Mathematical modelling of infectious disease

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Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic and help inform public health interventions. Models use basic assumptions or collected statistics along with mathematics to find parameters for various infectious diseases and use those parameters to calculate the effects of different interventions, like mass vaccination programmes. The modelling can help decide which intervention/s to avoid and which to trial, or can predict future growth patterns, etc.

History

The modeling of infectious diseases is a tool that has been used to study the mechanisms by which diseases spread, to predict the future course of an outbreak and to evaluate strategies to control an epidemic.

The first scientist who systematically tried to quantify causes of death was John Graunt in his book Natural and Political Observations made upon the Bills of Mortality, in 1662. The bills he studied were listings of numbers and causes of deaths published weekly. Graunt's analysis of causes of death is considered the beginning of the "theory of competing risks" which according to Daley and Gani is "a theory that is now well established among modern epidemiologists". 

The earliest account of mathematical modelling of spread of disease was carried out in 1760 by Daniel Bernoulli. Trained as a physician, Bernoulli created a mathematical model to defend the practice of inoculating against smallpox. The calculations from this model showed that universal inoculation against smallpox would increase the life expectancy from 26 years 7 months to 29 years 9 months. Daniel Bernoulli's work preceded the modern understanding of germ theory.

In the early 20th century, William Hamer and Ronald Ross applied the law of mass action to explain epidemic behaviour.

The 1920s saw the emergence of compartmental models. The Kermack–McKendrick epidemic model (1927) and the Reed–Frost epidemic model (1928) both describe the relationship between susceptible, infected and immune individuals in a population. The Kermack–McKendrick epidemic model was successful in predicting the behavior of outbreaks very similar to that observed in many recorded epidemics.

Assumptions

Models are only as good as the assumptions on which they are based. If a model makes predictions that are out of line with observed results and the mathematics is correct, the initial assumptions must change to make the model useful.

• Rectangular and stationary age distribution, i.e., everybody in the population lives to age L and then dies, and for each age (up to L) there is the same number of people in the population. This is often well-justified for developed countries where there is a low infant mortality and much of the population lives to the life expectancy.
• Homogeneous mixing of the population, i.e., individuals of the population under scrutiny assort and make contact at random and do not mix mostly in a smaller subgroup. This assumption is rarely justified because social structure is widespread. For example, most people in London only make contact with other Londoners. Further, within London then there are smaller subgroups, such as the Turkish community or teenagers (just to give two examples), who mix with each other more than people outside their group. However, homogeneous mixing is a standard assumption to make the mathematics tractable.

Types of epidemic models

Stochastic

"Stochastic" means being or having a random variable. A stochastic model is a tool for estimating probability distributions of potential outcomes by allowing for random variation in one or more inputs over time. Stochastic models depend on the chance variations in risk of exposure, disease and other illness dynamics.

Deterministic

When dealing with large populations, as in the case of tuberculosis, deterministic or compartmental mathematical models are often used. In a deterministic model, individuals in the population are assigned to different subgroups or compartments, each representing a specific stage of the epidemic. Letters such as M, S, E, I, and R are often used to represent different stages. 

The transition rates from one class to another are mathematically expressed as derivatives, hence the model is formulated using differential equations. While building such models, it must be assumed that the population size in a compartment is differentiable with respect to time and that the epidemic process is deterministic. In other words, the changes in population of a compartment can be calculated using only the history that was used to develop the model.

Reproduction number

The basic reproduction number (denoted by R₀) is a measure of how transferable a disease is. It is the average number of people that a single infectious person will infect over the course of their infection. This quantity determines whether the infection will spread exponentially, die out, or remain constant: if R₀ > 1, then each person on average infects more than one other person so the disease will spread; if R₀ < 1, then each person infects fewer than one person on average so the disease will die out; and if R₀ = 1, then each person will infect on average exactly one other person, so the disease will become endemic: it will move throughout the population but not increase or decrease. 

The basic reproduction number can be computed as a ratio of known rates over time: if an infectious individual contacts β other people per unit time, if all of those people are assumed to contract the disease, and if the disease has a mean infectious period of 1/γ, then the basic reproduction number is just R₀ = β/γ. Some diseases have multiple possible latency periods, in which case the reproduction number for the disease overall is the sum of the reproduction number for each transition time into the disease. For example, Blower et al model two forms of tuberculosis infection: in the fast case, the symptoms show up immediately after exposure; in the slow case, the symptoms develop years after the initial exposure (endogenous reactivation). The overall reproduction number is the sum of the two forms of contraction: R₀ = R₀^FAST + R₀^SLOW.

