An Introduction to Mathematical Epidemiology pp 215-248 | Cite as
Control Strategies
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Abstract
This chapter studies control strategies. Control strategies are listed and explained. It focuses on modeling vaccination in single-strain and multistrain diseases. Different modes of introducing vaccination in models are shown. Imperfect vaccination as a mechanism leading to backward bifurcation and strain replacement is explained. Strain replacement with perfect vaccination is demonstrated. Quarantine and isolation are discussed and included in a model. Introduction to optimal control theory is incorporated and the theory is illustrated on a specific example with optimal control treatment. Matlab code for computing the optimal control is included.
Keywords
Reproduction Number Vaccine Efficacy Vaccination Rate Optimal Control Theory Endemic Equilibrium9.1 Introduction
Measures for prevention and control of infectious diseases include vaccination, treatment, quarantine, isolation, and prophylaxis.
Prophylaxis is the series of measures taken to prevent a specific infectious disease. These measures can be as simple as hand-washing with soap and water, or wearing protective gear, or taking a medication to prevent a disease. Treatment is the use of an agent, procedure, or regimen, such as a drug, or bed rest in an attempt to cure or mitigate a disease. Nowadays, for most infectious diseases, medications exist that can cure or lessen the impact of the diseases, while improving the life of the patients. Diseases for which medications can offer a cure include malaria and tuberculosis. Diseases for which medications offer relief but not a cure include HIV and genital herpes.
Vaccination is the process through which killed (inactivated) or weakened microorganisms are placed into the body. Our immune system recognizes vaccine agents as foreign. That triggers an immune response, and antibodies against them are developed. As a result, if the same types of microorganisms enter the body again, they will be destroyed much faster by the antibodies. Thus, an individual that is immunized is protected against the disease. If a large majority of people are vaccinated, it is much more difficult for an outbreak of disease to occur, let alone spread. This effect is called herd immunity.
Achievements of vaccination in the United States^{a}
Disease | Baseline years | Cases/year | Cases in 1998 | % Decrease | |
Smallpox | 1900–1904 | 48,164 | 0 | 100 | |
Diphtheria | 1920–1922 | 175,885 | 1 | 100 | |
Pertussis | 1922–1925 | 147,271 | 6,279 | 95.7 | |
Tetanus | 1922–1926 | 1,314 | 34 | 97.4 | |
Poliomyelitis | 1951–1954 | 16,316 | 0 | 100 | |
Measles | 1958–1962 | 503,282 | 89 | 100 | |
Mumps | 1968 | 152,209 | 606 | 99.6 | |
Rubella | 1966–1968 | 47,745 | 345 | 99.3 | |
Hib | 1985 | 20,000 | 54+71 | 99.7 |
Vaccines do not guarantee complete protection from a disease. There remains the possibility that a vaccinated person may get the disease. Even if the host develops antibodies, some pathogens can mutate (the common cold and influenza viruses are highly efficient at this), and in any case, the immune system might still not be able to defeat the infection. The degree to which vaccinated individuals are protected against the disease is called efficacy of the vaccine.
Quarantine and isolation are two measures by which exposed or infectious individuals are removed from the population to prevent further spread of the infection. Quarantine is applied to seemingly healthy but potentially infected individuals, while isolation is applied to already infectious individuals. Isolation has been used and is being used to control many dangerous diseases. Quarantine is applied less often. It is one of the first response methods that can be used in an extreme emergency. Quarantine was implemented during the SARS epidemic of 2002–2003.
The reproduction number, computed for mathematical models involving control strategies, depends on the control strategies, and it is often called a controlled reproduction number.
9.2 Modeling Vaccination: Single-Strain Diseases
There are two points in which vaccination models can differ from one another. The first is that some models assume that vaccination is equivalent to going through the disease and treats vaccinated individuals as recovered individuals. Thus an SIR model can include vaccinated individuals without an additional class. Other models assume that vaccinated individuals have to be separated into a vaccinated class V. The second point of distinction is that some classes of models assume that individuals enter the system at a point of their life when they either get vaccinated or skip vaccination and enter the system as susceptibles. This is more or less accurate for school children. Other models allow for continuous vaccination of individuals while in the system.
9.2.1 A Model with Vaccination at Recruitment
Question: What fraction, p, of the population must be vaccinated so that the reproduction number of the disease is reduced below 1?
