Estimating the population attributable fraction for schizophrenia when Toxoplasma gondii is assumed absent in human populations

Accepted Manuscript Title: Estimating the population attributable fraction for schizophrenia when Toxoplasma gondii is assumed absent in human populations Author: Gary Smith PII: DOI: Reference:

S0167-5877(14)00347-X http://dx.doi.org/doi:10.1016/j.prevetmed.2014.10.009 PREVET 3669

To appear in:

PREVET

Received date: Revised date: Accepted date:

5-6-2014 15-9-2014 10-10-2014

Please cite this article as: Smith, G.,Estimating the population attributable fraction for schizophrenia when Toxoplasma gondii is assumed absent in human populations, Preventive Veterinary Medicine (2014), http://dx.doi.org/10.1016/j.prevetmed.2014.10.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Highlights:

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There is evidence that Toxoplasma gondii may be a component cause of schizophrenia We describe a new method for estimating the population attributable fraction (PAF) The PAF for schizophrenia in those exposed to T. gondii is tentatively 21.4% The PAF could be up to 30.6% in some countries although many uncertainties exist

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Estimating the population attributable fraction for schizophrenia when Toxoplasma

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gondii is assumed absent in human populations

3 Gary Smith

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School of Veterinary Medicine, University of Pennsylvania,

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New Bolton Center, 382 West Street Rd., Kennett Square, PA 19348, USA.

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Tel: 01 (610) 925 6312, fax: 01 (610) 925 6830, [email protected]

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ABSTRACT There is increasing evidence that infection with Toxoplasma gondii, a common parasite of people, cats and rodents, is associated with an increased risk of a diagnosis of

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schizophrenia. Although the claim that infection with T. gondii is one of the component

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causes of a diagnosis of schizophrenia remains contentious it is worth asking how

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important a causal association might be if only to inform our attitude to further work on

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the subject. The appropriate measure of importance is the population attributable fraction

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(PAF). The PAF is the proportion of diagnoses of schizophrenia that would not occur in a

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population if T. gondii infections were not present. The assumptions that underlie the

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derivation of the standard formula for measuring the PAF are violated in the specific

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instance of T. gondii and schizophrenia and so the conventional estimation method

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cannot be used. Instead, the PAF was estimated using a deterministic model of

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Toxoplasma gondii infection and schizophrenia occurrence in a hypothetical cohort of

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people at risk of both conditions. The incidence of infection with T. gondii in the cohort

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was assumed to be constant. Under these circumstances, the life-time mean population

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attributable fraction was estimated to be 21.4%, but it could not be ruled out that it could

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be as high as 30.6% or as low as 13.7% given the 95% confidence interval pertaining to

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the point estimate of the OR that was central to the calculation. These estimates (even

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the lowest) are higher than those obtained using the standard method for the same system

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and underscore the importance of understanding the limitations of conventional

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epidemiological formulae.

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Keywords: Toxoplasma, schizophrenia, mathematical model, population attributable

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fraction

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1. Introduction Toxoplasma gondii is a protozoan parasite that people acquire by ingesting contaminated food, water or soil – it is also vertically transmitted in people. Completion

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of the parasite life-cycle depends upon a predator-prey system involving felids (including

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domestic cats) and their rodent prey. The infection is common in cats and small rodents

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(De Craeye et al., 2008; Dabritz et al., 2007a; 2008) and spills over into many of the

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animals that we eat, including pigs, sheep, chickens and hunted wildlife species (Dubey

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and Jones, 2008; Dubey, 2009; Boughattas et al., 2014). Water-bodies and soil are also

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frequently contaminated with infective stages (Bowie et al., 1997, Eng et al., 1999;

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Dabritz et al., 2007b; Dabritz and Conrad, 2010). It is usual in veterinary teaching

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curricula to deal with this zoonotic infection in terms of the damaging neurological

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consequences of vertical infection in people (Torgerson and Mastroiacovo, 2013), the

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deaths attributable to toxoplasmic encephalitis in AIDS patients (Anon., 2006), and the

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epidemiologically interesting (but unusual) epidemics of acute toxoplasmosis (and its

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sequelae) that occur when people ingest oocysts in tap water (Bowie et al., 1997, Eng et

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al., 1999, dr Moura et al., 2006). However, the effect of T. gondii infection on human

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health and well-being may be much more pervasive than this rather narrow focus

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suggests.

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Infection with T. gondii alters rodent behavior in ways that can be interpreted as

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facilitating transmission in a predator-prey cycle: for example, infected rodents are more

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active at night and less neophobic; and uninfected rodents avoid areas contaminated with

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cat urine whereas infected rodents prefer them (Webster, 2001; Webster et al., 2013).

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Similarly, there is evidence that infection with T. gondii in people also results in

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increased activity, decreased reaction times, altered personality profiles and changes in

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sensory perception (Flegr et al. 1996; Flegr et al., 2011; Flegr 2013; and reviewed in

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Webster et al., 2013). In addition, Lafferty (2006) argued that the occurrence of specific

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national cultural traits was associated with the countrywide prevalence of T. gondii.

