J Environ Sci Public Health 2021;5 (2):251-272 DOI: 10.26502/jesph.96120128

Research Article Treatment, Duration of Infectiousness, and Disease

Thomas Caraco*

Department of Biological Sciences, University at Albany, Albany, NY, 12222, USA

*Corresponding Author: Thomas Caraco, Department of Biological Sciences, University at Albany, Albany, NY, 12222, USA

Received: 12 March 2021; Accepted: 22 March 2021; Published: 08 April 2021

Citation: Thomas Caraco. Antibiotic Treatment, Duration of Infectiousness, and Disease Transmission. Journal of Environmental Science and Public Health 5 (2021): 251-272.

Abstract infectious 's time-dependent probability of By curing infectious individuals, antibiotic therapy pathogen transmission, as well as the probabilistic must sometimes limit the spread of contagious disease duration of the infectious period. among hosts. But suppose that a diseased host stops transmitting due either to antibiotic cure or At the within-host scale, the model varies (1) to non-therapeutic removal (e.g., isolation or inoculum size, (2) bacterial self-regulation, (3) the mortality). An antibiotic`s suppression of within-host time between infection and initiation of therapy, and pathogen growth increases the likelihood of curing a (4) antibiotic efficacy. At the between-host scale the single infection and may also reduce the probability model varies (5) the size of groups randomly of non-therapeutic removal. If antibiotic treatment encountered in the infectious host’s environment. relaxes the total rate of infection removal sufficiently to extend the average duration of infectiousness, Results identify conditions where an antibiotic can between-host transmission can increase. That is, increase duration of a host`s infectiousness, and under some conditions, curing individuals with consequently increase the expected number of new can impact public health negatively (more . At lower antibiotic efficacy, therapy might new infections). To explore this counter-intuitive, but convert a rare, serious bacterial disease into a plausible effect, this paper assumes that a common, but treatable infection. deterministic within-host dynamics drives the

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Keywords: Group size; Infectious contacts; antibiotics increase host survival. Therapeutic Inoculum; Isolation; Pathogen extinction; Within-host recovery of a treated individual may imply a public- dynamics health benefit. If antibiotics shorten the infectious period, the count of infections per infection could 1. Introduction decline [4]. This interpretation follows from SIR Antibiotics are administered routinely to humans, compartment models, where neither the host-removal agricultural/pet animals, and certain plants [1-3]. rate nor the antibiotically-induced recovery rate Most commonly, antibiotic treatment is intended to depends explicitly on within-host pathogen density. control an individual`s bacterial infection [4]. Beyond That is, antibiotics are assumed to reduce duration of concerns about the evolution of resistance [5, 6], use the infectious period and to exert no effect on per- of antibiotics to treat infection presents challenging individual transmission intensity. By extension, questions, including optimizing trade-offs between antibiotics may then reduce pathogen transmission. antibacterial efficacy and toxicity to the treated host [7]. This study asks if antibiotic treatment of an However, antibiotic therapy might, in other cases, infection can have untoward consequences at the increase the expected length of the infectious period. population scale; the paper models an antibiotic's Transitions in host status must often depend on a direct impact on within-host pathogen dynamics and within-host dynamics [8, 11]. As infection progresses, resulting, indirect effects on between-host the pathogen density's trajectory should drive change transmission [8, 9]. in the rate of host removal while ill (e.g., isolation), the rate of recovery from disease, as well as the rate at The model assumes that the antibiotic`s suppression which infection is transmitted [12, 13]. For many of within-host bacterial density extends the average human bacterial infections, an individual can still waiting time for the host`s removal from transmit the pathogen after beginning antibiotic infectiousness via other processes (e.g., physical therapy [14]. Common infections remain isolation or disease mortality). The paper`s focal transmissible for a few days to two weeks [15]. question asks how varying the age of infection when Although not addressed here, sexually transmitted antibiotic treatment begins impacts both the duration disease may persist within a host for months after of disease and the intensity of transmission during the antibiotic therapy has begun [16]. Therapeutic host`s infectious period. When removal equates with reduction in pathogen density might eventually cure disease mortality, the results identify conditions under the host, while allowing the host to avoid isolation, which an antibiotic may simultaneously increase both etc. during treatment [17]. The result might be a survival of an infected individual and the expected longer period of infectious contacts and, number of secondary infections. consequently, increased secondary infections.

1.1 The infectious period This paper assumes that with or without antibiotic Efficacious antibiotics, by definition, reduce within- treatment, a diseased host`s infectious period may be host pathogen density [10]; for some infections, ended by a removal process that depends on within-