Endemic steady state

An infectious disease is said to be endemic when it can be sustained in a population without the need for external inputs. This means that, on average, each infected person is infecting exactly one other person (any more and the number of people infected will grow exponentially and there will be an epidemic, any less and the disease will die out). In mathematical terms, that is:

${\displaystyle \ R_{0}S\ =1.}$

The basic reproduction number (R₀) of the disease, assuming everyone is susceptible, multiplied by the proportion of the population that is actually susceptible (S) must be one (since those who are not susceptible do not feature in our calculations as they cannot contract the disease). Notice that this relation means that for a disease to be in the endemic steady state, the higher the basic reproduction number, the lower the proportion of the population susceptible must be, and vice versa. This expression has limitations concerning the susceptibility proportion, e.g. the R₀ equals 0.5 implicates S has to be 2, however this proportion exceeds to population size. 

Assume the rectangular stationary age distribution and let also the ages of infection have the same distribution for each birth year. Let the average age of infection be A, for instance when individuals younger than A are susceptible and those older than A are immune (or infectious). Then it can be shown by an easy argument that the proportion of the population that is susceptible is given by:

${\displaystyle S={\frac {A}{L}}.}$

We reiterate that L is the age at which in this model every individual is assumed to die. But the mathematical definition of the endemic steady state can be rearranged to give:

${\displaystyle S={\frac {1}{R_{0}}}.}$

Therefore, due to the transitive property:

${\displaystyle {\frac {1}{R_{0}}}={\frac {A}{L}}\Rightarrow R_{0}={\frac {L}{A}}.}$

This provides a simple way to estimate the parameter R₀ using easily available data.

For a population with an exponential age distribution,

${\displaystyle R_{0}=1+{\frac {L}{A}}.}$

This allows for the basic reproduction number of a disease given A and L in either type of population distribution.

Modelling epidemics

The SIR model is one of the more basic models used for modelling epidemics. There are many modifications to the model.

The SIR model

Diagram of the SIR model with initial values ${\textstyle S(0)=997,I(0)=3,R(0)=0}$, and rates for infection ${\textstyle \beta =0.4}$ and for recovery ${\textstyle \gamma =0.04}$

Animation of the SIR model with initial values ${\textstyle S(0)=997,I(0)=3,R(0)=0}$, initial rate for infection ${\textstyle \beta =0.5}$ and constant rate for recovery ${\textstyle \gamma =0.04}$. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as "flattening the curve") by appropriate measures (e. g. "social distancing"). This animation shows the impact of reducing the infection rate by 76 % (from ${\textstyle \beta =0.5}$ down to ${\textstyle \beta =0.12}$).

In 1927, W. O. Kermack and A. G. McKendrick created a model in which they considered a fixed population with only three compartments: susceptible, ${\displaystyle S(t)}$; infected, ${\displaystyle I(t)}$; and recovered, ${\displaystyle R(t)}$. The compartments used for this model consist of three classes:

• ${\displaystyle S(t)}$ is used to represent the individuals not yet infected with the disease at time t, or those susceptible to the disease of the population.
• ${\displaystyle I(t)}$ denotes the individuals of the population who have been infected with the disease and are capable of spreading the disease to those in the susceptible category.
• ${\displaystyle R(t)}$ is the compartment used for the individuals of the population who have been infected and then removed from the disease, either due to immunization or due to death. Those in this category are not able to be infected again or to transmit the infection to others.

The flow of this model may be considered as follows:

${\displaystyle {\color {blue}{{\mathcal {S}}\rightarrow {\mathcal {I}}\rightarrow {\mathcal {R}}}}}$

Using a fixed population, ${\displaystyle N=S(t)+I(t)+R(t)}$ in the three functions resolves that the value N should remain constant within the simulation. The model is started with values of S(t=0), I(t=0) and R(t=0). These are the number of people in the susceptible, infected and removed categories at time equals zero. Subsequently, the flow model updates the three variables for every time point with set values for ${\displaystyle \beta }$ and ${\displaystyle \gamma }$. The simulation first updates the infected from the susceptible and then the removed category is updated from the infected category for the next time point (t=1). This describes the flow persons between the three categories. During an epidemic the susceptible category is not shifted with this model, ${\displaystyle \beta }$ changes over the course of the epidemic and so does ${\displaystyle \gamma }$. These variables determine the length of the epidemic and would have to be updated with each cycle.