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Importantly, though, meta-analyses of 38 studies demonstrate that serological

evidence of infection with T. gondii is associated with an increased risk of a diagnosis of

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schizophrenia (OR 2.71; 95% CI 1.93–3.80, Torrey et al., 2007; Torrey et al., 2012) and

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subsequent research from around the world continues to support the finding (eg Wang et

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al., 2011). It should be noted that T. gondii is not the only infection that has been

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associated with a diagnosis of schizophrenia. For example, meta-analyses have

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implicated Chlamydophila psittici, C. pneumoniae, Borna Disease Virus, Human Herpes

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Virus 2 and Human Endogenous Retrovirus W among others (Yolken and Torrey, 2008;

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Arias et al., 2012) and it is possible that those with a diagnosis of schizophrenia simply

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may have an increased risk of various infections as a result of increased hospitalizations

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or disease associated life style factors. Nevertheless, the association between serological

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evidence of infection with T. gondii and a diagnosis of schizophrenia satisfies many of

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the criteria for a causal association (Hill, 1965): strength of association – the odds ratio

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exceeds that for genetic or other environmental factor identified to date (Torrey et al.,

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2007); temporality – the association has been demonstrated in populations in which

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infection with T. gondii is known to have occurred before a diagnosis of schizophrenia

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(Mortensen et al., 2007; Neibuhr et al., 2008; Pedersen et al., 2011); dose-response curve

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– the risk of schizophrenia increases with T. gondii-specific IgG level (Hinze-Selch et

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al., 2007; Pedersen et al., 2011), coherence – there are nascent, plausible hypotheses

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concerning mechanism (Hinze-Selch et al., 2007; Yolken et al., 2009; Henriquez et al.,

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2009; Hayes et al., 2014); and consistency – Mortensen et al. (2007) reported that of ―54

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studies which examined the issue, 49 reported that individuals with schizophrenia and

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other psychoses had a higher prevalence of antibodies to T. gondii when compared with

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controls…‖.

The claim that infection with T. gondii is one of the component causes (sensu

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Rothman and Greenland, 1998) of a diagnosis of schizophrenia is contentious (see

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discussions in Brown and Derkits, 2010; Brown, 2011) and one that is trivialized in the

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popular press (eg Anon., 2012). Nevertheless, it is worth asking how important a causal

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association might be if only to inform our attitude to further work on the subject. The

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appropriate question is what would be the lifetime reduction in the risk of a diagnosis of

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schizophrenia if we could prevent human infection with T. gondii? In other words, what

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is the proportion of diagnoses that would not occur in a population if T. gondii infections

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were not present? This parameter is usually called the population attributable fraction

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(PAF).

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The population attributable fraction estimated using the standard epidemiological

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formula is 13% (Brown and Derkits, 2010). However, it is probably inappropriate to use

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the conventional methodology in the particular case of schizophrenia and T. gondii (see

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Section 2.1) and so this paper suggests an alternative method of estimating the PAF based

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upon a deterministic, mathematical model of schizophrenia and T. gondii infections in a

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cohort of people.

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2. Methods

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2.1. Population attributable fraction (PAF) The PAF is defined as the proportion of cases that would not occur in a

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population if a particular risk factor were eliminated . The usual formula for the PAF is

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derived in Appendix 1 and discussed below

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PAF 

p( RR1  1) 1  p( RR1  1)

(1)

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The formula requires that we know the proportion (p) of the at-risk population exposed to

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the risk factor of interest and also the relative risk (RR1) of disease in the exposed

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fraction of the population over the period of interest. There are a number of problems

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related to the use of this formula (Rockhill et al., 1998) but in the present context the

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most serious of these are (a) that we need to know the life-time relative risk for a

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diagnosis of schizophrenia in those infected with T. gondii (which we do not), and (b)

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that the derivation of the formula (see Appendix 1) assumes that the proportion of the

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population infected with T. gondii remains constant as the cohort ages - and that the

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incidence of schizophrenia in the exposed and unexposed groups remains constant as the

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cohort ages. Neither of these last assumptions is true (Jones et al., 2001; Fromont et al.,

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2009; Bogren et al., 2010; Kodesh et al., 2012). Given these problems, the next section

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describes an alternative method of estimating the PAF based upon a determinsistic,

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mathematical model of schizophrenia and T. gondii infections in a cohort of people.

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2.2. Model of schizophrenia in a cohort of people Consider a cohort of people all born on the same day. This cohort lives under conditions typical of those in the USA or Western Europe. The fate of the cohort is

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followed for 100 years.

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The individuals in this cohort can exist in one or more of five states during their lifetimes: X = serologically negative for T. gondii/ no diagnosis of schizophrenia, Y =

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serologically positive for T. gondii / no diagnosis of schizophrenia, Z1 = serologically

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negative for T. gondii/ diagnosed with schizophrenia, Z2 = diagnosed with schizophrenia

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after becoming serologically positive for T. gondii, Z3 = became serologically positive for

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T. gondii after having been diagnosed with schizophrenia (Figure 1). The equations

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representing the rate of change in the number of people in each of the five states are as

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follows.

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dX  1 X   2 X  a1eb1t X dt

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dY  1 X  ( 2 .OR)Y  a1eb1tY dt

(3)

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dZ1   2 X  1Z1  a2 eb2t Z1 dt

(4)

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dZ 2  ( 2 .OR)Y  a2eb2t Z 2 dt

(5)

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dZ 3  1Z1  a2 eb2t Z 3 dt

(6)

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(2)

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Incidence in this model was defined as the instantaneous per capita rate of infection (for

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T. gondii) or disease onset (for schizophrenia). Thus, θ1 was the incidence (―force of

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infection‖) of T. gondii infection, and θ2 was the incidence of schizophrenia in those

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serologically negative for T. gondii. OR was the odds ratio that measures the increased

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incidence of a diagnosis of schizophrenia in those with serological evidence of infection

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with T. gondii. Mortality was assumed to be an exponentially increasing function of age,

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t years, of the form aiexp(bit) and was assumed to be different in those with (i = 2) and

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without (i = 1) a diagnosis of schizophrenia (Appendix 2, and Table 1).

