and η_d, the lower- and upper bounds ρ_d^- and ρ_d^+, and the fraction θ_d of R_0 caused by symptomatic cases depend only on R_0, R_0,d, and p_d. The values of these epidemiological quantities for scenarios 1 and 2 are summarized in Table <ref>. 



When it comes to the time evolution of the epidemic, the mean values and the distribution of L_a, L_d, ι_a, and ι_d start to matter as we will show using the M and RF assumption. We consider the following two scenarios. We let (L_d)=0, (ι_a)=0.10=(ι_d). Scenarios 1 and 2 differ only in the mean latent period (L_a) for asymptomatic cases: (L_a)=5 for scenario 1 and (L_a)=50000 for scenario 2. The reason for choosing such big differences in mean latent periods between symptomatics and asymptomatics is to clearly illustrate the behaviour described in Section <ref>. Furthermore, the scenarios are such that the RF model is close to the extreme model that yields the upper bound ρ_d^+ for all classes of models with the same p_d, R_0, and R_0,d. The epidemic curves for the two scenarios under the M and RF assumption are found in Figs. <ref> and <ref>. 



Note that, in both classes of models M and RF, according to theory, the fraction of susceptibles monotonically decreases over time and reaches the level 1-z, with z the final fraction of infecteds in the epidemic. Furthermore, in Fig. <ref>, we find that the RF model displays the extreme behaviour as described in Section <ref>. The level 1-η_k of the kth wave is indeed characterized by the equations (<ref>)-(<ref>). Next, we also note that the M model does not display any extreme behaviour under scenario 1. In other words, the mean latent period (L_a) is not long enough (relatively speaking). In particular, we find that the two classes of models display very different behaviour, i.e. a `multi-wave epidemic' and a `standard epidemic'. In Fig. <ref> we consider scenario 2 and both the RF and M model exhibit the extreme behaviour described in Sections <ref> and <ref>.



Evaluating ρ_d numerically yields 0.73 (M) and 0.74 (RF) for scenario 1 and 0.74 (M and RF) for scenario 2. The upper bound (for both scenarios and classes of models) is ρ_d^+=0.74; see Table <ref>. Hence we find that the two scenarios are both extreme enough that they yield ρ_d close to ρ_d^+. We find that there is a significant difference between ρ_d and θ_d=0.63, i.e. symptomatic cases play a different role for the beginning of an outbreak as in the final size of the epidemic. Furthermore, when it comes to ρ_d, the M or RF assumption has little effect in the two scenarios that we consider (this need not be true in general, see Section <ref>). But, we also find that the time evolution of the epidemic are very different between the Markov and Reed-Frost models (Figs. <ref> and <ref>).



 §.§ Numerical investigation of ρ_d

We consider some numerical examples where parameter values are chosen to illustrate our beliefs on the monotonicity of ρ_d that we stated in Section <ref>. We fix R_0=3.2, R_0,d=3.5, and we vary p_d. These parameter values yield lower- and upper bounds ρ_d^- and ρ_d^+ as functions of p_d as well as parameter θ_d in between the bounds; see Fig. <ref>. Fig. <ref> illustrates that the range between ρ_d^- and ρ_d^+ can be quite wide. Next, we consider ρ_d for the M and RF models. We let (L_a)=1, (L_d)=5, (ι_a)=1.4, and (ι_d)=2. The example of Fig. <ref> supports our first general belief, that ρ_d is an increasing function of p_d. 



For our second example, we fix p_d=0.5, R_0=3.8, R_0,d=3.5, (L_a)=1, (L_d)=5, (ι_a)=1.4, and vary λ_d (consequently we also vary (ι_d)); see Fig. <ref> for the result. Note that also R_0,a is fixed (at 4.1). Then ρ_d^+, ρ_d^-, θ_d, and p_d are constants as a function of λ_d. On the other hand, in this example, ρ_d(M) and ρ_d(RF) are both monotonically increasing functions of λ_d, which supports our second conjecture. 





 §.§ Illustration with norovirus and measles

We take two scenarios `norovirus' and `measles' that represent real-life epidemics with parameter values in reasonable ranges taken from literature. This section serves to illustrate the role of ρ_d. By no means do we try to incorporate the complexity of the two infections. 



The first scenario represents a norovirus outbreak in a hospital setting, and we take reference values based on <cit.>. From <cit.>, we obtain p_d=0.7, duration of shedding (ι_a)=19.3 days and (ι_d)=25.3 days, and mean latent periods (L_a)=1.5  days=(L_d). <cit.> report estimated reproduction numbers of R_0,a=0.85 and R_0,d=1.64 for asymptomatic and symptomatic cases respectively. Furthermore, from their data, we find p_d=0.7 consistent with <cit.>. Hence R_0=1.38.  

The second scenario represents measles in England in the vaccine era with subclinical infections. We base parameter values for the reproduction numbers, p_d, and mean infectious periods on <cit.>. Unvaccinated individuals that are infected are symptomatic and have a high reproduction number. Vaccinated individuals could potentially become asymptomatically infected but these have a very short infectious period and low reproduction number (in <cit.> the possibility of asymptomatic infection is related to antibody decay). The model in <cit.> is used to address a completely different question related to waning and boosting of immunity, and does not include any latent period. For our purposes we let mean latent periods be (L_a)=7  days=(L_d), coinciding with a reasonable incubation period <cit.>. Again, we do not try to fit our model to any real measles outbreak, but rather want to illustrate the use of the quantity ρ_d by considering reasonable parameter values.

Parameter values for both scenarios are summarized in Table <ref>. Note that both scenarios have R_0>1 and p_d R_0,d>1. The resulting epidemic curves are presented in Fig. <ref> for both the M and the RF model. Note that, for both scenarios, the curves of both the M and RF model are of a `standard epidemic'. Qualitatively, the two curves are similar. However, when it comes to the quantitative dynamics, we do find that the assumption on the distribution of the latent and infectious periods matter.