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host pathogen density. As a convenience, removal probabilistically [27-29]. At the within-host scale, the includes any event terminating infectious contacts model considers both density-independent and self- with susceptible hosts, prior to the antibiotic curing regulated pathogen growth. The host`s removal rate the disease. Social/physical isolation [18] and host and the infection-transmission intensity will depend mortality are dynamically equivalent removals in that directly on the time-dependent bacterial density. they end the infectious period. The model assumes Pathogen density increases monotonically from time that an antibiotic, by deterring within-host pathogen of infection until antibiotic treatment begins, given growth, increases the expected waiting time for persistence of the host`s infectious state. The removal, but an increase in antibiotic efficacy reduces antibiotic then reduces pathogen density until the host the time elapsing until the host is cured. This is cured or removed prior to completing therapy interaction affects the count of secondary infections; (whichever occurs first). disease reproduction numbers (before and after therapy begins) identify conditions where an Counts of secondary infections will require the antibiotic increases the spread of disease. temporal distribution of infectious contacts, since the transmission probability depends on the time- 1.2 Random encounters: susceptible groups dependent pathogen density [13, 30]. The results When infection is rare, random variation in the explore effects of antibiotics and inoculum size [31] number of contacts between diseased and susceptible on length of the infectious period, disease hosts influences whether the pathogen does or does reproduction numbers, and pathogen extinction. The not spread at the population scale [19, 20]. Therefore, last two results connect logically; the first addresses this paper treats reproduction numbers, i.e., infections mean infections per infection, and the second per infection, as random variables [21]. The concerns the variance in the count of new infections. environment governs social group size, which can affect contacts between infectious and susceptible 2. Within-Host Dynamics: Timing of hosts, and so impact infection transmission [22-24]. Antibiotic Treatment The model asks how the number of hosts per For many bacterial infections of vertebrates, little is encounter with an infectious individual (with the known about within-host pathogen growth [32]. In the product of encounter rate and group size fixed) laboratory, Pseudomonas aeruginosa readily infects impacts the variance in the count of secondary Drosophila melanogaster [33]; the pathogen increases infections; specifically, the paper asks how group size exponentially until the host dies or antibacterial impacts the probability that a rare infection fails to treatment begins [29, 34, 35]. In more complex host- invade a host population [25, 26]. pathogen systems, resource limitation or physical crowding must often decelerate pathogen growth 1.3 Organization within the host, implying self-regulation [36-39]. The model treats within-host pathogen dynamics Numerical results below compare ways in which the deterministically [2]. Removal from the infectious strength of self-regulation interacts with an antibiotic state and between-host transmission are modeled

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to influence duration of infectiousness, and intensity replication and mortality rates per unit density. The of pathogen transmission. latter rate may reflect a nonspecific host immune response [40]; the model does not include explicit

퐵푡 represents the within-host bacterial density at time immune dynamics, to focus on effects of antibiotic

t; 퐵0 is the inoculum size. Antibiotic treatment begins timing and efficacy. Under logistic self-regulation, the

at time 푡퐴 > 0. Table 1 defines symbols used in this per-unit growth rate becomes (푟 − 푐퐵푡) where c paper. represents intra-specific competition. For this case, the within-host density prior to treatment becomes: If the pathogen grows exponentially prior to

푟푡 푟 −푟푡 treatment, 퐵푡 = 퐵0 푒 for ≤ 푡퐴 . The intrinsic 퐵푡 = 푟⁄[푐 + ( − 푐) 푒 ] ; 푡 ≤ 푡퐴 퐵0 growth rate 푟 > 0 is the difference between bacterial

Symbols Definitions Within-host scale 푡 Time since infection; infection age 퐵푡 Bacterial density at time t 퐵0 Inoculum density r Pathogen’s intrinsic rate of increase c Pathogen intraspecific competition ∗ 훾퐴 Bacterial mortality due to antibiotic 푡퐴 Age of infection when antibiotic initiated 휃 Prefactor, pathogen density at time of cure 푡퐶 Age of infection when host cured Individual host scale ℎ푡 Removal rate of infectious host at time t 휙 Removal rate prefactor 휂 Infection severity parameter 퐿푡 Pr[Host infectious at time 푡 ≤ 푡퐶] Between-host scale 휆⁄퐺 Stochastic contact rate, group size G 휈푡 Probability of infection, given contact 휉 Infection susceptibility parameter 푝푡 Probability new infection occurs at time t 풫 Time averaged infection probability 푅1 Expected new infections before 푡퐴 푅2 Expected new infections on (푡퐴, 푡퐶) 푅0 푅1 + 푅2

Table 1: Definitions of model symbols, organized by scale.

where 퐵0 < 푟⁄푐; the inoculum should be smaller infection and initiation of antibiotic therapy. For both

than the ``carrying capacity.'' For the same (퐵0, 푟), the growth assumptions, 퐵푡퐴 represents the within-host self-regulated density, of course, never exceeds the density at initiation of therapy. exponentially growing density between time of

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Most antibiotics increase bacterial mortality [10, 41], equilibrium plasma concentration of the antibiotic is ∗ though some impede replication [37]. When a 퐴 = 퐷퐴⁄푘. bacterial population is treated with an efficacious antibiotic, bacterial density (at least initially) declines Bacterial mortality increases in a decelerating manner exponentially [42-44]. Hence, I assume that a as antibiotic concentration increases [41, 45]. Using a bactericidal antibiotic induces exponential decay of standard formulation [7]:

퐵푡.

훾(퐴푡) = 훾푚푎푥 퐴푡⁄(푎1/2 + 퐴푡); 푡 > 푡퐴 (1) 2.1 Host states

The host becomes infectious at time 푡 = 0, and where 훾(퐴푡) = 훾푚푎푥⁄2 when 퐴푡 = 푎1/2. Applying ∗ ∗ remains infectious until either removed or cured by the quasi-steady state assumption, let 훾퐴 = 훾(퐴 ).

the antibiotic (see Section 2.3). No secondary Since the antibiotic is efficacious, 훾퐴∗ > 푟. infections occur after removal or therapeutic cure, whichever occurs first. Hence, transmission can occur 2.3 Antibiotic treatment duration

during antibiotic therapy, prior to cure. If the host Antibiotic therapy begins at time 푡퐴. During remains infectious at time t, both the probability of treatment, within-host pathogen density declines as

disease transmission (given encounter with a 푑퐵푡⁄푑푡 = − (훾퐴∗ − 푟) 퐵푡. Then: susceptible) and the removal rate depend explicitly on [ ( ∗ ) ( )] ∗ within-host density 퐵푡. 퐵푡 = 퐵푡퐴 푒푥푝 − 훾퐴 − 푟 푡 − 푡퐴 ; 푡 > 푡퐴; 훾퐴 > 푟 (2)