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2.3. Model development and parameter estimation

The ordinary differential equations (2) to (6) were solved numerically using the

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fourth-order Runge-Kutta algorithm in Berkeley Madonna (Macy and Oster, 2001).

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Parameter values were estimated using the non-linear least-squares methods provided by

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Berkeley Madonna's ―curve fitter‖ option. Methods for estimating mortality and

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incidence are described in detail in Appendix 2. The goodness of fit of the final model to

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an independent data set was evaluated using STATA 12 (StataCorp, 2007).

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2.4. Estimating the Population Attributable Fraction In order to estimate the life-time proportion of schizophrenia cases that would not

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occur if there were no human T. gondii infections, we ran the fully parameterized model

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twice: first with the fitted value (θ1) of the incidence of T. gondii infection and then with

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θ1 = 0. In the first run we obtained the total number of diagnoses (C1) of schizophrenia in

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the cohort given the usual observed risk of infection with T. gondii. In the second run, the

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incidence of infection with T. gondii (θ1) had been set to zero and so we obtained the total

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number of diagnoses of schizophrenia (C2) we would expect in the absence of any human

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infections with T. gondii. By definition PAF = (C1-C2)/C1 (expressed here as a

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percentage).

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The point estimate of the odds ratio that measures the association between

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serological evidence of infection with T. gondii and a diagnosis of schizophrenia is OR =

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2.71 with 1.93–3.80 as the 95% confidence interval (Torrey et al., 2007; Torrey et al.,

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2012). Thus we cannot rule out the possibility that the odds ratio might be as low as 1.93

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or as high as 3.80. For this reason, the calculation of the PAF described above was

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repeated two more times with OR = 1.93 and 3.80 respectively - having first re-estimated

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the parameters of the function describing θ2 (Appendix 2, equations A16 - A18) in each

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case.

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2.5. Potential geographical differences in the PAF The age adjusted prevalence of T. gondii infection in countries in Western Europe

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and North America varies from about 7% (UK) to about 47% (Belgium)(Lafferty, 2006).

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These are all countries where the age-prevalence curve follows the same qualitative

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pattern as the one used here for the USA (eg Proctor and Banerjee, 1994; Kortbeek et al.,

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2004; Birgisdóttir et al., 2006; Pinto et al., 2012) and so equations (2) to (6) could be

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used to examine the plausible range for the PAF given variations in the overall age

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adjusted prevalence of infection in the cohort. All model parameters were as previously

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(OR = 2.71) except that the incidence of infection with T. gondii (θ1) was varied between

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0.001-0.02 per year. This range of values for θ1 generated overall age-adjusted

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prevalences of infection with T. gondii ranging between approximately 5-50%.

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3. Results

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3.1 Parameter values

The fitted survivorship curves for the general population and for those with a

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diagnosis of schizophrenia are shown in Figure 2a. The corresponding parameter values

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are a1 = 0.000055, b1 = 0.086221, and a2 = 0.00357797, b2 = 0.0404366 respectively

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(Table 1).

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The incidence (θ1) of infection with T. gondii was 0.0068/year. The age-

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prevalence curve generated by this value for incidence is shown in Figure 2b. The fitted

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and observed annual cumulative incidence (c(t), equation A16) of a diagnosis of

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schizophrenia is shown in Figure 2c. The corresponding values of the parameters that

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describe how the incidence (θ2) of schizophrenia changes with age (equations A17 and

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A18) in those not infected with T. gondii were γ1 = 0.0004212, γ2 = 19.2524, γ3 =

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2.5262, γ4 = 11.454 (OR set to 2.71, Table 1).

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3.2 Model Validation

We expect that a well fitted model of schizophrenia with appropriate model

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architecture should be able to generate an age-specific prevalence curve for schizophrenia

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that describes the observed data well. The calculated age-specific prevalence curve for

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schizophrenia is shown in Figure 3a together with data points taken from Proctor et al.,

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(1998), Wu et al.,(2006), Chien et al. (2004), Saha et al., (2005), Kodesh et al.( 2012)

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and Morgan et al.,(2012). We emphasize that these observed data points were not used in

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any of the fitting procedures described above. Based upon the graph shown in Figure 3a,

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and considering the large variation of the age-specific data points, the calculated

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prevalence curve appears to describe their general behavior rather well. More formally

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though, in evaluating the goodness of fit of the model to this independent data set we

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need to investigate to what extent a non-random structure is evident in the residual errors

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(the difference between the observed prevalence of a diagnosis of schizophrenia at each

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age and the values generate by the model). A random error structure is consistent with a

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good fit. A plot of residual errors (Figure 3c) revealed two outliers (one at age 20 and one

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at age 30). A review of the data suggested no a priori reason to exclude these outliers so

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two analyses were done: with and without the outliers. Without the outliers, the mean

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residual error was 0.00071 (variance = 0.0000077). With the outliers, the mean residual

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error was 0.0013 (variance = 0.000014). A skewness/kurtosis tests for normality (sktest,

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STATA12) without the outliers provided no reason to reject the null hypothesis that the

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distribution of the residuals was random (p = 0.434). However, the same test when

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applied with the outliers suggested that the distribution of the residuals was skewed (p =

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0.0138) but not kurtotic (p = 0.12). In the absence of other independent data on

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prevalence the goodness of fit analysis is indeterminate. One possible interpretation is

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that the model slightly underestimates the prevalence of schizophrenia but especially over

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those age classes (20-30 years) when the incidence of schizophrenia is changing most

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rapidly. Nevertheless, the model does provide a plausible explanation for why the

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prevalence of a diagnosis of schizophrenia declines in the older age classes: this is a

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result of the higher mortality of those with a diagnosis of schizophrenia.

