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Antibiotic Treatment, Duration of Infectiousness, and Disease Transmission

Article Information

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.

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Abstract

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

Keywords

Group size; Infectious contacts; Inoculum; Isolation; Pathogen extinction; Within-host dynamics

Group size articles; Infectious contacts articles; Inoculum articles; Isolation articles; Pathogen extinction articles; Within-host dynamicsarticles

Article Details

1. Introduction

Antibiotics are administered routinely to humans, agricultural/pet animals, and certain plants [1-3]. Most commonly, antibiotic treatment is intended to control an individual`s bacterial infection [4]. Beyond concerns about the evolution of resistance [5, 6], use of antibiotics to treat infection presents challenging questions, including optimizing trade-offs between antibacterial efficacy and toxicity to the treated host [7]. This study asks if antibiotic treatment of an infection can have untoward consequences at the population scale; the paper models an antibiotic's direct impact on within-host pathogen dynamics and resulting, indirect effects on between-host transmission [8, 9].

The model assumes that the antibiotic`s suppression of within-host bacterial density extends the average waiting time for the host`s removal from infectiousness via other processes (e.g., physical isolation or disease mortality). The paper`s focal question asks how varying the age of infection when antibiotic treatment begins impacts both the duration of disease and the intensity of transmission during the host`s infectious period. When removal equates with disease mortality, the results identify conditions under which an antibiotic may simultaneously increase both survival of an infected individual and the expected number of secondary infections.

1.1 The infectious period

Efficacious antibiotics, by definition, reduce within-host pathogen density [10]; for some infections, antibiotics increase host survival. Therapeutic recovery of a treated individual may imply a public-health benefit. If antibiotics shorten the infectious period, the count of infections per infection could decline [4]. This interpretation follows from SIR compartment models, where neither the host-removal rate nor the antibiotically-induced recovery rate depends explicitly on within-host pathogen density. That is, antibiotics are assumed to reduce duration of the infectious period and to exert no effect on per-individual transmission intensity. By extension, antibiotics may then reduce pathogen transmission.

However, antibiotic therapy might, in other cases, increase the expected length of the infectious period. Transitions in host status must often depend on a within-host dynamics [8, 11]. As infection progresses, the pathogen density's trajectory should drive change in the rate of host removal while ill (e.g., isolation), the rate of recovery from disease, as well as the rate at which infection is transmitted [12, 13]. For many human bacterial infections, an individual can still transmit the pathogen after beginning antibiotic therapy [14]. Common infections remain transmissible for a few days to two weeks [15]. Although not addressed here, sexually transmitted disease may persist within a host for months after antibiotic therapy has begun [16]. Therapeutic reduction in pathogen density might eventually cure the host, while allowing the host to avoid isolation, etc. during treatment [17]. The result might be a longer period of infectious contacts and, consequently, increased secondary infections.

This paper assumes that with or without antibiotic treatment, a diseased host`s infectious period may be ended by a removal process that depends on within-host pathogen density. As a convenience, removal includes any event terminating infectious contacts with susceptible hosts, prior to the antibiotic curing the disease. Social/physical isolation [18] and host mortality are dynamically equivalent removals in that they end the infectious period. The model assumes that an antibiotic, by deterring within-host pathogen growth, increases the expected waiting time for removal, but an increase in antibiotic efficacy reduces the time elapsing until the host is cured. This interaction affects the count of secondary infections; disease reproduction numbers (before and after therapy begins) identify conditions where an antibiotic increases the spread of disease.

1.2 Random encounters: susceptible groups

When infection is rare, random variation in the number of contacts between diseased and susceptible hosts influences whether the pathogen does or does not spread at the population scale [19, 20]. Therefore, this paper treats reproduction numbers, i.e., infections per infection, as random variables [21]. The environment governs social group size, which can affect contacts between infectious and susceptible hosts, and so impact infection transmission [22-24]. The model asks how the number of hosts per encounter with an infectious individual (with the product of encounter rate and group size fixed) impacts the variance in the count of secondary infections; specifically, the paper asks how group size impacts the probability that a rare infection fails to invade a host population [25, 26].