2.2 Antibiotic concentration and efficacy where only 퐵푡퐴 depends on the presence/absence of

Assumptions concerning antibiotic efficacy follow self-regulation. For the exponential case 퐵푡 = ∗ Austin et al. [37]. Given that the host remains 퐵0 푒푥푝[푟푡 − 훾퐴 (푡 − 푡퐴)] after treatment begins. For

infectious at 푡 > 푡퐴, the total loss rate per unit self-regulated pathogen growth:

bacterial density is 휇 + 훾(퐴푡), where 퐴푡 is plasma

∗ concentration of antibiotic, and 훾 maps 퐴 to bacterial 훾퐴푡퐴 푡 푟 푒 ∗ − (훾퐴−푟)푡 퐵푡 = [ 푟 ] 푒 (3) mortality per unit density. 푐푒푟푡퐴 + ( − 푐) 퐵0

Assume that the antibiotic is `dripped' at rate 퐷퐴. Given that the host is not otherwise removed, Plasma antibiotic concentration decays through both antibiotic treatment continues until the host is cured at

metabolism and excretion; let 푘퐴 represent the total time 푡퐶 > 푡퐴. A `cure' means that the within-host

decay rate. Then, 푑퐴푑⁄푑푡 = 퐷퐴 − 푘퐴푡, so that pathogen density has declined sufficiently that the −푘푡 퐴푡 = (퐷퐴⁄푘) (1 − 푒 ), for 푡 > 푡퐴. Antibiotic host no longer can transmit the pathogen; a cure need concentration generally approaches equilibrium faster not imply complete clearance of infection. No host

than the dynamics of bacterial growth or decline [37]. remains infectiousness beyond 푡퐶. In terms of

Then a quasi-steady state assumption implies the pathogen density, 퐵(푡푐) = 퐵0⁄휃, where 휃 ≥ 1. Hence infectiousness terminated by therapy always

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occurs at a within-host density less than (no greater removal rate at any time 푡 strictly increases with

than) the density where the host may first transmit the pathogen density 퐵푡. pathogen. For exponential pathogen growth, we have: The model takes removal as the first event of a ∗ ∗ 훾퐴 푡퐴 + 푙푛휃 퐵0⁄휃 = 퐵0 푒푥푝[푟푡퐶 − 훾퐴 (푡퐶 − 푡퐴)] ⇒ 푡퐶 = ∗ > 푡퐴 (4) nonhomogeneous Poisson process; ℎ푡 is the 훾퐴 − 푟 instantaneous rate of removal at time 푡 [47]. 퐿푡 is the probability that the host, infected at time 0, remains If the cure requires only that 퐵푡 return to the inoculum ∗ ∗ infectious at time 푡 ≤ 푡퐶. Prior to initiation of size, then 휃 = 1, and 푡퐶 = 훾퐴 푡퐴/(훾퐴 − 푟). Instead of defining cure via therapy as a density proportional to therapy:

퐵0, suppose that the host is cured if the within-host 푡 density declines to 퐵(푡 > 푡퐴) = 퐵̃ ≤ 퐵0 . Let 퐿푡 ≡ 푒푥푝 [− ∫ ℎ휏 푑휏] ; 푡 ≤ 푡퐴 (6) 0 ̃ ̃ 휃 = 퐵⁄퐵0. The associated maximal age of infection is 푡̃ = (훾∗ 푡 − 푙푛 휃)⁄(훾∗ − 푟) . 푡̃ depends on 훾∗, 푡 , 퐴 퐴 퐴 퐴 퐴 and (1 − 퐿푡) is the probability the host has been and 푟 as 푡 does, and numerical differences will be 퐶 removed before time 푡. ℎ푡 is the stochastic removal 휂 small unless 퐵0 and 퐵̃ differ greatly. rate at time 푡; assume ℎ푡 = 휙퐵푡 ; 휙, 휂 > 0. The parameter 휙 scales bacterial density to the timescale If pathogen growth self-regulates and 퐵(푡퐶) = of removal. Removal is more/less likely as bacterial

퐵0⁄휃, the host is cured at: density increases/decreases. For 푡 ≤ 푡퐴, ℎ푡 has the form of the Gompertz model for age-dependent ∗ 훾퐴 푡퐴 + 푙푛휃 퐵0 푟푡퐴 ∗ −1 푡퐶 = ∗ − 푙푛 [1 + (푒 − 1)] (훾퐴 − 푟) (5) mortality among adult humans [48]. ℎ푡 saturates 훾퐴 − 푟 푟⁄푐 (휂 < 1), increases linearly (휂 = 1), or accelerates (휂 > 1) with increasing pathogen density, depending 퐵(푡퐴) is smaller under self-regulated growth than under density independent growth; the antibiotic cures on the host-pathogen combination. the host faster under self-regulation. Numerical results below let 휃 = 1, implying a symmetry between the Suppose the pathogen grows exponentially before

time 푡퐴. Then the host remains infectious prior to within-host state at initial infectiousness (퐵0 > ̇ ̇ antibiotic treatment with probability: 0; 퐵 > 0) and the state at 푡퐶 (퐵푡퐶 = 퐵0; 퐵 < 0).