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3.2 The population attributable fraction (PAF) The PAF was 21.4% at the point estimate value of the odds ratio (OR = 2.71) that measures the increased incidence of a diagnosis of schizophrenia in those with

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serological evidence of infection with T. gondii (Figure 3b). At the lower bound of the

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95% confidence interval (OR = 1.93) the PAF was 13.7%, and, at the higher bound of the

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95% confidence interval (OR = 3.80) the PAF was 30.6%.

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The change in PAF as the national age-adjusted prevalence of T. gondii increases from 5-50% is shown in Figure 3d. It should be noted that these calculations apply only

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to those countries in Western Europe and North America in which the incidence T. gondii

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infection is independent of age. As shown in Figure 3d, the PAF increases from about 5-

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40% over the range considered.

d 4. Discussion and Conclusions

―Schizophrenia is a disabling group of brain disorders characterized by symptoms

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such as hallucinations, delusions, disorganized communication, poor planning, reduced

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motivation, and blunted affect‖ (Saha et al., 2005): the disease accounts for about 2.5%

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of all health care expenditures in the USA ($50 -62 billion annually) (Chien et al., 2004;

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Wu et al., 2005). A wide range of metabolic, familial, social and environmental risk

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factors for schizophrenia have been examined (reviewed by Brown, 2011; Torrey et al.

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2012) including infections. There is increasing interest in investigating T. gondii as one

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of the component causes of schizophrenia and a number of experimental models have

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been suggested (eg Webster et al., 2013). The purpose of the present paper was to assess

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the potential importance of T. gondii as one of the components of the etiology of

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schizophrenia by estimating the proportion of diagnoses of schizophrenia that would not

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occur if the human population could be protected from infection – and, in doing so, to

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illustrate an alternative to the conventional methodology for estimating the PAF

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The estimated PAF was 21.4%, but we cannot rule out that it might be as low as 13.7% or as high as 30.6% given the 95% confidence interval pertaining to the point

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estimate of the OR that was central to the calculation. These estimates of the PAF are

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higher than those derived using the conventional formula (Brown and Derkits, 2010), but

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their absolute values should be treated very cautiously. The estimation method took

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explicit account of the non-linearities (dependence on age and time) in the system, which

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the conventional methodology does not. These non-linearities include age dependent

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changes in the prevalence of T. gondii infection, in the incidence of a diagnosis of

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schizophrenia and, also, the time dependence of relative risk. This latter is often ignored,

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but should not be when assessing the reduction in the lifetime (100years) reduction in risk

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of a diagnosis of schizophrenia. However, these non-linearities also present formidable

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estimation problems as indicated by the uncertainties with regard to the goodness of fit of

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the final model and the estimates of PAF so obtained should be regarded as preliminary.

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The concept of the PAF originated with the work of Levin (1953) who was

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interested in lung cancer. Many applications since then have focused on similar

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conditions in which assumptions regarding constant incidence of disease and unvarying

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exposure to risk factors over the period of interest can more easily be justified. There are

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many instances though in which the risk factor of interest is itself an infectious disease -

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in which case the proportion exposed will not be constant. Veterinary examples include

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the development of lymphoma in cats infected with FIV (Magden et al., 2011), secondary

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bacterial infections in horses infected with Equine Herpes virus -1 or in cattle infected

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with Parainfluenza-3 virus, and myocarditis secondary to infection with canine

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parvovirus, encephalomyocarditis virus or equine infectious anemia virus (Merck

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Veterinary Manual, 2010). However, in all cases, whether one is using the conventional

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methodology to estimate the PAF or the modelling approach described here, the

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overriding assumption is that the increased risk associated with exposure to the risk factor

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has been measured without bias (Rothman and Greenland, 1998) - one more reason to

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treat the estimates of the PAF for schizophrenia reported here with caution.

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Preventing human infection with T. gondii will be difficult. There are no approved vaccines for use against T. gondii infection in people, although there are some promising

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experimental candidates (Dziadek et al., 2012; Chen et al., 2014; Tanaka et al., 2014).

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Similarly, there are no approved vaccines for use in food animals and only one in cats

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(Zhang et al., 2013) - and there remains the problem of oocysts shed by wild felids. There

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are no available drugs that are completely effective against T. gondii infections,

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especially persistent tissue cysts in hosts (Chen et al., 2014), although, interestingly,

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some antipsychotic drugs used to treat schizophrenia inhibit T. gondii development

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(Goodwin et al., 2011). Recommendations about behaviors intended to prevent human

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infection with T. gondii stress safe food handing and cooking (to reduce the risk of

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infection from tissue cysts) and environmental precautions (to reduce the risk of infection

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from oocysts in water or soil) (CDC, 2014), and there is evidence that diets involving

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less or no meat reduce the risk of infection (Roghman et al., 1999).

