Theoretical Epidemiology of
Infectious Diseases
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Anderson RM, May RM (1991) Infectious Diseases of Humans: Dynamics and Control.
Oxford Univ. Press
Diekmann O, Heesterbeek JAP (2000) Mathematical Epidemiology of Infectious
Diseases: Model Building, Analysis, and Interpretation. Wiley.
Ebert D, Herre EA (1996) The evolution of parasitic diseases. Parasitology Today, 12:
96-101.
Ewald PW (1994) Evolution of Infectious Disease. Oxford Univ. Press
Giesecke J (2002) Modern Infectious Disease Epidemiology. Arnold.
Mascie-Taylor CGN (1993) The Anthropology of Disease. Oxford Univ. Press
Rothman KJ (2012) Epidemiology: An Introduction 2nd Ed. Oxford Univ. Press.
Vynnycky E, White RG (2010) An Introduction to Infectious Disease Modelling. Oxford
Univ. Press
Wolfe ND, Dunavan CP, Diamond J (2007) Origins of major human infectious diseases.
Nature, 447: 279-283.
For COVID-19, please see, https://minato.sip21c.org/COVID-19-publichealthmeasure.pdf
8 April 2024
Minato Nakazawa

1What is human infectious diseases?
●"Infectious diseases" from ecological perspective– Among the symbiosis (mutualism, commensalism,

parasitism), a kind of parasitism– The life and reproduction of parasites depend on host's
life (differently by macro-/micro-)– Host-parasite co-evolution
Antimalarial genes in malaria endemic area
– Thalassemia in Eastern Mediterranean– Sickle cell anemia in Sub-Saharan Africa●
Hypoferremic adaptation hypothesis: relatively lower ironconcentration in serum than in liver found in malaria-endemic area●

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History of human infectious diseases

Source: Mascie-Taylor CGN (1993)

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Minato Nakazawa

The origin of human infectious diseases●

5 Stages from animal pathogen to specialized pathogen of humans (Wolfe et al. 2007)(http://www.nature.com/nature/journal/v447/n7142/full/nature05775.html)

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Minato Nakazawa

Descriptive epidemiology of infectious diseases●

Epidemic curves (TIME)–

Bar chart of (or lines connecting)the numbers of newly infected (ordied) patients by time after the
onset of outbreak

Fitted by mathematical models toestimate parameters●
Epidemiologic maps (PLACE)–

Cholera outbreak map of London by
John Snow– Recently using GIS
Sex/Age distribution (PERSON)
Seroepidemiology– Antibody titers' distribution
shows endemicity (experienceof infection)–●●

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Prevalence and Incidence
Infected

Recovered

Died

Majority●●●●

Available data is usually limited to prevalence (cross-sectionally, how much proportion
among the population is infected) or incidence (newly infected number among the observed population)– number of asymptomatic patients needs active detection screening
Diseases with high virulence are underestimated by cross-sectional study
Detection of asymptomatic cases is important for diseases with long latent period
Distribution of infection frequency can be obtained from retrospective study, but longitudinal cohort study is preferable.
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Minato Nakazawa

Basic elements which affect the transmissionof infectious diseases
Host condition: population (size, density, age-structure), gene(resistant, susceptible), nutritional status, socio-cultural
factors (network, behavior)●
Environmental condition: temperature, humidity and vector
animals (in the case of vector-borne infection)●
Parasite condition: host-specificity, lifespan, transmission
type, etc.
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Interaction: route of infection, evolution to optimal virulence
based on the interaction between infectiousness and virulence(Ebert and Herre, 1996), virulence decrease in directtransmission (like JC virus) vs no change in vector-borne
transmission (Ewald, 1994)●

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Minato Nakazawa

Route of infection
Route of transmission

Characteristics

Contact

Requires direct/indirect contact* Indirect = infected fomite, blood or body fluid* Direct = skin or sexual contact
Ingestion of contaminated food (outbreaks may be large
and dispersed, depending on distribution of food)

Food- or water-borne
Airborne (droplet, droplet
nuclei, micro-droplet)

Inhalation of contaminated air* Droplet = large droplets by cough* Droplet nuclei = Dried particle from droplets* Micro-droplets = small suspended droplets