휂 휂 휂 푡 휙퐵 휙퐵 퐿 = 푒푥푝 [− 휙퐵 ∫ 푒휂푟휏 푑휏] = 푒푥푝 [ 0 ]⁄푒푥푝 [ 푡 ] ; 푡 ≤ 푡 (7) 3. Duration of Infectious State 푡 0 0 휂푟 휂푟 퐴

Removal includes any event, other than antibiotic Simplifying: cure, that ends the host's infectious period. Removal

occurs probabilistically and the rate of removal 휙 퐿 = 푒푥푝 [− (퐵휂 − 퐵휂)] ; 푡 ≤ 푡 (8) depends on pathogen density. Noting that removal by 푡 휂푟 푡 0 퐴 mortality becomes more likely with the severity of ``pathogen burden'' [46], the model assumes that the

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퐿(푡 = 0) = 1, and persistence of infection declines as Using Eq (2), we have the probability that 푡 increases. For logistic pathogen growth prior to infectiousness persists to time 푡 during therapy, for treatment, we have: either presence or absence of self-regulation prior to therapy:

푡 휂 푑휏 퐿푡 = 푒푥푝 [− 휙푟 ∫ 휂] (9) 휂 휂 푟 푟휏 휙퐵 휙퐵 0 [푐 + ( − 푐) 푒 ] 퐿 = 퐿 푒푥푝 푡 ⁄푒푥푝 푡퐴 퐵0 푡 푡퐴 [ ∗ ] [ ∗ ] 휂(훾퐴 − 푟) 휂(훾퐴 − 푟) 휙 = 퐿 푒푥푝 [ (퐵휂 − 퐵휂)] ; 푡 > 푡 (12) 푡퐴 − ∗ 푡퐴 푡 퐴 휂(훾퐴 − 푟) For given (퐵0, 푟) the self-regulated density at

푡 ∈ (0, 푡퐴) must be lower than the unregulated

where 퐵푡퐴 > 퐵푡 , and 퐿푡퐴 is given by either Eq (8) or density. Consequently, 퐿푡 under exponential growth cannot exceed the corresponding probability when Eq (9), as appropriate. pathogen growth self-regulates; pathogen self- regulation increases the chance that the host survives The infectiousness-survival probabilities 퐿푡 collect until antibiotic treatment begins. consequences of model assumptions. Delaying initiation of therapy (i.e., increasing 푡 ) increases 퐵 퐴 푡퐴

3.1 Antibiotic therapy: removal vs cure and hence decreases 퐿푡퐴, the probability that During antibiotic treatment, a host has instantaneous infectiousness persists until treatment begins. removal rate: Rephrased, delaying the antibiotic increases the chance that the host is removed (and so stops

휂 ∗ ℎ = 휙퐵 푒−휂(훾퐴−푟)(푡−푡퐴); 푡 > 푡 (10) transmitting infection) before therapy begins. Since 푡 푡퐴 퐴

퐵푡퐴 increases with 푡퐴, the time required for the

where, again, only 퐵푡퐴 depends on the antibiotic to cure the host (푡퐶 − 푡퐴) must increase

presence/absence of self-regulation. The probability with 푡퐴. that the host remains infectious at time 푡, where

푡퐴 < 푡 < 푡퐶, is the probability of entering treatment Furthermore, since increasing 푡퐴 decreases 퐿푡퐴 and in the infectious state, 퐿 , times the probability of 푡퐴 increases (푡퐶 − 푡퐴), then 휕퐿푡퐶⁄휕푡퐴 < 0; the

avoiding removal from 푡퐴 to 푡 (given the state at 푡퐴). probability that the remains infectious until cured Using Eq (10), the probability that the host remains decreases with delayed initiation of treatment. These infectious during treatment is: effects always hold for exponential growth; they hold

for logistic growth when 퐵푡퐴 < 푟⁄푐, the carrying 푡 휂 ∗ 퐿 = 퐿 푒푥푝 [− 휙퐵 ∫ 푒−휂(훾퐴−푟)(휏−푡퐴) 푑휏] ; 푡 > 푡 (11) capacity. 푡 푡퐴 푡퐴 푎 푡 퐴

Increasing bacterial self-regulation moderates, but where 퐵푡퐴 and, consequently, 퐿푡푎 depend on does not reverse, these effects of 푡퐴. For 푟⁄푐 large presence/absence of self-regulation. enough, pathogen density 퐵푡 declines as 푐 increases. 퐴

Then 퐿푡퐴 must increase, and time needed for

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therapeutic cure (푡퐶 − 푡퐴) must decrease. Hence 푒푥푝[−휉퐵푡], where 휉 is the susceptibility parameter.

−휉퐵푡 휕퐿푡퐶⁄휕푐 ≥ 0; increasing the strength of self- Then 푝푡 = 퐿푡 (1 − 푒 ). 휈푡 decelerates with 퐵푡 regulation in pathogen growth never decreases the since infection of a single host saturates with probability that the host is cured therapeutically. propagule number [20, 53, 54]. Note that

휕휈푡⁄휕퐵푡 > 0 and 휕ℎ푡⁄휕퐵푡 > 0. An increase in The model's simple within-host dynamics allows the transmission probability, due to greater within-host rate of removal and (its complement) persistence of pathogen density, is countered by a greater removal the infectious state to depend clearly and explicitly on rate. within-host pathogen density. The dynamics of infection transmission, and so any public-health 4.1 New-infection probabilities: before and during implications, will also depend on within-host treatment pathogen density [49]. New infections occur randomly, both before and after treatment begins. To clarify effects of varying the

4. Transmission timing of therapy, let 푅1 represent the expected

The focal infective contacts susceptible hosts as number of new infections on (0, 푡퐴]; let 푅2 be the

groups. Each group has the same size G; often G = 1. expected number of new infections on (푡퐴, 푡퐶]. For Contacts occur as a Poisson process, with constant simplicity, refer to these respective time intervals as

probabilistic rate 휆⁄퐺, so that the expected number of the first and second period. 푅0 is the expected total

individuals contacted in any period does not depend number of new infections per infection; 푅0 = 푅1 +

on susceptible-host group size. Contact rate is also 푅2.

independent of time and pathogen state 퐵푡. Suppose that N such encounters with the infectious

A contact implies G independent Bernoulli trials, and host occur on some time interval (푡푥, 푡푦). By the the number of new infections, per contact, follows a Poisson’s memoryless property, the times of the binomial probability function with parameters G and encounters (as unordered random variables) are