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In a broader veterinary context, we may be doing our students (and their future clients) a disservice by prefacing our remarks about Toxoplasma infections by saying that

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―normally the infection in animals and humans is asymptomatic‖ (Zheng et al., 2013).

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This is true, but T. gondii is such a common infection that adverse sequelae in even a

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small proportion of infections translates into a large absolute number of cases. For

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example, it is estimated that schizophrenia affects a minimum of 0.5% of the US

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population (Wu et al., 2005). Were infection with T. gondii, in fact, one of the component

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causes of a diagnosis of schizophrenia that would amount to over 335,000 potentially

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preventable cases of schizophrenia in the USA over a single human lifetime (assuming a

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PAF of 21.4%).

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Finally, it should be noted that the calculations presented here apply only to

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countries in which the age-specific prevalence of infection with T. gondii increases

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almost linearly with age. There are many countries (or particular socio-economic classes)

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in which the age-specific prevalence of infection rises rapidly to an asymptotic value by

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late adolescence or early adult hood - the age-group in which the incidence schizophrenia

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is greatest (eg Bahia-Olivera et al., 2003; Fernandes et al., 2009; Pappas et al., 2009).

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Perhaps, this offers a partial explanation for the observation in Saha et al. (2005) that

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―when a broad definition of schizophrenia was used, the developing countries had higher

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incidence rates than the developed countries‖.

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Acknowledgements

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This research was made possible by support from the University of Pennsylvania School

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of Veterinary Medicine (which played no role in the design or execution of this study).

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Technical advice from Dr Chris Rorres was much appreciated.

332 Appendix 1

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A.1. Population Attributable Fraction

The population attributable fraction (PAF) is defined as the proportion of cases

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that would not occur in a population if a particular risk factor were eliminated. The

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derivation of the standard formula is described in detail to illustrate the assumptions upon

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which the derivation depends.

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Consider a source population of N people. In this source population there is a

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cohort of at-risk individuals R1 exposed to Risk Factor 1. The incidence of disease (ie the

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instantaneous, per capita rate of disease onset) in this cohort is a constant value , θ1. It

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follows that the rate of change in the number of people still at risk of disease is

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dR1  1 R1 dt

345

which has a standard solution

346

R1 (t )  R1 (0)e 1t

347

where t is the period of observation.

348

The risk of disease (ie the cumulative incidence) c1(t) over the interval, t, is therefore

Ac ce p

341

c1 (t )  349

( R1 (0)  R1 (t ))  1  e 1t R1 (0)

(A1)

(A2)

(A3)

17

Page 18 of 48

350

Using the same argument, for a cohort of R0 individuals from the same source population

351

not exposed to Risk Factor 1, we get

352

And so the relative risk of disease given exposure to Risk Factor 1 is

c1 (t ) 1  e 1t  c0 (t ) 1  e 0t

(A5)

us

RR1  354

(A4)

cr

353

( R0 (0)  R0 (t ))  1  e 0t R0 (0)

ip t

c0 (t ) 

Note that the relative risk is a function of the period of observation (t) and declines as the

356

value of t increases (Smith, 2013). A relative risk measured over an observation period of

357

(say) five years will be different from the life-time relative risk. Now consider a source population of N people, where a proportion, p, is exposed

M

358

an

355

to Risk Factor 1, and the remaining fraction (1-p) is not exposed to the risk factor. The

360

number of cases (C1) occurring over the interval, t, is

361

C1  pN (1  e 1t )  (1  p) N (1  e 0t )

362

If that Risk Factor 1 were removed, the number of cases (C0) occurring over the interval,

363

t, in the same source population would be

364

C 0  N (1  e 0t )

365

By definition, the population attributable fraction (PAF) is

366

C1  C 0 pN (1  e 1t )  (1  p) N (1  e 0t )  N (1  e 0t ) PAF   C1 pN (1  e 1t )  (1  p) N (1  e 0t )

367

If we divide top and bottom by N (1  e 0t ) and remember the formula for the relative

368

risk (equation A5) we get

Ac ce p

te

d

359

(A6)

(A7)

(A8)

369

18

Page 19 of 48

p( RR1  1) 1  p( RR1  1)

370

PAF 

371

Because the relative risk changes with period of observation (see equation A5), we need

372

to know the lifetime relative risk if we wish to estimate the PAF over a period equal to

373

the average lifetime of an individual. Also note that it has been assumed that the

374

proportion (p) exposed to the risk factor remains constant and the incidence of the disease

375

of interest is also constant.

us

cr

ip t

(A9)

376

378

an

377 Appendix 2

381

Mortality rates

The mortality experienced by members of the cohort modelled by equations (1) to

d

380

M

379

(6) (main text) will depend upon whether or not they are infected with T. gondii and

383

whether or not they have a diagnosis of schizophrenia. Those who are seropositive for T.

384

gondii will experience a higher death rate than those who are seronegative, and those

385

with a diagnosis of schizophrenia will experience a higher death rate than those without

386

such a diagnosis (see Mortality of those without a diagnosis of schizophrenia). There is

387

also evidence that the death rate in those who have a diagnosis of schizophrenia is higher

388

in that subset also seropositive for T. gondii (Dickerson et al., 2007). However, it is

389

argued below that the only differences in mortality large enough to matter with respect to

390

the measuring the PAF are the very substantial differences that exist between those with

391

and without a diagnosis of schizophrenia - irrespective of their T. gondii infection status.