Vector-borne

Dependent on biology of the vector (mosquito, tick, snail,etc), as well as the infectivity of the organism
Similar to contact infection; however, the contact may occur
in utero during pregnancy or at the time of delivery

Perinatal

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TYPES OF TRANSMISSION●

Host population constitutes reservoir for thepathogen
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Route

ExamplesDirectAirborne

Anthrax （炭疽） , chicken pox,
common cold, influenza,
measles, mumps, rubella,tuberculosis, whooping coughDirectcontact

Athlete’s foot （水虫） ,impetigo （とびひ） , warts （いぼ）Fecal-oral

Cholera, hepatitis A,
rotavirus, salmonella(=typhoid fever)Maternalfetal

Hepatitis B, syphilis

SexualChlamydia, gonorrhea,
hepatitis B, herpes, syphilis,
HPVIntermediate
host

Tapeworm (from eating
inadequately cooked pork)

Vectorborne

Bubonic plague (by fleas),malaria (by Anopheles
mosquitoes), typhus (by lice),
West Nile encephalitis (byCulex mosquitoes), yellow
fever (by Culex mosquitoes),dengue fever (by Aedes 9
mosquitoes)

Primary habitat for pathogen
Pathogen can survive and spread via other hosts thanhuman

Highly virulent pathogen cannot survive and spreadbecause of early death of host–

Variety of transmission pathway evolution●●●

Direct, person-to-person (communicable, contagious):
measles (host is only human) viable only for 2-3 hours indroplets
Via transmitting animal (vector): malaria (from infected
human with 5 types Plasmodium gametocytes toAnopheles mosquitoes, then sporozoites in salivary gland
moves to another human by the next biting). Most
vectors are arthropods
Zoonoses can spread animal reservoirs to humans
Indirect

Vector-borne: Equine encephalitis, plague– Directly from animal to human: Toxoplasmosis (from cat),ebola virus (from bat), flu (hosts are human, birds, andpigs), rabies (hosts are all warm-blooded animals)
CFR (Case Fatality Risk: Number of death due to that disease divided by the
number of diagnosed patients) of rabies is 100% if untreated (human is dead-endhost), but the virus can survive within other animals than humans–●

Transmission

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Different frequency distributions between the diseases with vectorborne and direct transmission by virulence (case fatality rates)

Source: Ewald (1994) [pp.38, Figure 3.1]
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The evolution of optimal virulence

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Source: Ebert & Herre (1996)

What makes pandemic?
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Definition of pandemic
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An epidemic of unusually high occurrence of disease.
An epidemic occurring worldwide or over a very wide area, crossing boundaries ofseveral countries and usually affecting a large number of people (Dictionary of
epidemiology)
An influenza pandemic occurs when a new influenza virus appears against which
the human population has no immunity, resulting in several, simultaneous
epidemics worldwide with enormous numbers of deaths and illness (WHO, before
pandemic H1N1 flu in 2009)
An influenza pandemic occurs when a new influenza virus appears against which thehuman population has no immunity, resulting in several, simultaneous epidemics
worldwide (WHO, after pandemic H1N1 flu in 2009)

Change of definition of pandemic flu by WHO was to answer to the critics thatthe pandemic declaration for H1N1 flu in 2009 was motivated by the ties between
WHO and pharmaceutical industry (though WHO denied such ties)
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HERD IMMUNITY AND BASIC REPRODUCTION
NUMBER (R0), effective reproduction number (Rt)●●●

The relative proportions of immune andSerial interval
susceptible persons in a population candetermine whether the infection will taket
t+1
hold in the community or die out quickly TIME
When substantial proportion is immune(herd immunity situation), an infected
person will be less likely to spread thepathogen
R =4
R0 (basic reproduction number) is theaverage number of secondary cases that
occur from a single index case in asusceptible population●

If R0 < 1, the outbreak will die out
unless fueled by external re-infections

Rt (effective reproduction number) is thevalue of reproduction number that takes
into account the mix of immunity andsocial interaction at any point in time asan outbreak progresses

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Primary mode oftransmission

R0

Measles

Airborne

15

Pertussis(whoopingcough)

Airborne droplet

15

Diphtheria

Saliva6

Smallpox

Social contact

6

Polio

Fecal-oral

6

RubellaAirborne droplet

6

Mumps

Airborne droplet

5

HIV/AIDS

Sexual contact

3

SARS

Airborne droplet

3

Ebola

Bodily fluids

2

1918 flu

Airborne droplet

2

t+2

0

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DiseaseR0=4
Rt=1

Susceptible

Transmission

2009 flu

Airborne droplet

1.5

ImmuneNo transmission

COVID-19

Airborne droplet

1.4-3?