푝푡. 푝푡 is the conditional probability that any distributed uniformly and independently over (푡푥, 푡푦) susceptible host j acquires the infection, given contact [55]. Uniformity identifies the time averaging for the

at time t. The model writes 푝푡 as a product: 푝푡 = conditional infection probability 푝푡. For the first 퐿푡 휈푡. 퐿푡 weighs ``births'' of new infections upon period, the unconditional probability of infection at

contact [49, 50]. 휈푡 is the conditional probability that contact is 풫1: any susceptible host j is infected at time t given that

the focal host remains infectious, and contact occurs. 1 푡퐴 풫 = ∫ 퐿 (1 − 푒−휉퐵휏 ) 푑휏 (13) 1 푡 휏 Both 퐿푡 and 휈푡 depend on within-host density 퐵푡. 퐴 0

Eq (13) applies to both exponential and self-regulated Given an encounter, the transmission probability 휈푡 assumes a dose-response relationship [30, 51]. growth. For the former case, we have:

Following a preferred model [52], 휈푡 = 1 −

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Let 푋1 be the random count of new infections during 휙퐵 풫 = (푒푥푝 [ 0]⁄푡 ) the first period, and 푋2 be the second-period count. 1 휂푟 퐴 휂 휂 From the time of infection until antibiotic treatment 푡 휙퐵 푡 휙퐵 × (∫ 퐴 푒푥푝 [− 휏 ] 푑휏 − ∫ 퐴 푒푥푝 [− 휏 − 휉퐵 ] 푑휏) (14) 0 휂푟 0 휂푟 휏 begins, 푅1 = 퐸[푋1] = 휆풫1푡퐴 and 푉[푋1] =

푅1[1 + 풫1(퐺 − 1)]. For the second period, 푅2 = For the second period, averaging uniformly yields 풫2, 퐸[푋2] = 휆풫2(푡퐶 − 푡퐴), and the variance of 푋2 is the infection probability after treatment begins. 풫2 has 푅2[1 + 풫2(퐺 − 1)]. the form as Eq (13), with averaging over (푡 − 푡 ). 퐶 퐴 For both exponential and logistic within-host growth, By construction, the expected number of infections we obtain: both before and after antibiotic treatment begins does

not depend on group size G. But each variance of the 휙퐵휂 퐿푡퐴 푡퐴 풫2 = ( ⁄푒푥푝 [ ∗ ]) number of new infections increases with group size. 푡퐶 − 푡퐴 휂(훾퐴 − 푟) 휂 휂 푡퐶 휙퐵휏 푡퐶 휙퐵휏 Finally, the total number of new infections per × (∫ 푒푥푝 [ ∗ ] 푑휏 − ∫ 푒푥푝 [ ∗ − 휉퐵휏] 푑휏) (15) 푡퐴 휂(훾퐴−푟) 푡퐴 휂(훾퐴−푟) infection has expectation 푅0 = 퐸[푋1 + 푋2] =

휆[풫1푡퐴 + 풫2(푡퐶 − 푡퐴)]. The variance of the total where 퐿푡퐴 is given above (and depends on the number of new infections is 푉[푋1 + 푋2] = 푅0 + presence/absence of self-regulation), and 퐵휏 is given (퐺 − 1)[풫1푅1 + 풫2푅2]. by Eq (2). 풫 and 풫 collect effects of within-host 1 2 density, modulated by antibiotic treatment, on Since group size affects only the variance of the between-host transmission of infection. Since the reproduction numbers, any increase G can increase within-host dynamics affects both persistence of the 푃푟[푋1 + 푋2 = 0], the probability of no new infectious state and the probability of transmitting infections, even though 푅0 > 1. No new infections infection upon contact, the strength of self-regulation requires that each 푋푧 = 0; 푧 = 1, 2. The probability of should impact the number of secondary infections per no pathogen transmission at a single encounter is infection. 퐺 (1 − 풫푧) , since outcomes for each susceptible are

mutually independent. Given N encounters in period 4.2 푹 ퟎ z, the conditional probability of no new infections For each of the two periods, the number of infections 퐺 푁 during that period is 푃푟{푋푧 = 0|푁} = [(1 − 풫푧) ] . sums a random number of random variables. Each Unconditionally: element of the sum is a binomial variable with

expectation 퐺풫푧 and variance 퐺풫푧(1 − 풫푧); 푧 = 1, 2. ∞ 퐺 푁 The number of encounters with susceptible hosts is a 푃푟[푋푧 = 0] = ∑[(1 − 풫푧) ] 푃푟[푁] (16) 푁=0 Poisson variable with expectation during the first

period (휆⁄퐺) 푡퐴, and expectation during the second 퐺 Since (1 − 풫푧) < 1, 푃푟[푋푧 = 0] is given by the period (휆⁄퐺) (푡퐶 − 푡퐴). probability generating function for N, evaluated at 퐺 (1 − 풫푧) . From above, N is Poisson with parameter

(휆⁄퐺) 푡퐴 during the first period, and:

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푃푟[푋 = 0] = 푒푥푝[(휆⁄퐺) 푡 ([1 − 풫 ]퐺 − 1)] (17) 1 퐴 1 푡퐴 delayed sufficiently) prevents initial spread of

infection (푅0 < 1).Why does increasing the time For the second period, 푃푟[푋2 = 0] = 푒푥푝[(휆⁄퐺)(푡퐶 − elapsing between infection and initial treatment (or no 퐺 푡퐴) ([1 − 풫2] − 1)]. Each probability of no infection treatment) sometimes reduce the chance that disease increases as G increases. The chance of no new will spread? Why does relaxed pathogen self- infections is, of course, the product of the independent regulation increase this effect? A small 푡퐴 implies a probabilities. low 퐵푡퐴; early treatment maintains a reduced within-

host density and a consequently reduced removal rate 5. Numerical Results for (푡 > 푡퐴). Figures (1) and (2) show results motivating this paper. Consider first how 푅0 varies with 푡퐴, at different The host's chance of being cured, rather than first levels of self-regulation. If antibiotic therapy begins being removed, increases when treatment begins immediately after the host becomes infectious relatively soon after infection. That is, therapy begun