Ac ce p

te

382

392

19

Page 20 of 48

393 394

Mortality of those without a diagnosis of schizophrenia: We assumed that the mortality of those without a diagnosis of schizophrenia was approximated by the mortality of the general population of the USA irrespective of

396

whether or not they were serologically positive for T. gondii. Two conditions needed to

397

be satisfied for this assumption to be justified. The first was that removing the mortality

398

associated with a diagnosis of schizophrenia made little difference to the survivorship

399

curve for the general population. This condition was satisfied because the substantial

400

additional mortality associated with a diagnosis of schizophrenia (see below) was offset

401

by the very low prevalence of the disease in the general population (0.5-0.8%) (Chien et

402

al., 2004; Saha et al., 2005; Wu et al., 2006; Phanthunane et al., 2010; Morgan et al.,

403

2012). The second condition was that the additional mortality associated with T. gondii

404

could also be neglected. In this case, we argued that although the prevalence of T. gondii

405

is relatively high (approximately 22.5% overall in the USA, Jones et al., 2001) the

406

additional mortality attributable to the infection is very low. Deaths due to T. gondii are

407

conventionally divided into those attributable to congenital toxoplasmosis - these

408

generally occur in the first year of life - and those attributable to acquired toxoplasmosis

409

(a life-time risk). The incidence of congenital toxoplasmosis is variously estimated to be

410

between 1–10 cases per 10,000 live births and approximately 10% of these will die with

411

the first year of life (Havelaar et al., 2007). This means that in a cohort of 100,000 (say)

412

people we can expect a maximum of 10 deaths due to congenital toxoplasmosis in the

413

first year of the life of the cohort. Deaths attributable to T. gondii infections acquired

414

after birth will also occur. For example, serological evidence of infection with T. gondii

415

is associated with an increased risk of traffic accidents (Yereli et al., 2006) and brain

Ac ce p

te

d

M

an

us

cr

ip t

395

20

Page 21 of 48

cancer (Vittecoq et al., 2012; Thomas et al., 2012), and toxoplasmic encephalitis and

417

other complications of T. gondii infection have been an important, if variable, cause of

418

illness and death among immune-compromised patients (Edvinsson et al., 2009). For

419

example, Jones et al. (2002) reported that in 1992 there were 1326 Toxoplasmosis related

420

deaths in the USA among those also infected with HIV. By 1998, this number had

421

decreased to 324. A little later, Scallen et al. (2011), using national estimates of inpatient

422

deaths from NIS (2000-2006) and ICD-9-CM code 130 (Toxoplasmosis) estimated that

423

there were an average 327 deaths annually in the USA attributable to T. gondii infections

424

(see also Hoffman et al., 2012). The decline in the absolute number of deaths attributable

425

to toxoplasmic encephalitis despite little or no change in the incidence of HIV infection is

426

a consequence of improved treatment regimens (Béraud et al., 2009). However, even if

427

we were to choose the highest number of death due to toxoplasmic encephalitis, prorated

428

for population size we would expect less than one person in a cohort of 100,000 to die of

429

toxoplasmic encephalitis. It follows that the total number of deaths directly caused by T.

430

gondii in the entire cohort of 100,000 people will probably be between 10 and 11.

431

Therefore despite the fact that T. gondii is a common infection, the absolute number of

432

deaths attributable to the infection is so small that it is not necessary here take them into

433

account. The values of the parameters (a1 and b1) which define the mortality of those

434

without a diagnosis of schizophrenia were estimated by fitting a Gompertz survival

435

model (Lenart, 2012) to survivorship data for the US population as a whole (Arias, 2011).

436

The fitted model was as follows.

Ac ce p

te

d

M

an

us

cr

ip t

416

437 438

dP(t )  a1eb1t P(t ) dt

(A10)

21

Page 22 of 48

439

Here, P(t) was the number of people in the cohort surviving to age t years.

440 441

Mortality of those with a diagnosis of schizophrenia Schizophrenia is associated with a substantial increase in mortality (Crump et al.,

443

2013; Gale et al., 2013). The median Standardized Mortality Ratio for all-cause mortality

444

in those with schizophrenia compared with the general population is of order 2.58 (Saha

445

et al., 2007). This means that schizophrenia is associated with a 10-25 year shorter life

446

expectancy compared with the general population (Suvisaari et al., 2013; Kodesh et al.,

447

2012). The excess mortality is greatest amongst the younger age groups (Simpson and

448

Tsuang, 1996; Kodesh et al., 2012). To my knowledge there is no published survivorship

449

curve for those with a diagnosis of schizophrenia. However, it is possible to construct a

450

survivorship curve based upon the age-dependent relative risks of death for those with a

451

diagnosis of schizophrenia that are recorded in Figure 5 of Kodesh et al., (2012). Kodesh

452

et al., (2012) based their work on the registry of a 2 million member Israeli health

453

maintenance organization with over 8,000 diagnoses of schizophrenia over a 6 year

454

period. We imagined a hypothetical cohort consisting only of individuals with a

455

diagnosis of schizophrenia. The number still alive (Z(t)) at age t was

457

cr

us

an

M

d

te

Ac ce p

456

ip t

442

Z (t )  Z (t  1)(1  cz (t ))

(A11)

458 459

where the cumulative incidence (ie risk) of death over the interval t-1 to 1 was

460

represented by cz(t). In order to construct a survivorship curve for the hypothetical cohort