1.3 Organization

The model treats within-host pathogen dynamics deterministically [2]. Removal from the infectious state and between-host transmission are modeled probabilistically [27-29]. At the within-host scale, the model considers both density-independent and self-regulated pathogen growth. The host`s removal rate and the infection-transmission intensity will depend directly on the time-dependent bacterial density. Pathogen density increases monotonically from time of infection until antibiotic treatment begins, given persistence of the host`s infectious state. The antibiotic then reduces pathogen density until the host is cured or removed prior to completing therapy (whichever occurs first).

Counts of secondary infections will require the temporal distribution of infectious contacts, since the transmission probability depends on the time-dependent pathogen density [13, 30]. The results explore effects of antibiotics and inoculum size [31] on length of the infectious period, disease reproduction numbers, and pathogen extinction. The last two results connect logically; the first addresses mean infections per infection, and the second concerns the variance in the count of new infections.

2. Within-Host Dynamics: Timing of Antibiotic Treatment

For many bacterial infections of vertebrates, little is known about within-host pathogen growth [32]. In the laboratory, Pseudomonas aeruginosa readily infects Drosophila melanogaster [33]; the pathogen increases exponentially until the host dies or antibacterial treatment begins [29, 34, 35]. In more complex host-pathogen systems, resource limitation or physical crowding must often decelerate pathogen growth within the host, implying self-regulation [36-39]. Numerical results below compare ways in which the strength of self-regulation interacts with an antibiotic to influence duration of infectiousness, and intensity of pathogen transmission.

Btrepresents the within-host bacterial density at time t; Bois the inoculum size. Antibiotic treatment begins at time tA>0. Table 1 defines symbols used in this paper.

If the pathogen grows exponentially prior to treatment, Bt= Boertfor ≤tA. The intrinsic growth rate  is the difference between bacterial replication and mortality rates per unit density. The latter rate may reflect a nonspecific host immune response [40]; the model does not include explicit immune dynamics, to focus on effects of antibiotic timing and efficacy. Under logistic self-regulation, the per-unit growth rate becomes (r-cBt)where c represents intra-specific competition. For this case, the within-host density prior to treatment becomes:

image

Table icon

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

where Bo < r/c; the inoculum should be smaller than the ``carrying capacity.'' For the same (Bor), the self-regulated density, of course, never exceeds the exponentially growing density between time of infection and initiation of antibiotic therapy. For both growth assumptions, BtArepresents the within-host density at initiation of therapy.

Most antibiotics increase bacterial mortality [10, 41], though some impede replication [37]. When a bacterial population is treated with an efficacious antibiotic, bacterial density (at least initially) declines exponentially [42-44]. Hence, I assume that a bactericidal antibiotic induces exponential decay of Bt.

2.1 Host states

The host becomes infectious at time t=0 and remains infectious until either removed or cured by the antibiotic (see Section 2.3). No secondary infections occur after removal or therapeutic cure, whichever occurs first. Hence, transmission can occur during antibiotic therapy, prior to cure. If the host remains infectious at time t, both the probability of disease transmission (given encounter with a susceptible) and the removal rate depend explicitly on within-host density Bt.

2.2 Antibiotic concentration and efficacy

Assumptions concerning antibiotic efficacy follow Austin et al. [37]. Given that the host remains infectious at t > tA the total loss rate per unit bacterial density isimage, where At is plasma concentration of antibiotic, and γ maps At to bacterial mortality per unit density.

Assume that the antibiotic is `dripped' at rate DA. Plasma antibiotic concentration decays through both metabolism and excretion; let KA represent the total decay rate. Then,image , so that , image Antibiotic concentration generally approaches equilibrium faster than the dynamics of bacterial growth or decline [37]. Then a quasi-steady state assumption implies the equilibrium plasma concentration of the antibiotic is image.

Bacterial mortality increases in a decelerating manner as antibiotic concentration increases [41, 45]. Using a standard formulation [7]:

image.

2.3 Antibiotic treatment duration

Antibiotic therapy begins at time . During treatment, within-host pathogen density declines as image

image occurs at a within-host density less than (no greater than) the density where the host may first transmit the pathogen. For exponential pathogen growth, we have:

image

3. Duration of Infectious State

Removal includes any event, other than antibiotic cure, that ends the host's infectious period. Removal occurs probabilistically and the rate of removal depends on pathogen density. Noting that removal by mortality becomes more likely with the severity of ``pathogen burden'' [46], the model assumes that the removal rate at any time t strictly increases with pathogen density Bt.

image

L(t = 0)and persistence of infection declines as t increases. For logistic pathogen growth prior to treatment, we have:

image

3.1 Antibiotic therapy: removal vs cure

During antibiotic treatment, a host has instantaneous removal rate:

image

Using Eq (2), we have the probability that infectiousness persists to time  during therapy, for either presence or absence of self-regulation prior to therapy:

image

The model's simple within-host dynamics allows the rate of removal and (its complement) persistence of the infectious state to depend clearly and explicitly on within-host pathogen density. The dynamics of infection transmission, and so any public-health implications, will also depend on within-host pathogen density [49].