(푡퐴 < 2 in Figure 1c) then 푅0 < 1 in both the at low 푡퐴 more likely cures the host, but (on average) presence and absence of self-regulation; the disease leaves the host infectious longer. The latter effect will likely fail to invade a susceptible population. But maximizes 푅0 at a lower 푡퐴 in both the exponential antibiotics are seldom administered at the onset of and weakly self-regulated examples. infectiousness [3, 39]. A relatively small delay in 푡퐴 allows the infection to spread. That is, reasonably Earlier initiation of treatment must reduce 푅1. For rapid initiation of antibiotic treatment allows 푅0 > 1, exponential and weakly self-regulated pathogen for both self-regulated and unregulated growth before growth, the spread of infection among hosts, for low

푡퐴. As 푡퐴 continues to increase, only strong self- 푡퐴, is due more to transmission during antibiotic regulation produces further, though quickly treatment; 푅0 and 푅2 reach their respective maxima at decelerating, increase in 푅0. More interestingly, for nearly the same 푡퐴 value. For any t, 푡퐴 < 푡 < 푡퐶, the both exponential growth and weak self-regulation, reduction in the infection probability 휈푡 due to the delaying therapy sufficiently leaves 푅0 < 1 again, antibiotic's regulation of within-host pathogen density inhibiting the spread of infection. For the exponential is more than compensated by the increase in 퐿푡, the example, results for larger 푡퐴 equate essentially to no probability that the host remains infectious. The focal

antibiotic therapy: (퐿푡 → 0; 푅2 = 0). 퐴 point is that 푅0 < 1 with no antibiotic therapy, though

푅0 can exceed 1 with therapy In this case relatively early initiation of antibiotic therapy increases the probability the host will be . When removal and therapeutic cure without removal cured (Figure. 1f) but allows the disease to advance depend differently on the within-host dynamics, this among hosts (푅 > 1). But no antibiotic therapy (or 0 non-obvious effect of 푡퐴 can occur.

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Figure 1: Antibiotic therapy can sometimes promote infection transmission. Each plot: solid line is exponential, −6 −5 dashed line is weak self-regulation (푐 = 10 ), dotted line is stronger self-regulation (푐 = 10 ). (a) 푅1 expected

infections before 푡퐴. (b) 푅2 expected infections after antibiotic started. (c) 푅0 total expected infections per infection. (d) Time required for treatment curing the host. (e) Probability host begins antibiotic therapy. (f) Probability host is 4 −5 ∗ cured before removal. 퐵0 = 10 , 푟 = 0.3, 휙 = 10 , 훾퐴 = 0.35, 휂 = 휃 = 1.0, 휆 = 0.2, 휉 = 1.0.

( ) Stronger self-regulation reduces 퐵푡퐴 and so lowers the common 푅0 > 1 , through treatable disease when

removal rate for 푡 > 푡퐴.The host is then more likely antibiotic therapy begins soon after initial infection. to remain infectious until cured. The example with strong self-regulation reduces the time-dependent Figure 2 verifies how increasing susceptible-host

removal rate enough that 푅0 increases monotonically group size increases the probability of no secondary

with increasing 푡퐴. infections, despite independence of 푅0 and group size. Larger groups increase the variance in the total count Consider the exponential case and suppose that of infections per infection. avoiding removal through the antibiotic treatment implies surviving disease; the host is either removed As a result, the probability of no new infections by mortality or cured by the antibiotic. Then, the (pathogen ``extinction'') increases strongly with G.

infected host obviously benefits from therapy. But Even for the 푡퐴 levels maximizing 푅0 in Figure 1, there can be a cost at the among-host scale as the sufficiently large group size (under both exponential

infection spreads. A rare (푅0 < 1), but virulent and weakly self-regulated growth) assures that infection in the absence of antibiotics can become a pathogen extinction is more likely than is spread of infection.

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Figure 2: Susceptible group size and disease-extinction probability. Each plot: 퐺 = 1 (solid line), 퐺 = 4 (dashed line), 퐺 = 10 (dotted line). Left column: Exponential pathogen growth. Right column: logistic growth (푐 = 10−5). Variance in the count of infections per infection increases when groups are larger but encountered less often. Probability of no secondary infections per infection (``extinction'') increases as group size increases.

5.1 Inoculum size, antibiotic efficacy, and 푹ퟎ probability that a host remains infectious until cured The preceding numerical results varied 푡 , and held 퐴 (퐿푡퐶); results were calculated for a smaller and larger

both inoculum size 퐵0 and antibiotic efficacy 푡퐴. ∗ (훾퐴 − 푟) constant.

For these parameter values, 푅0 reaches a maximum at Variation in inoculum size can impact within-host low antibiotic efficacy and small inoculum; the pathogen growth [56-58] and infectiousness [31, 51, pattern holds for both exponential pathogen growth ∗ 59]. Of course, increasing (훾퐴 − 푟) should increase (subplot a) and stronger self-regulation [subplot (e)].

the likelihood of curing, rather than removing, the Subplots (a) and (e) show results for 푡퐴 = 4; the host. surfaces have the same shape for both smaller and

larger 푡퐴 levels.