22

Page 23 of 48

461

of individuals with a diagnosis of schizophrenia we needed to estimate the successive

462

values of cz(t) using

463

cz t   RR (t )cg (t ) ,

(A12)

ip t

464

cr

465

where RR(t) was the relative risk of death for those with a diagnosis of schizophrenia

467

between ages t-1 and t (obtained from Kodesh et al., 2012), and cg(t) was the cumulative

468

incidence (ie risk) of death over the same age interval in those without a diagnosis of

469

schizophrenia. The prevalence of schizophrenia in the population studied by Kodesh et

470

al. (2012) was small (0.5%) so we assumed cg(t) was the same as for the general

471

population. Given a Gompertz survivorship curve (Lenart, 2102), it followed that

472

a  exp  1 (1  exp(b1t ) P(t  1)  P(t )  b1  c g (t )  1 P(t  1) a  exp  1 (1  exp(b1 (t  1)  b1 

(A13)

473

Ac ce p

te

d

M

an

us

466

474

(all parameters as defined for equation A10). The successive values of cg(t) obtained

475

using equation (A13) allowed us to calculate successive values of cz (using equation

476

A12), which, in turn, allowed us to construct a survivorship curve for the hypothetical

477

cohort (using equation A11) to which the following model was be fitted

478 479

dZ (t )  a2 eb2t Z (t ) dt

480

and the values if a2 and b2 estimated.

(A14)

23

Page 24 of 48

481 482 483

Incidence of infection with T. gondii (θ1) The incidence of infection with T. gondii was the assumed to be the same in those with and without a diagnosis of schizophrenia and did not vary with age. This incidence

485

(θ1) was estimated by fitting the following model to T. gondii age-prevalence data for the

486

USA (Jones et al., 2001).

cr

ip t

484

dX  1 X  a1 exp(b1t ) X dt dY  1 X  a1 exp(b1t ) X dt

an

488

us

487

M

489

(A15)

Equations (A15) describe a cohort of people all of the same age (t). The number of

491

people susceptible to infection with T. gondii was represented by X and the number of

492

people already infected was represented by Y. The values of the parameters, a1 and b1,

493

which define the age-dependent mortality of the uninfected and infected fractions had

494

been previously estimated using equation (A10).

496 497

te

Ac ce p

495

d

490

Incidence of a diagnosis of schizophrenia (θ2) The incidence of a diagnosis of schizophrenia is age-dependent (Bresnahan et al.,

498

2000; Bogren et al., 2010; Okkels et al., 2013). Among those serologically negative for

499

T. gondii this incidence is represented in the main text equations (2) to (6) by θ2. Among

500

those serologically positive for T. gondii the incidence is assumed to be estimated by

501

θ2.OR, where OR is the odds ratio measuring the increased incidence of schizophrenia in

502

those with serological evidence of infection with T. gondii (OR = 2.71, Torrey et al., 24

Page 25 of 48

2007; Torrey et al., 2012). The assumption that the incidence of a diagnosis of

504

schizophrenia in those serologically positive for T. gondii can be obtained by multiplying

505

θ2 by OR requires some explanation. Smith (2013) argued that an odds ratio was an

506

unbiased estimator of incidence rate ratio provided the disease of interest was in an

507

endemic steady state and the odds ratio was obtained in a case control study in which the

508

controls had been selected by density sampling. In density sampling the controls are

509

selected longitudinally throughout the course of the study as the cases are identified

510

(Pearce, 1993). Nowadays, most case control studies involve density sampling

511

(sometimes called ‗risk set sampling‘ or ‗sampling from the study base‘). Similar

512

arguments apply to certain cross-sectional studies that present evidence of association in

513

terms of odds ratios - for example, if the system is at equilibrium, subject selection is not

514

related to disease status, and the exposed and unexposed subjects differ only with respect

515

to the incidence of disease - Smith (2013). It follows that if we know (or can estimate) the

516

incidence (θ2) of schizophrenia in those not yet infected with T. gondii, the incidence of

517

schizophrenia in those with infected with T. gondii is θ2 multiplied by the relevant odds

518

ratio.

cr

us

an

M

d

te

Ac ce p

519

ip t

503

The literature on the incidence of schizophrenia approximates incidence (the

520

instantaneous per capita rate of diagnosis) by the cumulative incidence (c(t)) for each

521

successive year of age (Bresnahan et al., 2000; Bogren et al., 2010; Okkels et al., 2013).

522

The cumulative incidence is the risk of a diagnosis of schizophrenia over each successive

523

year and is estimated as the number of new cases of schizophrenia diagnosed in each year

524

age group divided by the number of people at risk at the beginning of the corresponding

525

12 month period. Using the population variables and parameters defined in main text

25

Page 26 of 48

526

equations (2) to (6), the cumulative incidence for each successive year of age could be

527

calculated as

528 c(t ) 

( 2 .OR)Y   2 X (X  Y)

ip t

529

(A 16)

cr

530

Here θ2, X and Y were all functions of age (t). The incidence of schizophrenia rises to a

532

maximum value in the early twenties and then declines (Figure 2c). The age incidence

533

curve is asymmetrical and preliminary work showed θ2 was best represented in equations

534

(2) to (6) (main text) and (A16, above) as an asymmetrical Gaussian function of age, t.

