Maximum Likelihood Inference for Univariate Delay Differential Equation Models with Multiple Delays

<table class="figure-group"><tr class="fig-image" id="a"><td><object data="https://static.hindawi.com/articles/complexity/volume-2017/6148934/figures/6148934.fig.001a.svgz" name="6148934.fig.001a" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(a) </b>Numerical solution of the EDDEM (<a href="https://static.hindawi.com/articles/complexity/volume-2017/6148934/figures/#EEq25">37</a>) using the fixed parameter values in <svg height="11.439pt" id="M279" style="vertical-align:-2.15067pt" version="1.1" viewbox="-0.0498162 -9.28833 39.3111 11.439" width="39.3111pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M290 -163V-135C183 -126 181 -122 181 -44V583C181 662 184 666 290 675V703H120V-163H290Z" id="g113-92"></path></g><g transform="matrix(.013,0,0,-0.013,4.485,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g113-49"></path></g><g transform="matrix(.013,0,0,-0.013,10.725,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z" id="g113-45"></path></g><g transform="matrix(.013,0,0,-0.013,15.868,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path></g><g transform="matrix(.013,0,0,-0.013,22.108,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g113-49"></path></g><g transform="matrix(.013,0,0,-0.013,28.348,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 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data="https://static.hindawi.com/articles/complexity/volume-2017/6148934/figures/6148934.fig.001b.svgz" name="6148934.fig.001b" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(b) </b>Numerical solution of the EDDEM (<a href="https://static.hindawi.com/articles/complexity/volume-2017/6148934/figures/#EEq25">37</a>) using the estimated parameter values. The stars are the simulated noisy data from (<a href="https://static.hindawi.com/articles/complexity/volume-2017/6148934/figures/#EEq1">1</a>) at <svg height="8.69875pt" id="M284" style="vertical-align:-0.3499298pt" version="1.1" viewbox="-0.0498162 -8.34882 60.0976 8.69875" width="60.0976pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z" id="g113-111"></path></g><g transform="matrix(.013,0,0,-0.013,9.432,0)"><path d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z" 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499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path></g></svg> equally spaced time points in <svg height="11.439pt" id="M285" style="vertical-align:-2.15067pt" version="1.1" viewbox="-0.0498162 -9.28833 33.0477 11.439" width="33.0477pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M290 -163V-135C183 -126 181 -122 181 -44V583C181 662 184 666 290 675V703H120V-163H290Z" id="g113-92"></path></g><g transform="matrix(.013,0,0,-0.013,4.485,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g113-49"></path></g><g transform="matrix(.013,0,0,-0.013,10.725,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z" id="g113-45"></path></g><g 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Complexity

fig1

Figure 1

Figure 1: Maximum Likelihood Inference for Univariate Delay Differential Equation Models with Multiple Delays 