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Note: If average R0 is same, control efficacy may largely differ by variance.

The nature of R0 and Rt●

The reproduction number reflects the biologic
potential of the infectious agent and the social
intercourse that leads to situations in whichtransmission might occur
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If directly transmitted disease patient is too
sick to move, there will be few contact withsusceptible host, results in low reproductive
number

R0 varies by population (due to behavioral
difference by age and so on)
Even if R0 is low, some social networks within
a population may form a subset with rapid
spread of infection. (eg. a few“superspreaders” such as needle-sharers
transmitting a blood-borne infection cansuffice to spark an outbreak)Superspreading is not always an attribute ofperson, sometimes a condition of the fieldsetting (in the case of COVID-19).
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While Rt>1, epidemic continues
Eventually Rt becomes 1 or below, because theproportion of susceptible people decreases or control
measures are implemented
If Rt=1 (endemic equilibrium), the prevalence of
infection remains level over time as new susceptibles areadded to the population to balance those who acquire
immunity
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Rt = 1 = R0 x ps, where ps is the proportion of the
population susceptible to infection at equilibrium, thus
R0 = 1/ps

Basic strategy to reduce transmission is isolation of
infected persons.
Related strategy is quarantine to restrict contacts amongpeople who are not yet ill but already contacted with
infected persons(For bioterrorism by smallpox, ring-vaccination is to be
conducted to reduce Rt)
In Japan, restriction of behavior to fill the conditions forsuperspreading events was taken to reduce Rt as acountermeasure against COVID-19 outbreak.
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Example of SARS, and if
vaccine would be available?
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The strategy of isolation and quarantine worked well against SARS
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If vaccine would be available, Rt depends on vaccine efficacy (Ve) and coverage (Vc)–

Rt = R0 (1 – Ve x Vc) ↔ Rt/R0 = 1 - Ve x Vc ↔ Vc = (1 - Rt/R0)/Ve–

Rt < 1 ↔ Vc > (1 – 1/R0)/Ve–

When R0 is large, to succeed in curtailing the epidemic, high efficacy and coverage are needed (If R 0 is 10
and Ve is 95%, Vc has to be larger than 95% needed to reduce R t below 1; (1 – 1/10)/0.95 = 0.947… ≈ 0.95)–

In the case of measles, R0 is 15. Even if Ve is 100%, Vc has to be larger than 93% to reduce Rt below 1; (1 –1/15)/1 = 0.933… ≈ 0.93
If R0 is 2 and Ve is 95%, Vc needs to be larger than 53% to reduce Rt below 1; (1 –1/2) /0.95 = 0.526… ≈ 0.53.

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SARS nearly became pandemic in 2003, rapidly spread from China to 37 countries (infected more than
8000 people, CFR was almost 10%).
Canadian officials quarantined more than 23000 people who had been in contact with SARS cases, about
100 persons for every identified case of SARS. Movement of those under quarantine was restricted until 10days after the last contact
SARS was emerging disease in 2003 and thus no vaccine existed

The same relationship may stand for not only vaccination but also naturally acquired immunity after
infection. Vaccine efficacy corresponds to the proportion of immunized by single infection andcoverage corresponds to the proportion of people ever infected and recovered (herd immunity
threshold)
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Terms for individual infection history
Incubation period
Latent or pre-infectious period

Clinical symptoms

Infectious period

First infection
Onset
Second infection

Two general types of infection network topology(1) Random link network(2) Scale free network* equal infection probability for each* host preferences* distribution of infection frequency* distribution obeying power law
is unimodal* superspreader exists

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Minato Nakazawa

16Simplest mathematical model (SI)
S

b

I

g

immune / deathdS= - b SI
dt
dI
= b SI - g I
dt
Kermack-McKendrick model (1927)
for plague outbreak in Bombay from
December 17, 1905 to July 21,
1906.