9.2.2 A Model with Continuous Vaccination
Most diseases for which vaccination is successful have a recovered (immune) stage. After all, vaccination works with the immune system more or less as the disease does, so if the disease does not provide immunity, how could vaccination? However, there are diseases for which it is justified to consider vaccination in addition to an SIS model, that is, a model where recovery brings the individual back to the susceptible class. One such disease is tuberculosis, which imparts very short-lived immunity. Another situation occurs with bacterial infections with Neisseria meningitidis and Streptococcus pneumoniae. Both these bacteria can exist in the host without causing disease, a scenario, called carriage. Both carriers and infected (sick) people can transmit the microorganism, so from the point of view of disease transmission, they can be considered indistinguishable and modeled with one class. Carriage and disease impart immunity against the disease but probably not so much against carriage. Thus individuals who become completely pathogen-free can be counted as susceptible (at least for carriage). In both cases, there are vaccines, at least against some variants of the microorganisms, and an SIS model with vaccination may be appropriate.
9.2.2.1 An SIS Model with Vaccination
Let V (t) denote the number of vaccinated individuals, and ψ the per capita vaccination rate. Vaccination is applied only to healthy individuals, so only susceptible individuals get vaccinated. In this model, we also take into account the fact that vaccines are rarely perfect, and some of the vaccinated individuals can become infected and infectious even though they have been vaccinated. That happens at a reduced transmission rate β δ, where 0 ≤ δ ≤ 1 is the reduction coefficient. If δ = 0, then vaccinated individuals cannot get infected, and the vaccine is perfect. This implies that the vaccine efficacy is ε = 1. If δ = 1, then vaccinated individuals get infected just like susceptible individuals, and the vaccine plays no protective role. In that case, the vaccine efficacy is ε = 0.
List of parameters, variables, and their meanings
Notation | Meaning | |
Λ | Birth/recruitment rate into the population | |
μ | Per capita natural death rate | |
β | Per capita transmission rate | |
γ | Per capita recovery rate | |
χ | Proportion of individuals who recover to the vaccinated class | |
1 −χ | Proportion of individuals who recover to the susceptible class | |
ψ | Per capita vaccination rate | |
\(\varepsilon = 1-\delta\) | Vaccine efficacy | |
S(t) | Number of susceptible individuals | |
I(t) | Number of infected individuals | |
V (t) | Number of vaccinated individuals |
9.2.2.2 The Reproduction Number and the Critical Vaccination Proportion
To compute the reproduction number, we compute the Jacobian at the disease-free equilibrium:
In interpreting the reproduction number, we notice that \(\frac{\beta SI} {N}\) gives the number of secondary infections of susceptible individuals per unit of time. The number of secondary infections of susceptible individuals per unit of time for one infectious individual will be \(\frac{\beta S} {N}\). The proportion of susceptibles in a disease-free population is \(\frac{S} {N} = s^{0} = \frac{\mu } {\mu +\psi }\). Since \(\frac{1} {\mu +\gamma }\) is the time spent as an infectious individual, the first term in \(\mathcal{R}(\psi )\), given by \(\frac{\beta \mu }{(\mu +\gamma )(\mu +\psi )}\), gives the number of secondary infections of susceptible individuals that one infected individual can produce in a disease-free population. Similarly, \(\frac{\beta \delta V I} {N}\) gives the number of secondary infections of vaccinated individuals per unit of time. The number of secondary infections of vaccinated individuals per unit of time for one infectious individual will be \(\frac{\beta \delta V } {N}\). The proportion of vaccinated individuals in a disease-free population is \(\frac{V } {N} = v^{0} = \frac{\psi } {\mu +\psi }\). Since \(\frac{1} {\mu +\gamma }\) is the time spent as an infectious individual, the second term in \(\mathcal{R}(\psi )\), given by \(\frac{\beta \delta \psi }{(\mu +\gamma )(\mu +\psi )}\), gives the number of secondary infections of vaccinated individuals that one infected individual can produce in a disease-free population.
Question: What is the critical proportion of individuals that should be vaccinated if the vaccine is continuously applied and imperfect?
We note that the formula above is an extension of the critical vaccination proportion to imperfect vaccines. If the vaccine is perfect, that is, if ε = 1, then we obtain the customary formula for the critical vaccination proportion for perfect vaccines.