535

Thus

M

an

us

531

 2   1e

540 541

2 5 2

Ac ce p

538 539

( t  2 ) 2

(A17)

te

537



d

536

where

 (t   2 )  ( 3   4 )  ( 4   3 ) * tanh   3   5  2

(A18)

542

Here γ1 is the maximum value of θ2, γ2 is the age (in years) at which that maximum value

543

occurs, and γ3 and γ4 determine the variance and degree of asymmetry of the age-

544

incidence curve. The value of θ2 was estimated by fitting the full model together with

545

equation (A16) to the age-dependent cumulative incidences reported in Bresnahan et al.

546

(2000), Bogren et al. (2010) and Okkels et al. (2013). Sex differences in the age-

26

Page 27 of 48

547

dependent cumulative incidences of a diagnosis of schizophrenia are reported by some

548

but not by others ((Bresnahan et al., 2000; Bogren et al., 2010; Okkels et al., 2013) and

549

were ignored here.

551

ip t

550 References

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us

553

cr

552

Toxoplasmosis AIDS Education and Training Centers (AETC) National Resource

555

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Anon., 2012. ―The Approval Matrix‖, New York Magazine, February 20th-27th, 2012.

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Arias, E., 2011. United States Life Tables, 2007. National Vital Statistics Reports. 59(9)

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Jongert, E., 2008. Prevalence of Toxoplasma gondii infection in Belgian house

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Flegr, J. 2013. Influence of latent Toxoplasma infection on human personality,

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Gale, C. R., Batty, G. D,. Osborn, D. P., Tynelius, P., Whitley, E,. Rasmussen, F., 2012.

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880 Table 1 Definition and parameter value

N

Number of people in the cohort (N = X+Y+Z1+Z2+Z2)

θ1

The incidence of infection with T. gondii (0.0068/year)

θ2

The incidence of a diagnosis of schizophrenia in those not infected with T. gondii γ1 = 0.0004212, γ2 = 19.2524, γ3 = 2.5262, γ4 = 11.454

Odds ratio measuring the increased incidence of schizophrenia in those with

us

OR

ip t

Symbol

cr

881 882

serological evidence of infection with T. gondii (2.71, 95% ci 1.98-3.80) Parameters defining the instantaneous per capita mortality in those without a diagnosis

an

a1, b1

of schizophrenia (a1 = 0.000055, b1 = 0.086221) a2, b2

Parameters defining the instantaneous per capita mortality in those with a diagnosis of

M

schizophrenia (a2 = 0.003578, b2 = 0.040437)

Survivors in a cohort representative of the general population alive at age t.

S(t)

Survivors in a cohort representative of those with a diagnosis of schizophrenia alive at age t.

Cumulative incidence (risk) of death over the age interval t to t+1 in the general population

Cumulative incidence (risk) of death over the age interval t to t+1 in those with a

Ac ce p

cs(t)

te

cg(t)

d

P(t)

diagnosis of schizophrenia

c(t)

RR(t)

Cumulative incidence of schizophrenia over the age interval t to t+1 Relative risk of death over the age interval t to t+1 in those with a diagnosis of schizophrenia

883 884

43

Page 44 of 48

Figure Captions

886

Figure 1. Flow chart of the Toxplasma/Schizophrenia model described by equations (2)

887

to (6). (X = serologically negative for T. gondii/ no diagnosis of schizophrenia, Y =

888

serologically positive for T. gondii / no diagnosis of schizophrenia, Z1 = serologically

889

negative for T. gondii/ diagnosed with schizophrenia, Z2 = diagnosed with schizophrenia

890

after becoming serologically positive for T. gondii, Z3 = became serologically positive for

891

T. gondii after having been diagnosed with schizophrenia).

us

cr

ip t

885

892

Figure 2. (a) The fitted (line) and observed survivorship curves for the general US

894

population (●) and for those with a diagnosis of schizophrenia (○); (b) The fitted and

895

observed age-specific prevalence of human T. gondii infection in the USA; (c) The fitted

896

and observed age-specific cumulative incidence of a diagnosis of schizophrenia (●)

897

males, (○) females;

M

d te

898

an

893

Figure 3. (a) The estimated (line) and observed (●) age-specific prevalence of

900

schizophrenia (data for prevalence from Proctor et al., 1998; Wu et al., 2000; Chien et

901

al., 2004; Saha et al., 2005; Kodesh et al., 2012; Morgan et al., 2012); (b) The cumulative

902

number of diagnoses of schizophrenia in a model cohort of 1000 people at risk of

903

infection with T. gondii (solid line) compared with the number of diagnoses in a similar

904

cohort at no risk of infection with T. gondii (dashed line); (c) The residual errors between

905

the observed and predicted prevalence of schizophrena (bin size 0.002); (d) The

906

estimated change in Population Attributable Fraction as the overall age-adjusted

907

prevalence of infection with T. gondii increases.

Ac ce p

899

44

Page 45 of 48

us

cr

i

Figure

X

Y

Ac

μ1

ed

θ1

θ2

ce pt

μ1

M an

With a diagnosis of schizophrenia

Z1

μ2

Not infected with T. gondii

θ1

Z3

θ2.OR

Z2

μ2

μ2

Infected with T. gondii

Page 46 of 48

Ac

ce pt

ed

M an

us

cr

i

Figure

Page 47 of 48

Reduction in lifetime risk 100(956-751)/956 = 21%

ce pt Ac

Frequency

ed

M an

us

cr

i

Figure

Residual errors

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