Figure 3 varies the inoculum 퐵0 and antibiotic ∗ mortality 훾퐴. Dependent quantities are 푅0 and the

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∗ Figure 3: Effects of varying 퐵0 and 훾퐴. (a) 푅0 for 푡퐴 = 4 (smaller 푡퐴); 퐵푡 exponential. 푅0 declines monotonically as

inoculum size 퐵0 increases; 푅0 also declines as antibiotic efficacy increases. (b) Δ푅0 is 푅0 for smaller 푡퐴 minus 푅0

for 푡퐴 = 8 (larger 푡퐴), exponential growth. For medium and larger 퐵0, combined with lower antibiotic efficacy,

earlier treatment generates more secondary infections. (c) Probability treated host is cured, smaller 푡퐴. (d)

Probability host cured, larger 푡퐴. (e) 푅0 for 푡퐴 = 4; 퐵푡 logistic. (f) Δ푅0 is 푅0 for smaller minus 푅0 for larger 푡퐴; 퐵푡 logistic. All plots: 푟 = 0.3, 휙 = 10−6, 휂 = 휃 = 1.0, 휉 = 0.7, and 휆 = 0.4. Plots e and f: 푐 = 10−5, stronger self- regulation.

Why does 푅0 decline as inoculum size increases? Any increasing the likelihood of early removal, a larger

increase in 퐵0 increases 퐵푡 for all 푡 ≤ 푡퐶. The inoculum can decrease the expected number of

removal rate ℎ푡 increases as a result, and the expected secondary infections. Increasing antibiotic efficacy

duration of infectiousness must consequently decline. decreases not only 푅0, but also the sensitivity of 푅0 to For these parameters, where susceptibility 휉 is variation in inoculum size. comparatively large, any increase in the transmission

probability 휈푡 with 퐵푡 does not compensate for the Subplots (c) and (d) of Figure 3 verify, for reduction in duration of infectiousness. Hence, by exponential growth, that the chance of the host

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remaining infectious until cured (i.e., avoiding survival until cured is, overall, much less sensitive to

removal) declines as 퐵0 increases. Note the clear variation in 푡퐴.

quantitative differences between the two 퐿푡퐶-surfaces. ∗ For any (퐵0, 훾퐴)-combination, the host's probability of 5.2 Group size, 푹ퟎ and pathogen `extinction'

remaining infectious is greater for low 푡퐴 (subplot c) Figure 4 shows, for exponential pathogen growth,

than for high 푡퐴 (subplot d). If removal equates with how varying 푅0 and susceptible-group size G affects the probability that the focal host transmits no mortality, so that 퐿푡퐶 becomes the host`s survival probability, the host should ‘prefer' earlier therapy. secondary infections. 푅0 was varied by varying 퐵0. Given G, pathogen-extinction probability never Subplot (b) in Figure 3 shows the difference between increases, and sometimes declines, as 푅0 increases. The decline is greatest when susceptible hosts are 푅0 values for two 푡퐴 levels; Δ푅0 is the difference encountered as solitaries, i.e., when the infection- between 푅0 values of a lesser and greater 푡퐴 level. For ∗ number variance is minimal. Given 푅0, the chance of greater antibiotic efficacy (훾퐴 ≥ 0.5), Δ푅0 ≤ 0. A stronger antibiotic allows earlier start of therapy to pathogen extinction increases strictly monotonically decrease the expected number of secondary as G increases. Each plot in Figure 4 includes regions infections, despite extending the duration of infection where, for sufficiently large group size, 푅0 > 1 but (by reducing the removal rate). pathogen extinction is more likely than not.

∗ 6. Discussion However, for lesser antibiotic efficacy (훾퐴 ≤ 0.4), This paper assumes that any increase in within-host Δ푅0 ≥ 0 for sufficiently large 퐵0. Earlier treatment pathogen density makes removal/mortality due to still increases duration of the infective state (i.e., 퐿푡퐶 infection more probable. Antibiotic therapy reduces increases), and now also increases 푅0. When small inoculum size is combined with lower antibiotic pathogen density and so lowers the instantaneous efficacy, the infected host benefits most, in terms of removal rate. Removal and therapeutic recovery via the chance of being cured, from earlier therapy (low antibiotics interact through their separate functional relationships with pathogen density, and this 푡퐴). interaction governs both duration of infectiousness But the consequence is accelerated spread of and disease-transmission probabilities during the infectious period. Model results show that antibiotic infection at the among-host scale, since Δ푅0 > 0. therapy may sometimes benefit the individual treated Sufficiently strong self-regulation of within-host while imposing costs (additional disease) at the growth can eliminate this effect (subplot f), since host public-health scale [60].

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Figure 4: Probability of no new infections. Each plot shows the probability of no new infection both before and

after antibiotic therapy begins. Chance of extinction shown as function of 푅0 and group size G; note directions of 3 4 ∗ ∗ axes. 푅0 varied by increasing inoculum size 퐵0 form 10 to 2 × 10 . Top row: 훾퐴 = 0.35. Bottom row: 훾퐴 = 0.7.

Left column: 푡퐴 = 4 (lower 푡퐴). Right column: 푡퐴 = 8 (greater). Pathogen extinction less likely as 푅0 increases; extinction always more likely as group size G increases. Increase in extinction due to larger group size increases at

greater 푅0. Each plot shows a substantial region where 푅0 > 1, but probability of pathogen extinction exceeds 0.5. All plots: 푟 = 0.3, 휙 = 10−5.5, 휂 = 휃 = 1.0, 휉 = 0.5, 휆 = 0.1.