Diseases and their eradication vaccination levels
Disease | \(\mathcal{R}_{0}\) | \(\hat{p}\),% | Vaccine efficacy^{a} | \(\hat{p}_{\varepsilon }\),% | |
Smallpox | 3–5 | 67–80 | 0.75^{b} | 89–100 | |
Measles | 12–13 | 92 | 0.75–0.95 | 97–100 | |
Mumps | 4–7 | 75–86 | 0.75–0.95 | 79–100 | |
Rubella | 6–7 | 83–86 | 0.75–0.95 | 87–100 | |
Chickenpox | 9–10 | 89–90 | 0.8–0.95 | 94–100 | |
Pertussis | 13–17 | 92–94 | 0.8–0.9 | – | |
Poliomyelitis | 6 | 83 | 0.9–0.99 | 84–92 | |
Diphtheria | 4–6 | 75–83 | 0.87–0.96 | 78–95 |
9.2.2.3 Backward Bifurcation in the Imperfect Vaccination Model
The critical threshold above gives only the proportion that has to be vaccinated so that the reproduction number in the presence of vaccination is below one. However, imperfect vaccines have the disadvantage that they lead to backward bifurcation, and endemic equilibria exist and are stable even when the reproduction number in the presence of vaccination is below one. The main reason for the backward bifurcation is the fact that imperfect vaccination creates two classes of susceptible individuals with different susceptibilities—the naive susceptible and the vaccinated susceptible.
Imperfect vaccines lead to backward bifurcation. It is not hard to see that in the model above, backward bifurcation does not occur if the vaccine is perfect, δ = 0. Also, if there is no vaccination ψ = 0, then backward bifurcation does not occur. In this case, it can be seen that if \(\mathcal{R}_{0} <1\), the disease-free equilibrium is globally stable.
9.3 Vaccination and Genetic Diversity of Microorganisms
When a pathogen is represented by several variants, they may not all be included in the vaccine. The strains that are included in the vaccine are called vaccine strains. The number of strains included in the vaccine is called vaccine valency. For instance, the flu vaccine is trivalent, that is, it contains three strains.
The immunity that a vaccine creates is specific to those strains that are included in the vaccine. The vaccine may provide partial immunity, or no immunity at all, to strains that are not included in the vaccine. That makes impossible the eradication of diseases whose causative agents mutate and that are represented by multiple variants.
- 1.
Vaccination is carried against the dominant subtype. For instance, Haemophilus influenzae is represented by six serotypes: a, b, c, d, e, f, but before vaccination was instituted, serotype b caused most disease. Vaccination is now carried out against serotype b.
- 2.
Vaccination is carried out against several strains that account for most cases. For instance, Streptococcus pneumoniae is represented by more than 90 serotypes, but only 23 of the most common ones are included in the polysaccharide vaccine.
- 3.
When possible, vaccination is carried out against all subtypes (possibly one by one). For instance, poliomyelitis (caused by poliovirus, PV) is represented by three serotypes. Vaccination against each one is necessary, but polio has been nearly eradicated.
Reported increases in nonvaccine strains after vaccination [109]
Disease | Vaccine | Increase in | Region | |
H. influenzae | Hib | Nontype b | Alaska | |
Hib | Type f | m. states, US | ||
conj. Hib | Type a | Brazil | ||
conj. Hib | Noncapsulated | UK | ||
S. pneumoniae | PCV-7 | NVT | Finland | |
PCV-7 | NVT (carriage) | US | ||
PCV-7 | Serogroups 15 and 33 | US PMPSG, US | ||
PCV-7 | NVT (AOM) | Pittsburgh | ||
PPV-23 | 12F^{∗}, 7F, 22F, 7C | Alaska | ||
N. meningitidis | A-C vaccine | Serogroup B | Austria | |
A-C vaccine | Serogroup B | Europe | ||
A-C vaccine | Serogroup B | Cuba |
Since differential effectiveness of the vaccine leads to replacement, vaccine developers have tried to make vaccines less differentially effective. One way to do that is to include (if possible) more strains in the vaccine. That has been the case with pneumococcal polysaccharide vaccine, which originally contained very few serotypes of Streptococcus pneumoniae but now contains 23. That is still many fewer than the 90 serotypes that exist. A new approach is to target surface proteins that are common in all 90 serotypes.
Question: Suppose we can produce a vaccine that is perfect with respect to all strains. Will we eliminate strain replacement?