4. Transmission

image

4.1 New-infection probabilities: before and during treatment

image image image image

5. Numerical Results

image image image image

Consider the exponential case and suppose that avoiding removal through the antibiotic treatment implies surviving disease; the host is either removed by mortality or cured by the antibiotic. Then, the infected host obviously benefits from therapy. But there can be a cost at the among-host scale as the infection spreads. A rare (Ro < 1), but virulent infection in the absence of antibiotics can become a common (Ro > 1), through treatable disease when antibiotic therapy begins soon after initial infection.

Figure 2 verifies how increasing susceptible-host group size increases the probability of no secondary infections, despite independence of Ro and group size. Larger groups increase the variance in the total count of infections per infection.

As a result, the probability of no new infections (pathogen ``extinction'') increases strongly with G. Even for the tA levels maximizing Ro in Figure 1, sufficiently large group size (under both exponential and weakly self-regulated growth) assures that pathogen extinction is more likely than is spread of infection.

image

5.1 Inoculum size, antibiotic efficacy, and

image image image

5.2 Group size,  and pathogen `extinction'

Figure 4 shows, for exponential pathogen growth, how varying Ro and susceptible-group size G affects the probability that the focal host transmits no secondary infections. Ro was varied by varying Bo. Given G, pathogen-extinction probability never increases, and sometimes declines, as Ro increases. The decline is greatest when susceptible hosts are encountered as solitaries, i.e., when the infection-number variance is minimal. Given Ro, the chance of pathogen extinction increases strictly monotonically as G increases. Each plot in Figure 4 includes regions where, for sufficiently large group size, Ro > 1 but pathogen extinction is more likely than not.

6. Discussion

This paper assumes that any increase in within-host pathogen density makes removal/mortality due to infection more probable. Antibiotic therapy reduces pathogen density and so lowers the instantaneous removal rate. Removal and therapeutic recovery via antibiotics interact through their separate functional relationships with pathogen density, and this interaction governs both duration of infectiousness and disease-transmission probabilities during the infectious period. Model results show that antibiotic therapy may sometimes benefit the individual treated while imposing costs (additional disease) at the public-health scale [60].

image

The model was motivated by two observations. First, adults and children routinely take antibiotics (often accompanied by fever-reducing medicine) for upper respiratory infections, and then return to work or school as soon as symptoms begin to subside. Sometimes these presentees [61] remain infectious after beginning antibiotic treatment, and they transmit the associated pathogen [15]. Removal (remaining home while infectious) would diminish transmission, though at some inconvenience to the focal infective. A survey conducted within the last decade suggests that each week nearly imageemployees in the U.S. go to work sick [62], fearing lost wages or loss of employment [17]. Tension between pursuit of income and measures intended to curb the spread of infectious disease has become common during pandemic [63].

The second observation concerns self-medication in chimpanzees (Pan troglodytes). Chimpanzees consume a diverse plant diet, and at times select plants with antiparasitic properties [64]. When infested by intestinal nematodes, a chimpanzee will withdraw from its social group, and while isolated will eat plants with chemical and/or physical characteristics that usually reduce its parasite load [18, 65]. As symptoms moderate, the still-parasitized individual can return to the group [66] where its presence may promote transmission of the parasite. Plausibly, self-medication increases survival of the first chimpanzee, and indirectly increases the parasitism within the group. The next several subsections suggest a few questions about the way antibiotics may impact linkage between within-host pathogen growth and among-host transmission.

6.1 Bacteria

Genetic resistance to antibiotics, whether arising de novo or acquired via plasmids, challenges control of bacterial disease [4, 6, 67, 68]. Phenotypic tolerance presents related, intriguing questions [44]. Some genetically homogeneous bacterial populations consist of two phenotypes; one grows faster and exhibits antibiotic sensitivity, while the other grows more slowly and can persist after exposure to an antibiotic [43]. Phenotypes are not fixed; individual lineages may transition between the two forms [39]. An antibiotic's effect on densities of the two forms might easily extend the duration of infectiousness, but the probability of transmission, given contact, might decline as the frequency of the persistent type increases.