The model was motivated by two observations. First, and measures intended to curb the spread of infectious adults and children routinely take antibiotics (often disease has become common during [63]. accompanied by fever-reducing medicine) for upper respiratory infections, and then return to work or The second observation concerns self-medication in school as soon as symptoms begin to subside. chimpanzees (Pan troglodytes). Chimpanzees Sometimes these presentees [61] remain infectious consume a diverse plant diet, and at times select after beginning antibiotic treatment, and they transmit plants with antiparasitic properties [64]. When the associated pathogen [15]. Removal (remaining infested by intestinal nematodes, a chimpanzee will home while infectious) would diminish transmission, withdraw from its social group, and while isolated though at some inconvenience to the focal infective. will eat plants with chemical and/or physical A survey conducted within the last decade suggests characteristics that usually reduce its parasite load that each week nearly 3 × 106 employees in the U.S. [18, 65]. As symptoms moderate, the still-parasitized go to work sick [62], fearing lost wages or loss of individual can return to the group [66] where its employment [17]. Tension between pursuit of income presence may promote transmission of the parasite.

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Plausibly, self-medication increases survival of the This paper asks if variation in the time elapsing first chimpanzee, and indirectly increases the between initial infectiousness and the start of parasitism within the group. The next several antibiotic therapy could affect outcomes at the

subsections suggest a few questions about the way individual and population scale. Hence, 푡퐴 was treated antibiotics may impact linkage between within-host as an independent variable [68]. Extending the model pathogen growth and among-host transmission. could treat the time therapy begins as a positive

random variable. Since 푅0 depends nonlinearly on 푡퐴, 6.1 Bacteria randomization of the delay to treatment should Genetic resistance to antibiotics, whether arising de produce new qualitative predictions. In some

novo or acquired via plasmids, challenges control of applications 푡퐴 might be a symptom-driven function bacterial disease [4, 6, 67, 68]. Phenotypic tolerance of within-host density [7, 39]. Faster within-host presents related, intriguing questions [44]. Some growth, given inoculum size, would presumably genetically homogeneous bacterial populations induce earlier treatment. In this case, the consist of two phenotypes; one grows faster and presence/absence of pathogen self-regulation might exhibits antibiotic sensitivity, while the other grows prove important at both the within-host and between- more slowly and can persist after exposure to an host scales [60]. antibiotic [43]. Phenotypes are not fixed; individual lineages may transition between the two forms [39]. 6.3 Infected host An antibiotic's effect on densities of the two forms This paper neglects immune responses so that the might easily extend the duration of infectiousness, but duration of treatment, given cure by the antibiotic, the probability of transmission, given contact, might depends explicitly on the antibiotic's efficacy and the decline as the frequency of the persistent type age of infection when treatment begins. Incorporating increases. both a constitutive and inducible immune response should be straightforward. The constitutive response 6.2 Antibiotic administration imposes a constant, density-independent mortality If an antibiotic is delivered periodically as a pulse, rate on the pathogen. This response (common to rather than dripped, the therapeutically induced vertebrates and invertebrates) is innately fixed; its mortality of the pathogen can depend on time since effect can be inferred by varying this paper's pathogen the previous administration [44]. Complexity of the growth rate r. Induced immune responses impose impact on the within-host dynamics could then density-dependent regulation of pathogen growth; depend on the difference between the antibiotic's pathogen and induced densities are sometimes decay rate and the pathogen's rate of decline. Some coupled as a resource-consumer interaction [40]. authors refer to an ``inoculum effect,'' suggesting that The timing of antibiotic therapy might be modulated antibiotic efficacy can vary inversely with bacterial so that the current infection might be eliminated just density. That is, the per unit density bacterial slowly enough to prompt a lasting immunological mortality effected by a given antibiotic concentration memory, a ‘’ against future exposure to declines as bacterial density increases [10].

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the same pathogen [69]. Antibiotic dosing might be  Strong pathogen self-regulation increases the optimized similarly [68]. probability that the host remains infectious until therapeutically cured, and decreases the 6.4 Transmission time elapsing between initiation of treatment This paper assumes a constant (probabilistic) rate of and cure. infectious contact with susceptible hosts. The number  Treatment with a less efficacious antibiotic of contacts available may be limited, so that each soon after infection can increase the transmission event depletes the local-susceptible pool. probability of curing the disease, but also can Regular networks capture this effect for spatially increase the expected count of secondary detailed transmission [54], and networks with a infections. However, early treatment with a random number of links per host do the same when strong antibiotic can both increase the social preferences drive transmission [20]. For these likelihood of curing the disease and reduce cases, contact structure of the susceptible population the count of secondary infections. Antibiotics

can affect both 푅0 and the likelihood of pathogen may almost always benefit the individual extinction when rare [28]. treated, but the consequence for public health may not be so uniform. Contact avoidance may sometimes be more important  If hosts are moderately to highly susceptible than contact depletion [12]. If susceptible hosts to infection, duration of the infectious state recognize correlates of infectiousness, they can avoid and the expected count of secondary individuals or locations where transmission is likely infections decline as inoculum size increases. [70]. If antibiotics extend the period of infectiousness  When susceptible hosts are grouped, and and reduce symptom severity, correlates of larger groups are encountered less infectiousness might be more difficult to detect. frequently, the social structuring increases the variance of the secondary infection count 7. Conclusion and, consequently, increases the probability The results indicate several interrelated predictions, of no new infection. summarized here.  The expected count of secondary infections Note that the predictions do not depend on whether is often a single-peaked function of the time removal equates with isolation (usually faster) or host since infection when therapy begins. But mortality (usually slower). sufficiently strong pathogen self-regulation

can imply that 푅0 increases montonically Acknowledgements with time elapsing until therapy begins. Thanks to I-N Wang for both discussing bacteria-  Less efficacious antibiotics may increase the antibiotic interactions. Several readers offered careful, expected count of secondary infections insightful comments on the model’s assumptions. beyond the level anticipated without antibiotic intervention.

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