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SIR model for flu epidemic
dS/dt = ˗ βSI + δR●
dI/dt = βSI ˗ γI●
dR/dt = γI ˗ δR●
Note: None of S, I, and R canbe less than 0. It should beconsidered in numericalsimulation.
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Estimating params
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800
600
infected

400
200– β: {I(1)-I(0)}/S(0)Example: The data in English boys
boarding school in 1978●

– δ: negligible (loss of immunity)
https://minato.sip21c.org/tiid/flu-sir-2020.R
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Minato Nakazawa

0– γ: 1/{mean days of recovery}
0

2

4

6

8

time (days)

10

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THE REED-FROST EPIDEMIC MODEL●

Assumptions
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C(t+1) = S(t) {1 – (1 – p)C(t)}
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There is random mixing, with contact between infected
people and susceptible people within the population during
each time period
There is a uniform, fixed probability that a contact between
an infected person and a susceptible person would result intransmission
An infection is always followed by immunity
The population is isolated from other populations
These conditions remain constant with time
C(t): the number of newly infected people at time tS(t): the number of susceptible people at time tp: the probability that within one time period an infected
person will transmit the infection to a susceptible person withwhom there is contact

Reed-Frost projection of epidemic curve for infected,
susceptible, and immune sub-populations among 100 people
with one initial infected person and an effective contact
probability of 4% (high R0 in upper panel) and 1.5% (low R0 in
lower panel). The time scale is measured in generation times
If R0 is small, susceptible people remain after the epidemic.
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Minato Nakazawa

https://minato.sip21c.org/tiid/ReedFrost.R
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INFECTIOUS DISEASEEPIDEMIOLOGY INVESTIGATIONS●●

Several types of epidemiologic studies are unique to the investigation of infectious
diseaseFour types are worthy to mention––

Contact-tracing studies: In the early stage of an epidemic, it may be possible to interrupt
person-to-person transmission enough to bring R0 below 1 by isolation, treatment andquarantine of patients (shoe-leather approach).
Outbreak investigation: When a local epidemic occurs, documenting outbreak and investigating
its origin and propagation. Often detective work such as identification of the cause of diarrhea
outbreak as the church supper on the potato salad.
●––(Note: AIDS is an acronym of Acquired Immuno-Deficiency Syndrome, not Acute)

Seroprevalence surveys (sero-epidemiology): Distribution of prevalence of a specific disease isreflected in the distribution of the antibody titers against that disease agent in the blood, since
the antibody titers remain for several months after infection. It can assess the vulnerability of a
population to existing infectious agents, for finding susceptible subgroups
Vaccine trials: A randomized trial of preventive measure is called a field trial (Chapter 5). It’smuch more difficult than clinical trials. One of the reasons is the outcome (prevented) is rare.
(eg.) Salk vaccine trial. Infection of polio virus was popular but paralysis symptoms were rare.

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OUTLOOK FOR INFECTIOUS
DISEASE EPIDEMIOLOGY●●●

In the very beginning period after the invention of antibiotics, human misunderstood that the ultimate defense
against infection from bacteria was found
For vaccines, human also misunderstood that viral illness might be tamed and possible to eradicate as smallpox
However,
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Infectious disease epidemiology is a frontier that has observed 2 remarkable triumphs––●

High reproductive rate of microorganisms and their ability to mutate have enabled them to evade many of ourtechnologically driven defenses
Widespread and unnecessary use of antibiotics produced antibiotic-resistant bacteria
Increasing urbanization and intercontinental travel added risk of communicating infectious diseases
Social and medical practices opened new routes of transmission
Eradication of smallpox
Near-elimination of poliomyelitis

The hope of eradication of other diseases: malaria is candidate but challenging––––

Quinine, chloroquine, artemisinine, and other chemotherapy were effective to cure patients but resistancedeveloped
DDT and other insecticides were effective to reduce anopheles mosquitoes but those were toxic for environment
and mosquitoes got resistance
Development of highly effective vaccine is very difficult because life-cycle of malaria parasite is very complex
and multi-stage and P. falciparum can escape from antibody by distributing junk antigens
Previously nonhuman (simian) malaria, P. knowlesi switched the host from monkeys to human

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