Next, we consider the fraction ρ_d of cases that were caused by symptomatic individuals, given that they were ultimately infected in Table <ref>. Note that lower- and upper bounds ρ_d^- and ρ_d^+ are only dependent on p_d, R_0, and R_0,d and are calculated using (<ref>) and (<ref>). We find that (i) for both the scenarios that the RF and the M model assumption yield similar values for ρ_d (ii) the upper- and lower-bounds ρ_d^+ and ρ_d^- are narrow for the norovirus scenario but wider for the measles scenario, and (iii) for both scenarios ρ_d and θ_d are close to each other for both the RF and the M model assumption. Furthermore, while the measles scenario has only a small proportion p_d=0.1 of infected individuals that become symptomatic, both in the beginning of the epidemic and in the final size, they play a much bigger role than the asymptomatic cases. Indeed, we find that symptomatic cases contribute between ρ_d^-=0.57 and ρ_d^+=0.86 of all transmissions that occur in the epidemic. In the norovirus scenario, this difference in the fractions p_d and ρ_d is not as extreme. Still we find that the contribution of symptomatic cases to the transmissions is between ρ_d^-=0.81 and ρ_d^+=0.85, which is larger than would be expected based on p_d=0.7 alone. We find that symptomatic cases cause most of the infections in the norovirus scenario. This is consistent with the findings of <cit.> that symptomatic cases contributed most to the spread of infection in the outbreak settings that they analysed (incidentally, the outbreak duration is not consistent with <cit.>).





§ CONCLUSION AND DISCUSSION

In this paper we considered an SEIR model with symptomatic and asymptomatic cases and looked at the fraction ρ_d of the infected individuals that was caused by symptomatic cases. The quantity ρ_d gives an indication of the importance of symptomatic cases in the transmission process in an epidemic. In general, it is hard to make statements about ρ_d as timing of transmission events matter: a susceptible individual that gets infected by an asymptomatic case can no longer become infected by a symptomatic case. We were only able to derive an explicit expression for ρ_d for the simple situation where the latent and infectious period distributions of symptomatic and asymptomatic coincide. 

Instead we used this timing issue to our advantage. We considered extreme settings for the latent and infectious periods of symptomatics and asymptomatics. The current paper is accompanied by a twin paper <cit.> in which the arguments and reasoning are made precise (in a more general setting than we consider here). The extreme setting allowed us to derive an upper and lower bound ρ_d^+ and ρ_d^- for ρ_d. The bounds ρ_d^+ and ρ_d^- are the bounds for fixed R_0,d, R_0,a and p_d but unspecified latent and infectious periods. Although the expressions for ρ_d^+ and ρ_d^- are still implicit (depending on some implicit final size equations) they are relatively simple and easily obtained. We studied ρ_d and the bounds numerically in Section <ref>. It was seen that the biggest range of possible values of ρ_d, i.e. the biggest difference between ρ_d^- and ρ_d^+, is for the situation where both symptomatics and asymptomatics separately have big enough reproduction numbers to produce an outbreak on their own: p_dR_0,d>1 and p_aR_0,a>1 as seen in Fig. <ref>.

We stated two rather natural beliefs for the general model, namely that ρ_d increases with increasing fraction p_d of infecteds that become symptomatic, and that ρ_d increases with increasing infectious contact rate λ_d from symptomatic cases. Unfortunately we were not able to prove either statement, also not using the techniques employed in <cit.>, at least not in a straightforward way. Instead we pose these two statements as open problems for future work.

We paid special attention to two special classes of models, namely the class of Markov models (with latent and infectious periods exponentially distributed) and continuous-time Reed-Frost models (with deterministic latent and infectious periods). We specifically considered the epidemic curve in the extreme setting with fixed mean latent and infectious periods in Sections <ref> and <ref> and found large differences in the qualitative behaviour of the two classes of models. In other words, not only the means but also the distributions matter! At this point, we hope that Sections <ref> and <ref> and Figs. <ref> and <ref> have convinced readers that the choice of distribution functions matter. This fact is of course not new and has been observed before in different settings (e.g. <cit.>). The distribution of the infectivity profile can matter a lot. However, how much it matters depends on the aspects of the epidemic model that one considers. In case of R_0, it is the mean infectious period that matters and not its distribution. In case of the Malthusian parameter (or as we have seen in this paper, the epidemic curve), the choice of distributions can matter a lot. Assuming exponential distributions for mathematical convenience could prove dangerous when not also considering the possible influences of this assumption. The qualitative outcome can be dramatically different depending on other choices. The critical reader will now point out that parameter values of Figs. <ref> and <ref> probably do not fit any reasonable known disease. This is a valid criticism. However, our first aim was to illustrate possible effects. Moreover, much more reasonable parameter values can be chosen such that the assumption of constant periods create a multi-wave epidemic while the Markov assumption yields a `standard' epidemic curve (Fig. <ref>). Finally, while one can argue about the realism of the simple models and parameter values used in this text, we hope to have reminded the reader that conclusions are always based on the assumptions that one makes. Furthermore, we hope one keeps in mind that it is worth exploring what the possible differences are by assuming something other than exponential distributions. 

Studying the end of the epidemic and the question of who the infector was is more involved than studying R_0 for the beginning of the epidemic. We managed to gain insights by providing bounds. This increased understanding of the transmission dynamics may help us in deciding the role that asymptomatic cases can play, and how their role may differ throughout an epidemic outbreak.



 §.§ Acknowledgements

The authors are supported by the Swedish Research Council (VR) [grant 2015-05015 (TB and KYL) and grant 2016-04566 (PT)].




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