6.2 Antibiotic administration

If an antibiotic is delivered periodically as a pulse, rather than dripped, the therapeutically induced mortality of the pathogen can depend on time since the previous administration [44]. Complexity of the impact on the within-host dynamics could then depend on the difference between the antibiotic's decay rate and the pathogen's rate of decline. Some authors refer to an ``inoculum effect,'' suggesting that antibiotic efficacy can vary inversely with bacterial density. That is, the per unit density bacterial mortality effected by a given antibiotic concentration declines as bacterial density increases [10].

This paper asks if variation in the time elapsing between initial infectiousness and the start of antibiotic therapy could affect outcomes at the individual and population scale. Hence,  was treated as an independent variable [68]. Extending the model could treat the time therapy begins as a positive random variable. Since  depends nonlinearly on , randomization of the delay to treatment should produce new qualitative predictions. In some applications  might be a symptom-driven function of within-host density [7, 39]. Faster within-host growth, given inoculum size, would presumably induce earlier treatment. In this case, the presence/absence of pathogen self-regulation might prove important at both the within-host and between-host scales [60].

6.3 Infected host

This paper neglects immune responses so that the duration of treatment, given cure by the antibiotic, depends explicitly on the antibiotic's efficacy and the age of infection when treatment begins. Incorporating both a constitutive and inducible immune response should be straightforward. The constitutive response imposes a constant, density-independent mortality rate on the pathogen. This response (common to vertebrates and invertebrates) is innately fixed; its effect can be inferred by varying this paper's pathogen growth rate r. Induced immune responses impose density-dependent regulation of pathogen growth; pathogen and induced densities are sometimes coupled as a resource-consumer interaction [40].

The timing of antibiotic therapy might be modulated so that the current infection might be eliminated just slowly enough to prompt a lasting immunological memory, a ‘vaccination’ against future exposure to the same pathogen [69]. Antibiotic dosing might be optimized similarly [68].

6.4 Transmission

This paper assumes a constant (probabilistic) rate of infectious contact with susceptible hosts. The number of contacts available may be limited, so that each transmission event depletes the local-susceptible pool. Regular networks capture this effect for spatially detailed transmission [54], and networks with a random number of links per host do the same when social preferences drive transmission [20]. For these cases, contact structure of the susceptible population can affect both  and the likelihood of pathogen extinction when rare [28].

Contact avoidance may sometimes be more important than contact depletion [12]. If susceptible hosts recognize correlates of infectiousness, they can avoid individuals or locations where transmission is likely [70]. If antibiotics extend the period of infectiousness and reduce symptom severity, correlates of infectiousness might be more difficult to detect.

7. Conclusion

The results indicate several interrelated predictions, summarized here.

  1. The expected count of secondary infections is often a single-peaked function of the time since infection when therapy begins. But sufficiently strong pathogen self-regulation can imply that increases montonically with time elapsing until therapy begins.
  2. Less efficacious antibiotics may increase the expected count of secondary infections beyond the level anticipated without antibiotic intervention.
  3. Strong pathogen self-regulation increases the probability that the host remains infectious until therapeutically cured, and decreases the time elapsing between initiation of treatment and cure.
  4. Treatment with a less efficacious antibiotic soon after infection can increase the probability of curing the disease, but also can increase the expected count of secondary infections. However, early treatment with a strong antibiotic can both increase the likelihood of curing the disease and reduce the count of secondary infections. Antibiotics may almost always benefit the individual treated, but the consequence for public health may not be so uniform.
  5. If hosts are moderately to highly susceptible to infection, duration of the infectious state and the expected count of secondary infections decline as inoculum size increases.
  6. When susceptible hosts are grouped, and larger groups are encountered less frequently, the social structuring increases the variance of the secondary infection count and, consequently, increases the probability of no new infection.

Note that the predictions do not depend on whether removal equates with isolation (usually faster) or host mortality (usually slower).

Acknowledgements

Thanks to I-N Wang for both discussing bacteria-antibiotic interactions. Several readers offered careful, insightful comments on the model’s assumptions.

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