To obtain the invasion reproduction number of strain two, we differentiate the
9.4 Modeling Quarantine and Isolation
Epidemic models with isolation have been considered with respect to different diseases. Isolation has been found to destabilize the dynamics and lead to oscillations [61, 73] (see Chap. 7). As a result, isolation has been suggested as a potential intrinsic mechanism responsible for the recurrent outbreaks of childhood diseases [61].
Parameter meanings and parameter values [120]
Parameter | Parameter meaning | Value | |
Λ | Recruitment rate | 240 people/day | |
β | Transmission rate | 0.25 per day | |
ρ | Quarantine rate | 1/10 per day | |
μ | Natural death rate | 1/(70*365) per day | |
γ | Rate of developing symptoms | 1/6 per day | |
σ | Isolation rate | 1/5 per day | |
η _{1} | Rate of return to susceptible class | 1/10 per day | |
η _{2} | Rate of progression to infectiousness | 1/6.5 per day | |
r _{1} | Recovery rate for isolated individuals | 1/20 per day | |
r _{2} | Recovery rate for infectious individuals | 1/25 per day | |
q | Reduction of infectivity of exposed individuals | 0.8 (variable) |
Optimal periods for quarantine and isolation
Strategy | q = 0. 5 | q = 0. 8 | |
1∕ρ | 8.19 days | 5.53 days | |
1∕σ | 4.67 days | 4.13 days |
This table suggests that in the case q = 0. 8, the contact tracing and quarantining should improve dramatically from 10 days to 5.5 days, while isolation should improve from 5 days to 4 days, in order for the disease to be eliminated.
9.5 Optimal Control Strategies
In previous sections, we considered control strategies to be constant in time, but in reality, control strategies are variable in time. The mathematical theory used to derive optimal control strategies that vary in time is called optimal control theory. In this section, we introduce the basic principles and illustrate them with examples.
9.5.1 Basic Theory of Optimal Control
The variable u(t) is called control, and in the presence of the control, the solution x(t) depends on the control. The trajectory that corresponds to the control u(t) is called a corresponding response of the system.
The first question that must be addressed is whether an optimal control pair (x^{∗}(t), u^{∗}(t)) exists. The question of existence is settled by the following theorem [143]:
Theorem 9.1 (Filippov–Cesari Existence Theorem).
- 1.
N(t, x ) is convex for every (t, x ).
- 2.
A is compact.
- 3.
There exists a constant K > 0 such that || x (t)||≤ K for all t ∈ (0,T) and all admissible pairs ( x , u ).
Then there exists an optimal pair ( x ^{∗} (t), u ^{∗} (t)), where \(\mathbf{u}^{{\ast}}(t) \in \mathcal{A}\) .
The Pontryagin minimum principle is as follows.
Theorem 9.2 (Pontryagin’s Minimum Principle).
The following theorem gives sufficient conditions for the existence and uniqueness of the optimal pair [143]:
Theorem 9.3 (Mangasarian Theorem).
- 1.
A is convex.
- 2.
The partial derivative ∂g∕∂u _{j} and ∂f _{i} ∕∂u _{j} all exist and are continuous.
- 3.
The pair ( x ^{∗} (t), u ^{∗} (t)) satisfies all conditions of the Pontryagin minimum principle.
- 4.
H(t, x , u ) is concave down in ( x , u ) for all t ∈ [0,T].
Then the pair ( x ^{∗} (t), u ^{∗} (t)) solves the problem. If H(t, x , u ) is strictly concave down in ( x , u ), then the solution is unique.
There are several excellent books that introduce optimal control theory applied to biological systems [13, 94]. We illustrate the application of the existence theorem and Pontryagin’s minimum principle to finding the optimal control in the next subsection.
9.5.2 Examples
In this subsection we consider two examples of application of optimal control to epidemic models. The first example is an SIS model with treatment.
9.5.2.1 SIS Model with Treatment
We first prove the existence of an optimal control pair. We use the Filippov–Cesari theorem.
Proof.
Matlab requires use of numerical methods to solve the system of differential equations. The forward–backward sweep method [94] is often employed in this case. It combines the forward application of a fourth-order Runge–Kutta method for the original system with the backward application of a fourth-order Runge–Kutta method for the adjoint system. The Matlab code for system (9.26) is included in the appendix.
9.5.2.2 Two-Strain Model with Vaccination
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