${\displaystyle {\frac {dS}{dt}}=-{\frac {\beta SI}{N}}}$

${\displaystyle {\frac {dI}{dt}}={\frac {\beta SI}{N}}-\gamma I}$

${\displaystyle {\frac {dR}{dt}}=\gamma I}$

Several assumptions were made in the formulation of these equations: First, an individual in the population must be considered as having an equal probability as every other individual of contracting the disease with a rate of ${\displaystyle a}$ and an equal number ${\displaystyle b}$ of people that an individual makes contact with per unit time. Then, let ${\displaystyle \beta }$ be the multiplication of ${\displaystyle a}$ and ${\displaystyle b}$. This is the transmission probability times the contact rate. Besides, an infected individual makes contact with ${\displaystyle b}$ persons per unit time whereas only a fraction, ${\displaystyle {\frac {S}{N}}}$ of them are susceptible.Thus, we have every infective can infect ${\displaystyle {\frac {abS}{N}}}$---${\displaystyle {\frac {\beta S}{N}}}$ susceptible persons,and therefore, the whole number of susceptibles infected by infectives per unit time is ${\displaystyle {\frac {\beta SI}{N}}}$. For the second and third equations, consider the population leaving the susceptible class as equal to the number entering the infected class. However, a number equal to the fraction (${\displaystyle \gamma }$ which represents the mean recovery/death rate, or ${\displaystyle 1/\gamma }$ the mean infective period) of infectives are leaving this class per unit time to enter the removed class. These processes which occur simultaneously are referred to as the Law of Mass Action, a widely accepted idea that the rate of contact between two groups in a population is proportional to the size of each of the groups concerned. Finally, it is assumed that the rate of infection and recovery is much faster than the time scale of births and deaths and therefore, these factors are ignored in this model.

Steady-state solutions

The expected duration of susceptibility will be ${\displaystyle \operatorname {E} [\min(T_{L}\mid T_{S})]}$ where ${\displaystyle T_{L}}$ reflects the time alive (life expectancy) and ${\displaystyle T_{S}}$ reflects the time in the susceptible state before becoming infected, which can be simplified to:

${\displaystyle \operatorname {E} [\min(T_{L}\mid T_{S})]=\int _{0}^{\infty }e^{-(\mu +\delta )x}\,dx={\frac {1}{\mu +\delta }},}$

such that the number of susceptible persons is the number entering the susceptible compartment ${\displaystyle \mu N}$ times the duration of susceptibility:

${\displaystyle S={\frac {\mu N}{\mu +\lambda }}.}$

Analogously, the steady-state number of infected persons is the number entering the infected state from the susceptible state (number susceptible, times rate of infection ${\displaystyle \lambda ={\tfrac {\beta I}{N}},}$ times the duration of infectiousness ${\displaystyle {\tfrac {1}{\mu +v}}}$:

${\displaystyle I={\frac {\mu N}{\mu +\lambda }}\lambda {\frac {1}{\mu +v}}.}$

Other compartmental models

There are many modifications of the SIR model, including those that include births and deaths, where upon recovery there is no immunity (SIS model), where immunity lasts only for a short period of time (SIRS), where there is a latent period of the disease where the person is not infectious (SEIS and SEIR), and where infants can be born with immunity (MSIR).

Infectious disease dynamics

Mathematical models need to integrate the increasing volume of data being generated on host-pathogen interactions. Many theoretical studies of the population dynamics, structure and evolution of infectious diseases of plants and animals, including humans, are concerned with this problem.

Research topics include:

• transmission, spread and control of infection
• epidemiological networks
• spatial epidemiology
• persistence of pathogens within hosts
• intra-host dynamics
• immuno-epidemiology
• virulence
• Strain (biology) structure and interactions
• antigenic shift
• phylodynamics
• pathogen population genetics
• evolution and spread of resistance
• role of host genetic factors
• statistical and mathematical tools and innovations
• role and identification of infection reservoirs

Mathematics of mass vaccination

If the proportion of the population that is immune exceeds the herd immunity level for the disease, then the disease can no longer persist in the population. Thus, if this level can be exceeded by vaccination, the disease can be eliminated. An example of this being successfully achieved worldwide is the global smallpox eradication, with the last wild case in 1977. The WHO is carrying out a similar vaccination campaign to eradicate polio.

The herd immunity level will be denoted q. Recall that, for a stable state:

${\displaystyle R_{0}\cdot S=1.}$

In turn,

${\displaystyle R_{0}={\frac {N}{S}}={\frac {\mu N\operatorname {E} (T_{L})}{\mu N\operatorname {E} [\min(T_{L},T_{S})]}}={\frac {\operatorname {E} (T_{L})}{\operatorname {E} [\min(T_{L},T_{S})]}},}$

which is approximately:

${\displaystyle {\frac {\operatorname {\operatorname {E} } (T_{L})}{\operatorname {\operatorname {E} } (T_{S})}}=1+{\frac {\lambda }{\mu }}={\frac {\beta N}{v}}.}$

S will be (1 − q), since q is the proportion of the population that is immune and q + S must equal one (since in this simplified model, everyone is either susceptible or immune). Then:

${\displaystyle {\begin{aligned}&R_{0}\cdot (1-q)=1,\\[6pt]&1-q={\frac {1}{R_{0}}},\\[6pt]&q=1-{\frac {1}{R_{0}}}.\end{aligned}}}$

Remember that this is the threshold level. If the proportion of immune individuals exceeds this level due to a mass vaccination programme, the disease will die out.

We have just calculated the critical immunisation threshold (denoted q_c). It is the minimum proportion of the population that must be immunised at birth (or close to birth) in order for the infection to die out in the population.

${\displaystyle q_{c}=1-{\frac {1}{R_{0}}}.}$

Because the fraction of the final size of the population p that is never infected can be defined as:

${\displaystyle \lim _{t\to \infty }S(t)=e^{-\int _{0}^{\infty }\lambda (t)\,dt}=1-p.}$

Hence,

${\displaystyle p=1-e^{-\int _{0}^{\infty }\beta I(t)\,dt}=1-e^{-R_{0}p}.}$

Solving for ${\displaystyle R_{0}}$, we obtain:

${\displaystyle R_{0}={\frac {-\ln(1-p)}{p}}.}$

When mass vaccination cannot exceed the herd immunity

If the vaccine used is insufficiently effective or the required coverage cannot be reached (for example due to popular resistance), the programme may fail to exceed q_c. Such a programme can, however, disturb the balance of the infection without eliminating it, often causing unforeseen problems.

Suppose that a proportion of the population q (where q < q_c) is immunised at birth against an infection with R₀ > 1. The vaccination programme changes R₀ to R_q where

${\displaystyle R_{q}=R_{0}(1-q)}$

This change occurs simply because there are now fewer susceptibles in the population who can be infected. R_q is simply R₀ minus those that would normally be infected but that cannot be now since they are immune.

As a consequence of this lower basic reproduction number, the average age of infection A will also change to some new value A_q in those who have been left unvaccinated.

Recall the relation that linked R₀, A and L. Assuming that life expectancy has not changed, now:

${\displaystyle R_{q}={\frac {L}{A_{q}}},}$

${\displaystyle A_{q}={\frac {L}{R_{q}}}={\frac {L}{R_{0}(1-q)}}.}$

But R₀ = L/A so:

${\displaystyle A_{q}={\frac {L}{(L/A)(1-q)}}={\frac {AL}{L(1-q)}}={\frac {A}{1-q}}.}$

Thus the vaccination programme will raise the average age of infection, another mathematical justification for a result that might have been intuitively obvious. Unvaccinated individuals now experience a reduced force of infection due to the presence of the vaccinated group. 

However, it is important to consider this effect when vaccinating against diseases that are more severe in older people. A vaccination programme against such a disease that does not exceed q_c may cause more deaths and complications than there were before the programme was brought into force as individuals will be catching the disease later in life. These unforeseen outcomes of a vaccination programme are called perverse effects.

When mass vaccination exceeds the herd immunity

If a vaccination programme causes the proportion of immune individuals in a population to exceed the critical threshold for a significant length of time, transmission of the infectious disease in that population will stop. This is known as elimination of the infection and is different from eradication.

Elimination

Interruption of endemic transmission of an infectious disease, which occurs if each infected individual infects less than one other, is achieved by maintaining vaccination coverage to keep the proportion of immune individuals above the critical immunisation threshold.

Eradication

Reduction of infective organisms in the wild worldwide to zero. So far, this has only been achieved for smallpox and rinderpest. To get to eradication, elimination in all world regions must be achieved.